Theorem psdmul

Description: Product rule for power series. An outline is available at https://github.com/icecream17/Stuff/blob/main/math/psdmul.pdf . (Contributed by SN, 25-Apr-2025)

	Ref	Expression
Hypotheses	psdmul.s	⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
	psdmul.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
	psdmul.p	⊢ + = ( +g ‘ 𝑆 )
	psdmul.m	⊢ · = ( .r ‘ 𝑆 )
	psdmul.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
	psdmul.r	⊢ ( 𝜑 → 𝑅 ∈ CRing )
	psdmul.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐼 )
	psdmul.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
	psdmul.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐵 )
Assertion	psdmul	⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐹 · 𝐺 ) ) = ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) + ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		psdmul.s	⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
2		psdmul.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
3		psdmul.p	⊢ + = ( +g ‘ 𝑆 )
4		psdmul.m	⊢ · = ( .r ‘ 𝑆 )
5		psdmul.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
6		psdmul.r	⊢ ( 𝜑 → 𝑅 ∈ CRing )
7		psdmul.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐼 )
8		psdmul.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
9		psdmul.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐵 )
10		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
11		eqid	⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
12	6	crngringd	⊢ ( 𝜑 → 𝑅 ∈ Ring )
13	12	ringcmnd	⊢ ( 𝜑 → 𝑅 ∈ CMnd )
14	13	adantr	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑅 ∈ CMnd )
15		simpr	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
16		eqid	⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
17	16	psrbagsn	⊢ ( 𝐼 ∈ 𝑉 → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
18	5 17	syl	⊢ ( 𝜑 → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
19	18	adantr	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
20	16	psrbagaddcl	⊢ ( ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
21	15 19 20	syl2anc	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
22	16	psrbaglefi	⊢ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∈ Fin )
23	21 22	syl	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∈ Fin )
24		eqid	⊢ ( .g ‘ 𝑅 ) = ( .g ‘ 𝑅 )
25	6	crnggrpd	⊢ ( 𝜑 → 𝑅 ∈ Grp )
26	25	grpmndd	⊢ ( 𝜑 → 𝑅 ∈ Mnd )
27	26	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → 𝑅 ∈ Mnd )
28	16	psrbagf	⊢ ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → 𝑑 : 𝐼 ⟶ ℕ0 )
29	28	adantl	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 : 𝐼 ⟶ ℕ0 )
30	7	adantr	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑋 ∈ 𝐼 )
31	29 30	ffvelcdmd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 )
32		peano2nn0	⊢ ( ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 → ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℕ0 )
33	31 32	syl	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℕ0 )
34	33	adantr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℕ0 )
35		eqid	⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
36	12	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → 𝑅 ∈ Ring )
37	1 10 16 2 8	psrelbas	⊢ ( 𝜑 → 𝐹 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
38	37	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → 𝐹 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
39		elrabi	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } → 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
40	39	adantl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
41	38 40	ffvelcdmd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( 𝐹 ‘ 𝑢 ) ∈ ( Base ‘ 𝑅 ) )
42	1 10 16 2 9	psrelbas	⊢ ( 𝜑 → 𝐺 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
43	42	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → 𝐺 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
44		eqid	⊢ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } = { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) }
45	16 44	psrbagconcl	⊢ ( ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
46	21 45	sylan	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
47		elrabi	⊢ ( ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
48	46 47	syl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
49	43 48	ffvelcdmd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ∈ ( Base ‘ 𝑅 ) )
50	10 35 36 41 49	ringcld	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ∈ ( Base ‘ 𝑅 ) )
51	10 24 27 34 50	mulgnn0cld	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
52		disjdifr	⊢ ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∩ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) = ∅
53	52	a1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∩ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) = ∅ )
54		1nn0	⊢ 1 ∈ ℕ0
55		0nn0	⊢ 0 ∈ ℕ0
56	54 55	ifcli	⊢ if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ ℕ0
57	56	nn0ge0i	⊢ 0 ≤ if ( 𝑖 = 𝑋 , 1 , 0 )
58	29	ffvelcdmda	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℕ0 )
59	58	nn0red	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℝ )
60	56	nn0rei	⊢ if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ ℝ
61	60	a1i	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ ℝ )
62	59 61	addge01d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( 0 ≤ if ( 𝑖 = 𝑋 , 1 , 0 ) ↔ ( 𝑑 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) )
63	57 62	mpbii	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
64	63	ralrimiva	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ∀ 𝑖 ∈ 𝐼 ( 𝑑 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
65	29	ffnd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 Fn 𝐼 )
66	54 55	ifcli	⊢ if ( 𝑦 = 𝑋 , 1 , 0 ) ∈ ℕ0
67	66	elexi	⊢ if ( 𝑦 = 𝑋 , 1 , 0 ) ∈ V
68		eqid	⊢ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) )
69	67 68	fnmpti	⊢ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) Fn 𝐼
70	69	a1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) Fn 𝐼 )
71	5	adantr	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐼 ∈ 𝑉 )
72		inidm	⊢ ( 𝐼 ∩ 𝐼 ) = 𝐼
73	65 70 71 71 72	offn	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) Fn 𝐼 )
74		eqidd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) = ( 𝑑 ‘ 𝑖 ) )
75		eqeq1	⊢ ( 𝑦 = 𝑖 → ( 𝑦 = 𝑋 ↔ 𝑖 = 𝑋 ) )
76	75	ifbid	⊢ ( 𝑦 = 𝑖 → if ( 𝑦 = 𝑋 , 1 , 0 ) = if ( 𝑖 = 𝑋 , 1 , 0 ) )
77	56	elexi	⊢ if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ V
78	76 68 77	fvmpt	⊢ ( 𝑖 ∈ 𝐼 → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑖 ) = if ( 𝑖 = 𝑋 , 1 , 0 ) )
79	78	adantl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑖 ) = if ( 𝑖 = 𝑋 , 1 , 0 ) )
80	65 70 71 71 72 74 79	ofval	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
81	65 73 71 71 72 74 80	ofrfval	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ ∀ 𝑖 ∈ 𝐼 ( 𝑑 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) )
82	64 81	mpbird	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
83	82	adantr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
84	5	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐼 ∈ 𝑉 )
85	16	psrbagf	⊢ ( 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → 𝑘 : 𝐼 ⟶ ℕ0 )
86	85	adantl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑘 : 𝐼 ⟶ ℕ0 )
87	29	adantr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 : 𝐼 ⟶ ℕ0 )
88	16	psrbagf	⊢ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) : 𝐼 ⟶ ℕ0 )
89	21 88	syl	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) : 𝐼 ⟶ ℕ0 )
90	89	adantr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) : 𝐼 ⟶ ℕ0 )
91		nn0re	⊢ ( 𝑞 ∈ ℕ0 → 𝑞 ∈ ℝ )
92		nn0re	⊢ ( 𝑟 ∈ ℕ0 → 𝑟 ∈ ℝ )
93		nn0re	⊢ ( 𝑠 ∈ ℕ0 → 𝑠 ∈ ℝ )
94		letr	⊢ ( ( 𝑞 ∈ ℝ ∧ 𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ ) → ( ( 𝑞 ≤ 𝑟 ∧ 𝑟 ≤ 𝑠 ) → 𝑞 ≤ 𝑠 ) )
95	91 92 93 94	syl3an	⊢ ( ( 𝑞 ∈ ℕ0 ∧ 𝑟 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ) → ( ( 𝑞 ≤ 𝑟 ∧ 𝑟 ≤ 𝑠 ) → 𝑞 ≤ 𝑠 ) )
96	95	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑞 ∈ ℕ0 ∧ 𝑟 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ) → ( ( 𝑞 ≤ 𝑟 ∧ 𝑟 ≤ 𝑠 ) → 𝑞 ≤ 𝑠 ) )
97	84 86 87 90 96	caoftrn	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑘 ∘r ≤ 𝑑 ∧ 𝑑 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) → 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
98	83 97	mpan2d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑘 ∘r ≤ 𝑑 → 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
99	98	ss2rabdv	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
100		undifr	⊢ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↔ ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∪ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) = { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
101	99 100	sylib	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∪ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) = { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
102	101	eqcomd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } = ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∪ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
103	10 11 14 23 51 53 102	gsummptfidmsplit	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
104		eqid	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
105		ovex	⊢ ( ℕ0 ↑m 𝐼 ) ∈ V
106	105	rabex	⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∈ V
107	106	rabex	⊢ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∈ V
108	107	a1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∈ V )
109		ovex	⊢ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ∈ V
110		eqid	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) = ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) )
111	109 110	fnmpti	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) Fn { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) }
112	111	a1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) Fn { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
113		fvexd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 0g ‘ 𝑅 ) ∈ V )
114	112 23 113	fndmfifsupp	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) finSupp ( 0g ‘ 𝑅 ) )
115	10 104 24 108 50 114 14 33	gsummulg	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) )
116		difrab	⊢ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) = { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ 𝑘 ∘r ≤ 𝑑 ) }
117	116	eleq2i	⊢ ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↔ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ 𝑘 ∘r ≤ 𝑑 ) } )
118		breq1	⊢ ( 𝑘 = 𝑢 → ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
119		breq1	⊢ ( 𝑘 = 𝑢 → ( 𝑘 ∘r ≤ 𝑑 ↔ 𝑢 ∘r ≤ 𝑑 ) )
120	119	notbid	⊢ ( 𝑘 = 𝑢 → ( ¬ 𝑘 ∘r ≤ 𝑑 ↔ ¬ 𝑢 ∘r ≤ 𝑑 ) )
121	118 120	anbi12d	⊢ ( 𝑘 = 𝑢 → ( ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ 𝑘 ∘r ≤ 𝑑 ) ↔ ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ 𝑢 ∘r ≤ 𝑑 ) ) )
122	121	elrab	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ 𝑘 ∘r ≤ 𝑑 ) } ↔ ( 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ 𝑢 ∘r ≤ 𝑑 ) ) )
123	16	psrbagf	⊢ ( 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → 𝑢 : 𝐼 ⟶ ℕ0 )
124	123	ffnd	⊢ ( 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → 𝑢 Fn 𝐼 )
125	124	adantl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑢 Fn 𝐼 )
126	73	adantr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) Fn 𝐼 )
127	5	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐼 ∈ 𝑉 )
128		eqidd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) = ( 𝑢 ‘ 𝑖 ) )
129	65	adantr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 Fn 𝐼 )
130	66	a1i	⊢ ( 𝑦 ∈ 𝐼 → if ( 𝑦 = 𝑋 , 1 , 0 ) ∈ ℕ0 )
131	68 130	fmpti	⊢ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) : 𝐼 ⟶ ℕ0
132	131	a1i	⊢ ( 𝜑 → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) : 𝐼 ⟶ ℕ0 )
133	132	ffnd	⊢ ( 𝜑 → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) Fn 𝐼 )
134	133	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) Fn 𝐼 )
135		eqidd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) = ( 𝑑 ‘ 𝑖 ) )
136	78	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑖 ) = if ( 𝑖 = 𝑋 , 1 , 0 ) )
137	129 134 127 127 72 135 136	ofval	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
138	125 126 127 127 72 128 137	ofrfval	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) )
139	125 129 127 127 72 128 135	ofrfval	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∘r ≤ 𝑑 ↔ ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
140	139	notbid	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ¬ 𝑢 ∘r ≤ 𝑑 ↔ ¬ ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
141		rexnal	⊢ ( ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ↔ ¬ ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) )
142	140 141	bitr4di	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ¬ 𝑢 ∘r ≤ 𝑑 ↔ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
143	138 142	anbi12d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ 𝑢 ∘r ≤ 𝑑 ) ↔ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) )
144	31	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 )
145	123	adantl	⊢ ( ( 𝜑 ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑢 : 𝐼 ⟶ ℕ0 )
146	7	adantr	⊢ ( ( 𝜑 ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑋 ∈ 𝐼 )
147	145 146	ffvelcdmd	⊢ ( ( 𝜑 ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ‘ 𝑋 ) ∈ ℕ0 )
148	147	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ‘ 𝑋 ) ∈ ℕ0 )
149	148	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( 𝑢 ‘ 𝑋 ) ∈ ℕ0 )
150		nn0nlt0	⊢ ( ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 → ¬ ( 𝑑 ‘ 𝑋 ) < 0 )
151	144 150	syl	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ¬ ( 𝑑 ‘ 𝑋 ) < 0 )
152	29	adantr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 : 𝐼 ⟶ ℕ0 )
153	152	ffvelcdmda	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℕ0 )
154	153	nn0cnd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℂ )
155	154	addridd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑑 ‘ 𝑖 ) + 0 ) = ( 𝑑 ‘ 𝑖 ) )
156	155	breq2d	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + 0 ) ↔ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
157	156	biimpd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + 0 ) → ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
158		ifnefalse	⊢ ( 𝑖 ≠ 𝑋 → if ( 𝑖 = 𝑋 , 1 , 0 ) = 0 )
159	158	oveq2d	⊢ ( 𝑖 ≠ 𝑋 → ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) = ( ( 𝑑 ‘ 𝑖 ) + 0 ) )
160	159	breq2d	⊢ ( 𝑖 ≠ 𝑋 → ( ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ↔ ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + 0 ) ) )
161	160	imbi1d	⊢ ( 𝑖 ≠ 𝑋 → ( ( ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) → ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ↔ ( ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + 0 ) → ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) )
162	157 161	syl5ibrcom	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑖 ≠ 𝑋 → ( ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) → ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) )
163	162	imp	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 ≠ 𝑋 ) → ( ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) → ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
164	163	impancom	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) → ( 𝑖 ≠ 𝑋 → ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
165	164	necon1bd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) → ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → 𝑖 = 𝑋 ) )
166	165	ancrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) → ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) )
167	166	ex	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) → ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) ) )
168	167	ralimdva	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) → ∀ 𝑖 ∈ 𝐼 ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) ) )
169	168	anim1d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) → ( ∀ 𝑖 ∈ 𝐼 ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) )
170	169	imp	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( ∀ 𝑖 ∈ 𝐼 ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
171		rexim	⊢ ( ∀ 𝑖 ∈ 𝐼 ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → ∃ 𝑖 ∈ 𝐼 ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) )
172	171	imp	⊢ ( ( ∀ 𝑖 ∈ 𝐼 ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) → ∃ 𝑖 ∈ 𝐼 ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
173		fveq2	⊢ ( 𝑖 = 𝑋 → ( 𝑢 ‘ 𝑖 ) = ( 𝑢 ‘ 𝑋 ) )
174		fveq2	⊢ ( 𝑖 = 𝑋 → ( 𝑑 ‘ 𝑖 ) = ( 𝑑 ‘ 𝑋 ) )
175	173 174	breq12d	⊢ ( 𝑖 = 𝑋 → ( ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ↔ ( 𝑢 ‘ 𝑋 ) ≤ ( 𝑑 ‘ 𝑋 ) ) )
176	175	notbid	⊢ ( 𝑖 = 𝑋 → ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ↔ ¬ ( 𝑢 ‘ 𝑋 ) ≤ ( 𝑑 ‘ 𝑋 ) ) )
177	176	ceqsrexbv	⊢ ( ∃ 𝑖 ∈ 𝐼 ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ↔ ( 𝑋 ∈ 𝐼 ∧ ¬ ( 𝑢 ‘ 𝑋 ) ≤ ( 𝑑 ‘ 𝑋 ) ) )
178	177	simprbi	⊢ ( ∃ 𝑖 ∈ 𝐼 ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) → ¬ ( 𝑢 ‘ 𝑋 ) ≤ ( 𝑑 ‘ 𝑋 ) )
179	172 178	syl	⊢ ( ( ∀ 𝑖 ∈ 𝐼 ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) → ¬ ( 𝑢 ‘ 𝑋 ) ≤ ( 𝑑 ‘ 𝑋 ) )
180	31	adantr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 )
181	180	nn0red	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ‘ 𝑋 ) ∈ ℝ )
182	148	nn0red	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ‘ 𝑋 ) ∈ ℝ )
183	181 182	ltnled	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑑 ‘ 𝑋 ) < ( 𝑢 ‘ 𝑋 ) ↔ ¬ ( 𝑢 ‘ 𝑋 ) ≤ ( 𝑑 ‘ 𝑋 ) ) )
184	183	biimpar	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ¬ ( 𝑢 ‘ 𝑋 ) ≤ ( 𝑑 ‘ 𝑋 ) ) → ( 𝑑 ‘ 𝑋 ) < ( 𝑢 ‘ 𝑋 ) )
185	179 184	sylan2	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( 𝑑 ‘ 𝑋 ) < ( 𝑢 ‘ 𝑋 ) )
186	170 185	syldan	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( 𝑑 ‘ 𝑋 ) < ( 𝑢 ‘ 𝑋 ) )
187		breq2	⊢ ( ( 𝑢 ‘ 𝑋 ) = 0 → ( ( 𝑑 ‘ 𝑋 ) < ( 𝑢 ‘ 𝑋 ) ↔ ( 𝑑 ‘ 𝑋 ) < 0 ) )
188	186 187	syl5ibcom	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( ( 𝑢 ‘ 𝑋 ) = 0 → ( 𝑑 ‘ 𝑋 ) < 0 ) )
189	151 188	mtod	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ¬ ( 𝑢 ‘ 𝑋 ) = 0 )
190	189	neqned	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( 𝑢 ‘ 𝑋 ) ≠ 0 )
191		elnnne0	⊢ ( ( 𝑢 ‘ 𝑋 ) ∈ ℕ ↔ ( ( 𝑢 ‘ 𝑋 ) ∈ ℕ0 ∧ ( 𝑢 ‘ 𝑋 ) ≠ 0 ) )
192	149 190 191	sylanbrc	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( 𝑢 ‘ 𝑋 ) ∈ ℕ )
193		elfzo0	⊢ ( ( 𝑑 ‘ 𝑋 ) ∈ ( 0 ..^ ( 𝑢 ‘ 𝑋 ) ) ↔ ( ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 ∧ ( 𝑢 ‘ 𝑋 ) ∈ ℕ ∧ ( 𝑑 ‘ 𝑋 ) < ( 𝑢 ‘ 𝑋 ) ) )
194	144 192 186 193	syl3anbrc	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( 𝑑 ‘ 𝑋 ) ∈ ( 0 ..^ ( 𝑢 ‘ 𝑋 ) ) )
195		fzostep1	⊢ ( ( 𝑑 ‘ 𝑋 ) ∈ ( 0 ..^ ( 𝑢 ‘ 𝑋 ) ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ( 0 ..^ ( 𝑢 ‘ 𝑋 ) ) ∨ ( ( 𝑑 ‘ 𝑋 ) + 1 ) = ( 𝑢 ‘ 𝑋 ) ) )
196	194 195	syl	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ( 0 ..^ ( 𝑢 ‘ 𝑋 ) ) ∨ ( ( 𝑑 ‘ 𝑋 ) + 1 ) = ( 𝑢 ‘ 𝑋 ) ) )
197	149	nn0red	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( 𝑢 ‘ 𝑋 ) ∈ ℝ )
198	33	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℕ0 )
199	198	nn0red	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℝ )
200	7	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑋 ∈ 𝐼 )
201		iftrue	⊢ ( 𝑖 = 𝑋 → if ( 𝑖 = 𝑋 , 1 , 0 ) = 1 )
202	174 201	oveq12d	⊢ ( 𝑖 = 𝑋 → ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) = ( ( 𝑑 ‘ 𝑋 ) + 1 ) )
203	173 202	breq12d	⊢ ( 𝑖 = 𝑋 → ( ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ↔ ( 𝑢 ‘ 𝑋 ) ≤ ( ( 𝑑 ‘ 𝑋 ) + 1 ) ) )
204	203	rspcv	⊢ ( 𝑋 ∈ 𝐼 → ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) → ( 𝑢 ‘ 𝑋 ) ≤ ( ( 𝑑 ‘ 𝑋 ) + 1 ) ) )
205	200 204	syl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) → ( 𝑢 ‘ 𝑋 ) ≤ ( ( 𝑑 ‘ 𝑋 ) + 1 ) ) )
206	205	imp	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 ‘ 𝑋 ) ≤ ( ( 𝑑 ‘ 𝑋 ) + 1 ) )
207	206	adantrr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( 𝑢 ‘ 𝑋 ) ≤ ( ( 𝑑 ‘ 𝑋 ) + 1 ) )
208	197 199 207	lensymd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ¬ ( ( 𝑑 ‘ 𝑋 ) + 1 ) < ( 𝑢 ‘ 𝑋 ) )
209	208	intn3an3d	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ¬ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℕ0 ∧ ( 𝑢 ‘ 𝑋 ) ∈ ℕ ∧ ( ( 𝑑 ‘ 𝑋 ) + 1 ) < ( 𝑢 ‘ 𝑋 ) ) )
210		elfzo0	⊢ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ( 0 ..^ ( 𝑢 ‘ 𝑋 ) ) ↔ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℕ0 ∧ ( 𝑢 ‘ 𝑋 ) ∈ ℕ ∧ ( ( 𝑑 ‘ 𝑋 ) + 1 ) < ( 𝑢 ‘ 𝑋 ) ) )
211	209 210	sylnibr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ¬ ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ( 0 ..^ ( 𝑢 ‘ 𝑋 ) ) )
212	196 211	orcnd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) = ( 𝑢 ‘ 𝑋 ) )
213	143 212	sylbida	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ 𝑢 ∘r ≤ 𝑑 ) ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) = ( 𝑢 ‘ 𝑋 ) )
214	213	anasss	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ 𝑢 ∘r ≤ 𝑑 ) ) ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) = ( 𝑢 ‘ 𝑋 ) )
215	122 214	sylan2b	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ 𝑘 ∘r ≤ 𝑑 ) } ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) = ( 𝑢 ‘ 𝑋 ) )
216	117 215	sylan2b	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) = ( 𝑢 ‘ 𝑋 ) )
217	216	oveq1d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) = ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
218	217	mpteq2dva	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) = ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) )
219	218	oveq2d	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) )
220	16	psrbaglefi	⊢ ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∈ Fin )
221	220	adantl	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∈ Fin )
222	26	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑅 ∈ Mnd )
223	33	adantr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℕ0 )
224	12	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑅 ∈ Ring )
225		elrabi	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
226	37	adantr	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐹 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
227	226	ffvelcdmda	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝐹 ‘ 𝑢 ) ∈ ( Base ‘ 𝑅 ) )
228	225 227	sylan2	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝐹 ‘ 𝑢 ) ∈ ( Base ‘ 𝑅 ) )
229	42	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝐺 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
230	29	adantr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑑 : 𝐼 ⟶ ℕ0 )
231	230	ffvelcdmda	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℕ0 )
232	231	nn0cnd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℂ )
233	225 123	syl	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → 𝑢 : 𝐼 ⟶ ℕ0 )
234	233	adantl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑢 : 𝐼 ⟶ ℕ0 )
235	234	ffvelcdmda	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ∈ ℕ0 )
236	235	nn0cnd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ∈ ℂ )
237	56	nn0cni	⊢ if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ ℂ
238	237	a1i	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ ℂ )
239	232 236 238	subadd23d	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( ( 𝑑 ‘ 𝑖 ) − ( 𝑢 ‘ 𝑖 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) = ( ( 𝑑 ‘ 𝑖 ) + ( if ( 𝑖 = 𝑋 , 1 , 0 ) − ( 𝑢 ‘ 𝑖 ) ) ) )
240	232 238 236	addsubassd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) − ( 𝑢 ‘ 𝑖 ) ) = ( ( 𝑑 ‘ 𝑖 ) + ( if ( 𝑖 = 𝑋 , 1 , 0 ) − ( 𝑢 ‘ 𝑖 ) ) ) )
241	239 240	eqtr4d	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( ( 𝑑 ‘ 𝑖 ) − ( 𝑢 ‘ 𝑖 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) = ( ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) − ( 𝑢 ‘ 𝑖 ) ) )
242	241	mpteq2dva	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑖 ∈ 𝐼 ↦ ( ( ( 𝑑 ‘ 𝑖 ) − ( 𝑢 ‘ 𝑖 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) − ( 𝑢 ‘ 𝑖 ) ) ) )
243		eqid	⊢ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } = { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 }
244	16 243	psrbagconcl	⊢ ( ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f − 𝑢 ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
245		elrabi	⊢ ( ( 𝑑 ∘f − 𝑢 ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → ( 𝑑 ∘f − 𝑢 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
246	244 245	syl	⊢ ( ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f − 𝑢 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
247	246	adantll	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f − 𝑢 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
248	16	psrbagf	⊢ ( ( 𝑑 ∘f − 𝑢 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → ( 𝑑 ∘f − 𝑢 ) : 𝐼 ⟶ ℕ0 )
249	247 248	syl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f − 𝑢 ) : 𝐼 ⟶ ℕ0 )
250	249	ffnd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f − 𝑢 ) Fn 𝐼 )
251	69	a1i	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) Fn 𝐼 )
252	5	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝐼 ∈ 𝑉 )
253	230	ffnd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑑 Fn 𝐼 )
254	234	ffnd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑢 Fn 𝐼 )
255		eqidd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) = ( 𝑑 ‘ 𝑖 ) )
256		eqidd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) = ( 𝑢 ‘ 𝑖 ) )
257	253 254 252 252 72 255 256	ofval	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑖 ) = ( ( 𝑑 ‘ 𝑖 ) − ( 𝑢 ‘ 𝑖 ) ) )
258	78	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑖 ) = if ( 𝑖 = 𝑋 , 1 , 0 ) )
259	250 251 252 252 72 257 258	offval	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑑 ∘f − 𝑢 ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( ( ( 𝑑 ‘ 𝑖 ) − ( 𝑢 ‘ 𝑖 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) )
260		simplr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
261	18	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
262	260 261 20	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
263	262 88	syl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) : 𝐼 ⟶ ℕ0 )
264	263	ffnd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) Fn 𝐼 )
265	253 251 252 252 72 255 258	ofval	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
266	264 254 252 252 72 265 256	offval	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) = ( 𝑖 ∈ 𝐼 ↦ ( ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) − ( 𝑢 ‘ 𝑖 ) ) ) )
267	242 259 266	3eqtr4d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑑 ∘f − 𝑢 ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) )
268	16	psrbagaddcl	⊢ ( ( ( 𝑑 ∘f − 𝑢 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑑 ∘f − 𝑢 ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
269	247 261 268	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑑 ∘f − 𝑢 ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
270	267 269	eqeltrrd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
271	229 270	ffvelcdmd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ∈ ( Base ‘ 𝑅 ) )
272	10 35 224 228 271	ringcld	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ∈ ( Base ‘ 𝑅 ) )
273	10 24 222 223 272	mulgnn0cld	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
274		disjdifr	⊢ ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∩ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) = ∅
275	274	a1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∩ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) = ∅ )
276		simpl	⊢ ( ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) → 𝑘 ∘r ≤ 𝑑 )
277	276	a1i	⊢ ( 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → ( ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) → 𝑘 ∘r ≤ 𝑑 ) )
278	277	ss2rabi	⊢ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 }
279	278	a1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
280		undifr	⊢ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↔ ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∪ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) = { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
281	279 280	sylib	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∪ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) = { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
282	281	eqcomd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } = ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∪ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) )
283	10 11 14 221 273 275 282	gsummptfidmsplit	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
284		eldifi	⊢ ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
285	7	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑋 ∈ 𝐼 )
286		eqidd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑋 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑋 ) = ( 𝑑 ‘ 𝑋 ) )
287		eqidd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑋 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑋 ) = ( 𝑢 ‘ 𝑋 ) )
288	253 254 252 252 72 286 287	ofval	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑋 ∈ 𝐼 ) → ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) = ( ( 𝑑 ‘ 𝑋 ) − ( 𝑢 ‘ 𝑋 ) ) )
289	285 288	mpdan	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) = ( ( 𝑑 ‘ 𝑋 ) − ( 𝑢 ‘ 𝑋 ) ) )
290	284 289	sylan2	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) = ( ( 𝑑 ‘ 𝑋 ) − ( 𝑢 ‘ 𝑋 ) ) )
291	290	oveq2d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( 𝑢 ‘ 𝑋 ) + ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) ) = ( ( 𝑢 ‘ 𝑋 ) + ( ( 𝑑 ‘ 𝑋 ) − ( 𝑢 ‘ 𝑋 ) ) ) )
292	234 285	ffvelcdmd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑢 ‘ 𝑋 ) ∈ ℕ0 )
293	284 292	sylan2	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( 𝑢 ‘ 𝑋 ) ∈ ℕ0 )
294	293	nn0cnd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( 𝑢 ‘ 𝑋 ) ∈ ℂ )
295	31	nn0cnd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ‘ 𝑋 ) ∈ ℂ )
296	295	adantr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( 𝑑 ‘ 𝑋 ) ∈ ℂ )
297	294 296	pncan3d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( 𝑢 ‘ 𝑋 ) + ( ( 𝑑 ‘ 𝑋 ) − ( 𝑢 ‘ 𝑋 ) ) ) = ( 𝑑 ‘ 𝑋 ) )
298	291 297	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( 𝑢 ‘ 𝑋 ) + ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) ) = ( 𝑑 ‘ 𝑋 ) )
299	298	oveq1d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( ( 𝑢 ‘ 𝑋 ) + ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) ) + 1 ) = ( ( 𝑑 ‘ 𝑋 ) + 1 ) )
300	249 285	ffvelcdmd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) ∈ ℕ0 )
301	284 300	sylan2	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) ∈ ℕ0 )
302	301	nn0cnd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) ∈ ℂ )
303		1cnd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → 1 ∈ ℂ )
304	294 302 303	addassd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( ( 𝑢 ‘ 𝑋 ) + ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) ) + 1 ) = ( ( 𝑢 ‘ 𝑋 ) + ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ) )
305	299 304	eqtr3d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) = ( ( 𝑢 ‘ 𝑋 ) + ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ) )
306	305	oveq1d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) = ( ( ( 𝑢 ‘ 𝑋 ) + ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
307	26	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → 𝑅 ∈ Mnd )
308		peano2nn0	⊢ ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) ∈ ℕ0 → ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ∈ ℕ0 )
309	300 308	syl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ∈ ℕ0 )
310	284 309	sylan2	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ∈ ℕ0 )
311	284 272	sylan2	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ∈ ( Base ‘ 𝑅 ) )
312	10 24 11	mulgnn0dir	⊢ ( ( 𝑅 ∈ Mnd ∧ ( ( 𝑢 ‘ 𝑋 ) ∈ ℕ0 ∧ ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ∈ ℕ0 ∧ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ∈ ( Base ‘ 𝑅 ) ) ) → ( ( ( 𝑢 ‘ 𝑋 ) + ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) = ( ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ( +g ‘ 𝑅 ) ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) )
313	307 293 310 311 312	syl13anc	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( ( 𝑢 ‘ 𝑋 ) + ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) = ( ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ( +g ‘ 𝑅 ) ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) )
314	306 313	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) = ( ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ( +g ‘ 𝑅 ) ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) )
315	314	mpteq2dva	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) = ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ( +g ‘ 𝑅 ) ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) )
316	315	oveq2d	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ( +g ‘ 𝑅 ) ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
317		difssd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
318	221 317	ssfid	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∈ Fin )
319	10 24 222 292 272	mulgnn0cld	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
320	284 319	sylan2	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
321	10 24 222 309 272	mulgnn0cld	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
322	284 321	sylan2	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
323		eqid	⊢ ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) = ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
324		eqid	⊢ ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) = ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
325	10 11 14 318 320 322 323 324	gsummptfidmadd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ( +g ‘ 𝑅 ) ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
326	316 325	eqtrd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
327	7	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → 𝑋 ∈ 𝐼 )
328	65	adantr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → 𝑑 Fn 𝐼 )
329		elrabi	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } → 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
330	329 124	syl	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } → 𝑢 Fn 𝐼 )
331	330	adantl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → 𝑢 Fn 𝐼 )
332	5	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → 𝐼 ∈ 𝑉 )
333		eqidd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∧ 𝑋 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑋 ) = ( 𝑑 ‘ 𝑋 ) )
334		eqidd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∧ 𝑋 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑋 ) = ( 𝑢 ‘ 𝑋 ) )
335	328 331 332 332 72 333 334	ofval	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∧ 𝑋 ∈ 𝐼 ) → ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) = ( ( 𝑑 ‘ 𝑋 ) − ( 𝑢 ‘ 𝑋 ) ) )
336	327 335	mpdan	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) = ( ( 𝑑 ‘ 𝑋 ) − ( 𝑢 ‘ 𝑋 ) ) )
337		fveq1	⊢ ( 𝑘 = 𝑢 → ( 𝑘 ‘ 𝑋 ) = ( 𝑢 ‘ 𝑋 ) )
338	337	eqeq1d	⊢ ( 𝑘 = 𝑢 → ( ( 𝑘 ‘ 𝑋 ) = 0 ↔ ( 𝑢 ‘ 𝑋 ) = 0 ) )
339	119 338	anbi12d	⊢ ( 𝑘 = 𝑢 → ( ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ↔ ( 𝑢 ∘r ≤ 𝑑 ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ) )
340	339	elrab	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↔ ( 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑢 ∘r ≤ 𝑑 ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ) )
341	340	simprbi	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } → ( 𝑢 ∘r ≤ 𝑑 ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) )
342	341	simprd	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } → ( 𝑢 ‘ 𝑋 ) = 0 )
343	342	adantl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → ( 𝑢 ‘ 𝑋 ) = 0 )
344	343	oveq2d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → ( ( 𝑑 ‘ 𝑋 ) − ( 𝑢 ‘ 𝑋 ) ) = ( ( 𝑑 ‘ 𝑋 ) − 0 ) )
345	31	adantr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 )
346	345	nn0cnd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → ( 𝑑 ‘ 𝑋 ) ∈ ℂ )
347	346	subid1d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → ( ( 𝑑 ‘ 𝑋 ) − 0 ) = ( 𝑑 ‘ 𝑋 ) )
348	336 344 347	3eqtrrd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → ( 𝑑 ‘ 𝑋 ) = ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) )
349	348	oveq1d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) = ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) )
350	349	oveq1d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) = ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
351	350	mpteq2dva	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) = ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) )
352	351	oveq2d	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) )
353	326 352	oveq12d	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) = ( ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
354	25	adantr	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑅 ∈ Grp )
355	106	rabex	⊢ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∈ V
356	355	difexi	⊢ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∈ V
357	356	a1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∈ V )
358	320	fmpttd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) : ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ⟶ ( Base ‘ 𝑅 ) )
359		ovex	⊢ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ V
360	359 323	fnmpti	⊢ ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) Fn ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } )
361	360	a1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) Fn ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) )
362	361 318 113	fndmfifsupp	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) finSupp ( 0g ‘ 𝑅 ) )
363	10 104 14 357 358 362	gsumcl	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
364	322	fmpttd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) : ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ⟶ ( Base ‘ 𝑅 ) )
365		ovex	⊢ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ V
366	365 324	fnmpti	⊢ ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) Fn ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } )
367	366	a1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) Fn ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) )
368	367 318 113	fndmfifsupp	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) finSupp ( 0g ‘ 𝑅 ) )
369	10 104 14 357 364 368	gsumcl	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
370	106	rabex	⊢ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ∈ V
371	370	a1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ∈ V )
372	278	sseli	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } → 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
373	372 321	sylan2	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
374	373	fmpttd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) : { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ⟶ ( Base ‘ 𝑅 ) )
375		eqid	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) = ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
376	365 375	fnmpti	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) Fn { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) }
377	376	a1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) Fn { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } )
378	221 279	ssfid	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ∈ Fin )
379	377 378 113	fndmfifsupp	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) finSupp ( 0g ‘ 𝑅 ) )
380	10 104 14 371 374 379	gsumcl	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
381	10 11 354 363 369 380	grpassd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ) )
382	283 353 381	3eqtrd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ) )
383	219 382	oveq12d	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ) ) )
384	103 115 383	3eqtr3d	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ) ) )
385	8	adantr	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐹 ∈ 𝐵 )
386	9	adantr	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐺 ∈ 𝐵 )
387	1 2 35 4 16 385 386 21	psrmulval	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝐹 · 𝐺 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) )
388	387	oveq2d	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 · 𝐺 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) )
389	107	difexi	⊢ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∈ V
390	389	a1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∈ V )
391		eldifi	⊢ ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
392	39 123	syl	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } → 𝑢 : 𝐼 ⟶ ℕ0 )
393	392	adantl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → 𝑢 : 𝐼 ⟶ ℕ0 )
394	7	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → 𝑋 ∈ 𝐼 )
395	393 394	ffvelcdmd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( 𝑢 ‘ 𝑋 ) ∈ ℕ0 )
396	10 24 27 395 50	mulgnn0cld	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
397	391 396	sylan2	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
398	397	fmpttd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) : ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ⟶ ( Base ‘ 𝑅 ) )
399		eqid	⊢ ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) = ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
400	359 399	fnmpti	⊢ ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) Fn ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
401	400	a1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) Fn ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
402		difssd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
403	23 402	ssfid	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∈ Fin )
404	401 403 113	fndmfifsupp	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) finSupp ( 0g ‘ 𝑅 ) )
405	10 104 14 390 398 404	gsumcl	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
406	10 11 354 369 380	grpcld	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
407	10 11 354 405 363 406	grpassd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ) ) )
408	384 388 407	3eqtr4d	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 · 𝐺 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ) )
409	408	mpteq2dva	⊢ ( 𝜑 → ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 · 𝐺 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) = ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ) ) )
410	1 2 4 12 8 9	psrmulcl	⊢ ( 𝜑 → ( 𝐹 · 𝐺 ) ∈ 𝐵 )
411	1 2 16 5 6 7 410	psdval	⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐹 · 𝐺 ) ) = ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 · 𝐺 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) )
412	25	grpmgmd	⊢ ( 𝜑 → 𝑅 ∈ Mgm )
413	1 2 5 412 7 8	psdcl	⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ∈ 𝐵 )
414	1 2 4 12 413 9	psrmulcl	⊢ ( 𝜑 → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) ∈ 𝐵 )
415	1 2 5 412 7 9	psdcl	⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ∈ 𝐵 )
416	1 2 4 12 8 415	psrmulcl	⊢ ( 𝜑 → ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) ∈ 𝐵 )
417	1 2 11 3 414 416	psradd	⊢ ( 𝜑 → ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) + ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) ) = ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) ∘f ( +g ‘ 𝑅 ) ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) ) )
418	1 10 16 2 414	psrelbas	⊢ ( 𝜑 → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
419	418	ffnd	⊢ ( 𝜑 → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
420	1 10 16 2 416	psrelbas	⊢ ( 𝜑 → ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
421	420	ffnd	⊢ ( 𝜑 → ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
422	106	a1i	⊢ ( 𝜑 → { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∈ V )
423		inidm	⊢ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∩ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
424	413	adantr	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ∈ 𝐵 )
425	1 2 35 4 16 424 386 15	psrmulval	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) ‘ 𝑑 ) = ( 𝑅 Σg ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) ) )
426	355	a1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∈ V )
427	12	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑅 ∈ Ring )
428		elrabi	⊢ ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → 𝑏 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
429	1 10 16 2 413	psrelbas	⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
430	429	adantr	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
431	430	ffvelcdmda	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑏 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ∈ ( Base ‘ 𝑅 ) )
432	428 431	sylan2	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ∈ ( Base ‘ 𝑅 ) )
433	42	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝐺 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
434	16 243	psrbagconcl	⊢ ( ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f − 𝑏 ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
435	434	adantll	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f − 𝑏 ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
436		elrabi	⊢ ( ( 𝑑 ∘f − 𝑏 ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → ( 𝑑 ∘f − 𝑏 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
437	435 436	syl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f − 𝑏 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
438	433 437	ffvelcdmd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ∈ ( Base ‘ 𝑅 ) )
439	10 35 427 432 438	ringcld	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ∈ ( Base ‘ 𝑅 ) )
440	439	fmpttd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) : { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ⟶ ( Base ‘ 𝑅 ) )
441		ovex	⊢ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ∈ V
442		eqid	⊢ ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) = ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) )
443	441 442	fnmpti	⊢ ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) Fn { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 }
444	443	a1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) Fn { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
445	444 221 113	fndmfifsupp	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) finSupp ( 0g ‘ 𝑅 ) )
446		eqid	⊢ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
447		df-of	⊢ ∘f + = ( 𝑚 ∈ V , 𝑛 ∈ V ↦ ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) )
448		vex	⊢ 𝑢 ∈ V
449	448	a1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑢 ∈ V )
450		ssv	⊢ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ⊆ V
451	450	a1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ⊆ V )
452		ssv	⊢ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ⊆ V
453	452	a1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ⊆ V )
454	447 449 451 453	elimampo	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↔ ∃ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) ) )
455	454	biimpa	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ∃ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) )
456		elrabi	⊢ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → 𝑚 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
457	16	psrbagf	⊢ ( 𝑚 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → 𝑚 : 𝐼 ⟶ ℕ0 )
458	457	ffund	⊢ ( 𝑚 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → Fun 𝑚 )
459	456 458	syl	⊢ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → Fun 𝑚 )
460	459	funfnd	⊢ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → 𝑚 Fn dom 𝑚 )
461	460	ad2antrl	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → 𝑚 Fn dom 𝑚 )
462		velsn	⊢ ( 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ↔ 𝑛 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) )
463		funmpt	⊢ Fun ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) )
464		funeq	⊢ ( 𝑛 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) → ( Fun 𝑛 ↔ Fun ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
465	463 464	mpbiri	⊢ ( 𝑛 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) → Fun 𝑛 )
466	465	funfnd	⊢ ( 𝑛 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) → 𝑛 Fn dom 𝑛 )
467	462 466	sylbi	⊢ ( 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } → 𝑛 Fn dom 𝑛 )
468	467	ad2antll	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → 𝑛 Fn dom 𝑛 )
469		vex	⊢ 𝑚 ∈ V
470	469	dmex	⊢ dom 𝑚 ∈ V
471	470	a1i	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → dom 𝑚 ∈ V )
472		vex	⊢ 𝑛 ∈ V
473	472	dmex	⊢ dom 𝑛 ∈ V
474	473	a1i	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → dom 𝑛 ∈ V )
475		eqid	⊢ ( dom 𝑚 ∩ dom 𝑛 ) = ( dom 𝑚 ∩ dom 𝑛 )
476		eqidd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑜 ∈ dom 𝑚 ) → ( 𝑚 ‘ 𝑜 ) = ( 𝑚 ‘ 𝑜 ) )
477		eqidd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑜 ∈ dom 𝑛 ) → ( 𝑛 ‘ 𝑜 ) = ( 𝑛 ‘ 𝑜 ) )
478	461 468 471 474 475 476 477	offval	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑚 ∘f + 𝑛 ) = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) )
479	478	eqeq2d	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑚 ∘f + 𝑛 ) ↔ 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) ) )
480		elsni	⊢ ( 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } → 𝑛 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) )
481	480	oveq2d	⊢ ( 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } → ( 𝑚 ∘f + 𝑛 ) = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
482	481	eqeq2d	⊢ ( 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } → ( 𝑢 = ( 𝑚 ∘f + 𝑛 ) ↔ 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
483	482	ad2antll	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑚 ∘f + 𝑛 ) ↔ 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
484	5	ad3antrrr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝐼 ∈ 𝑉 )
485	456 457	syl	⊢ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → 𝑚 : 𝐼 ⟶ ℕ0 )
486	485	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑚 : 𝐼 ⟶ ℕ0 )
487	131	a1i	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) : 𝐼 ⟶ ℕ0 )
488		nn0cn	⊢ ( 𝑞 ∈ ℕ0 → 𝑞 ∈ ℂ )
489		nn0cn	⊢ ( 𝑟 ∈ ℕ0 → 𝑟 ∈ ℂ )
490		nn0cn	⊢ ( 𝑠 ∈ ℕ0 → 𝑠 ∈ ℂ )
491		addsubass	⊢ ( ( 𝑞 ∈ ℂ ∧ 𝑟 ∈ ℂ ∧ 𝑠 ∈ ℂ ) → ( ( 𝑞 + 𝑟 ) − 𝑠 ) = ( 𝑞 + ( 𝑟 − 𝑠 ) ) )
492	488 489 490 491	syl3an	⊢ ( ( 𝑞 ∈ ℕ0 ∧ 𝑟 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ) → ( ( 𝑞 + 𝑟 ) − 𝑠 ) = ( 𝑞 + ( 𝑟 − 𝑠 ) ) )
493	492	adantl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ ( 𝑞 ∈ ℕ0 ∧ 𝑟 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ) → ( ( 𝑞 + 𝑟 ) − 𝑠 ) = ( 𝑞 + ( 𝑟 − 𝑠 ) ) )
494	484 486 487 487 493	caofass	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( 𝑚 ∘f + ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
495		simpr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → 𝑖 ∈ 𝐼 )
496	56	a1i	⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ ℕ0 )
497	68 76 495 496	fvmptd3	⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑖 ) = if ( 𝑖 = 𝑋 , 1 , 0 ) )
498	133 133 5 5 72 497 497	offval	⊢ ( 𝜑 → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) )
499	498	oveq2d	⊢ ( 𝜑 → ( 𝑚 ∘f + ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝑚 ∘f + ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) ) )
500	499	ad3antrrr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝑚 ∘f + ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) ) )
501	237	subidi	⊢ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) = 0
502	501	mpteq2i	⊢ ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) = ( 𝑖 ∈ 𝐼 ↦ 0 )
503		fconstmpt	⊢ ( 𝐼 × { 0 } ) = ( 𝑖 ∈ 𝐼 ↦ 0 )
504	502 503	eqtr4i	⊢ ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) = ( 𝐼 × { 0 } )
505	504	oveq2i	⊢ ( 𝑚 ∘f + ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝑚 ∘f + ( 𝐼 × { 0 } ) )
506		0zd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 0 ∈ ℤ )
507	488	addridd	⊢ ( 𝑞 ∈ ℕ0 → ( 𝑞 + 0 ) = 𝑞 )
508	507	adantl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑞 ∈ ℕ0 ) → ( 𝑞 + 0 ) = 𝑞 )
509	484 486 506 508	caofid0r	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝐼 × { 0 } ) ) = 𝑚 )
510	505 509	eqtrid	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) ) = 𝑚 )
511	494 500 510	3eqtrd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = 𝑚 )
512		simpr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
513	511 512	eqeltrd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
514		oveq1	⊢ ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
515	514	eleq1d	⊢ ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↔ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
516	513 515	syl5ibrcom	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
517	516	adantrr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
518	483 517	sylbid	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑚 ∘f + 𝑛 ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
519	479 518	sylbird	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
520	519	rexlimdvva	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ∃ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
521	455 520	mpd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
522		simpr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
523	5	mptexd	⊢ ( 𝜑 → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ V )
524		elsng	⊢ ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ V → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ↔ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
525	523 524	syl	⊢ ( 𝜑 → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ↔ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
526	68 525	mpbiri	⊢ ( 𝜑 → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } )
527	526	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } )
528	447	mpofun	⊢ Fun ∘f +
529	528	a1i	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → Fun ∘f + )
530		xpss	⊢ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ⊆ ( V × V )
531	470	inex1	⊢ ( dom 𝑚 ∩ dom 𝑛 ) ∈ V
532	531	mptex	⊢ ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) ∈ V
533	532	rgen2w	⊢ ∀ 𝑚 ∈ V ∀ 𝑛 ∈ V ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) ∈ V
534	447	dmmpoga	⊢ ( ∀ 𝑚 ∈ V ∀ 𝑛 ∈ V ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) ∈ V → dom ∘f + = ( V × V ) )
535	533 534	mp1i	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → dom ∘f + = ( V × V ) )
536	530 535	sseqtrrid	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ⊆ dom ∘f + )
537	522 527 529 536	elovimad	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) )
538	5	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → 𝐼 ∈ 𝑉 )
539		elrabi	⊢ ( 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → 𝑣 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
540	16	psrbagf	⊢ ( 𝑣 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → 𝑣 : 𝐼 ⟶ ℕ0 )
541	539 540	syl	⊢ ( 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → 𝑣 : 𝐼 ⟶ ℕ0 )
542	541	ad2antll	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → 𝑣 : 𝐼 ⟶ ℕ0 )
543	131	a1i	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) : 𝐼 ⟶ ℕ0 )
544	492	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) ∧ ( 𝑞 ∈ ℕ0 ∧ 𝑟 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ) → ( ( 𝑞 + 𝑟 ) − 𝑠 ) = ( 𝑞 + ( 𝑟 − 𝑠 ) ) )
545	538 542 543 543 544	caofass	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( 𝑣 ∘f + ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
546	133	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) Fn 𝐼 )
547	78	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑖 ) = if ( 𝑖 = 𝑋 , 1 , 0 ) )
548	546 546 538 538 72 547 547	offval	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) )
549	548	oveq2d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( 𝑣 ∘f + ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝑣 ∘f + ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) ) )
550	504	oveq2i	⊢ ( 𝑣 ∘f + ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝑣 ∘f + ( 𝐼 × { 0 } ) )
551		0zd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → 0 ∈ ℤ )
552		nn0cn	⊢ ( 𝑝 ∈ ℕ0 → 𝑝 ∈ ℂ )
553	552	addridd	⊢ ( 𝑝 ∈ ℕ0 → ( 𝑝 + 0 ) = 𝑝 )
554	553	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) ∧ 𝑝 ∈ ℕ0 ) → ( 𝑝 + 0 ) = 𝑝 )
555	538 542 551 554	caofid0r	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( 𝑣 ∘f + ( 𝐼 × { 0 } ) ) = 𝑣 )
556	550 555	eqtrid	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( 𝑣 ∘f + ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) ) = 𝑣 )
557	545 549 556	3eqtrrd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → 𝑣 = ( ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
558		oveq1	⊢ ( 𝑢 = ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
559	558	eqeq2d	⊢ ( 𝑢 = ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑣 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ 𝑣 = ( ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
560	557 559	syl5ibrcom	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( 𝑢 = ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → 𝑣 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
561	18	ad3antrrr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
562	16	psrbagaddcl	⊢ ( ( 𝑚 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
563	456 561 562	syl2an2	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
564	16	psrbagf	⊢ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) : 𝐼 ⟶ ℕ0 )
565	563 564	syl	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) : 𝐼 ⟶ ℕ0 )
566	565	adantrr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) : 𝐼 ⟶ ℕ0 )
567		feq1	⊢ ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 : 𝐼 ⟶ ℕ0 ↔ ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) : 𝐼 ⟶ ℕ0 ) )
568	566 567	syl5ibrcom	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → 𝑢 : 𝐼 ⟶ ℕ0 ) )
569	483 568	sylbid	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑚 ∘f + 𝑛 ) → 𝑢 : 𝐼 ⟶ ℕ0 ) )
570	479 569	sylbird	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) → 𝑢 : 𝐼 ⟶ ℕ0 ) )
571	570	rexlimdvva	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ∃ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) → 𝑢 : 𝐼 ⟶ ℕ0 ) )
572	455 571	mpd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑢 : 𝐼 ⟶ ℕ0 )
573	572	adantrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → 𝑢 : 𝐼 ⟶ ℕ0 )
574	573	ffvelcdmda	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ∈ ℕ0 )
575	574	nn0cnd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ∈ ℂ )
576	237	a1i	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) ∧ 𝑖 ∈ 𝐼 ) → if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ ℂ )
577	575 576	npcand	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) ∧ 𝑖 ∈ 𝐼 ) → ( ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) = ( 𝑢 ‘ 𝑖 ) )
578	577	mpteq2dva	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( 𝑖 ∈ 𝐼 ↦ ( ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( 𝑢 ‘ 𝑖 ) ) )
579	573	ffnd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → 𝑢 Fn 𝐼 )
580	579 546 538 538 72	offn	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) Fn 𝐼 )
581		eqidd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) = ( 𝑢 ‘ 𝑖 ) )
582	579 546 538 538 72 581 547	ofval	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
583	580 546 538 538 72 582 547	offval	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) )
584	573	feqmptd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → 𝑢 = ( 𝑖 ∈ 𝐼 ↦ ( 𝑢 ‘ 𝑖 ) ) )
585	578 583 584	3eqtr4rd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → 𝑢 = ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
586		oveq1	⊢ ( 𝑣 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
587	586	eqeq2d	⊢ ( 𝑣 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 = ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ 𝑢 = ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
588	585 587	syl5ibrcom	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( 𝑣 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → 𝑢 = ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
589	560 588	impbid	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( 𝑢 = ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ 𝑣 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
590	446 521 537 589	f1o2d	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) : ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) –1-1-onto→ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
591	10 104 14 426 440 445 590	gsumf1o	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) ) = ( 𝑅 Σg ( ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) ∘ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) )
592	553	adantl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑝 ∈ ℕ0 ) → ( 𝑝 + 0 ) = 𝑝 )
593	484 486 506 592	caofid0r	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝐼 × { 0 } ) ) = 𝑚 )
594	505 593	eqtrid	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) ) = 𝑚 )
595	494 500 594	3eqtrd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = 𝑚 )
596	595 512	eqeltrd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
597	596 515	syl5ibrcom	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
598	597	adantrr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
599	483 598	sylbid	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑚 ∘f + 𝑛 ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
600	479 599	sylbird	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
601	600	rexlimdvva	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ∃ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
602	455 601	mpd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
603		eqidd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
604		eqidd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) = ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) )
605		fveq2	⊢ ( 𝑏 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) = ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
606		oveq2	⊢ ( 𝑏 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑑 ∘f − 𝑏 ) = ( 𝑑 ∘f − ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
607	606	fveq2d	⊢ ( 𝑏 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) = ( 𝐺 ‘ ( 𝑑 ∘f − ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) )
608	605 607	oveq12d	⊢ ( 𝑏 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) = ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) )
609	602 603 604 608	fmptco	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) ∘ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) )
610	5	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝐼 ∈ 𝑉 )
611	6	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑅 ∈ CRing )
612	7	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑋 ∈ 𝐼 )
613	8	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝐹 ∈ 𝐵 )
614		elrabi	⊢ ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
615	602 614	syl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
616	1 2 16 610 611 612 613 615	psdcoef	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( ( ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) )
617	572	ffnd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑢 Fn 𝐼 )
618	131	a1i	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) : 𝐼 ⟶ ℕ0 )
619	618	ffnd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) Fn 𝐼 )
620		eqidd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑋 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑋 ) = ( 𝑢 ‘ 𝑋 ) )
621		iftrue	⊢ ( 𝑦 = 𝑋 → if ( 𝑦 = 𝑋 , 1 , 0 ) = 1 )
622		1ex	⊢ 1 ∈ V
623	621 68 622	fvmpt	⊢ ( 𝑋 ∈ 𝐼 → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑋 ) = 1 )
624	623	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑋 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑋 ) = 1 )
625	617 619 610 610 72 620 624	ofval	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑋 ∈ 𝐼 ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = ( ( 𝑢 ‘ 𝑋 ) − 1 ) )
626	612 625	mpdan	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = ( ( 𝑢 ‘ 𝑋 ) − 1 ) )
627	626	oveq1d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) + 1 ) = ( ( ( 𝑢 ‘ 𝑋 ) − 1 ) + 1 ) )
628		nn0sscn	⊢ ℕ0 ⊆ ℂ
629	628	a1i	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ℕ0 ⊆ ℂ )
630	572 629	fssd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑢 : 𝐼 ⟶ ℂ )
631	630 612	ffvelcdmd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑢 ‘ 𝑋 ) ∈ ℂ )
632		1cnd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 1 ∈ ℂ )
633	631 632	npcand	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( ( 𝑢 ‘ 𝑋 ) − 1 ) + 1 ) = ( 𝑢 ‘ 𝑋 ) )
634	627 633	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) + 1 ) = ( 𝑢 ‘ 𝑋 ) )
635	617 619 610 610 72	offn	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) Fn 𝐼 )
636		eqidd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) = ( 𝑢 ‘ 𝑖 ) )
637	78	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑖 ) = if ( 𝑖 = 𝑋 , 1 , 0 ) )
638	617 619 610 610 72 636 637	ofval	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
639	572	ffvelcdmda	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ∈ ℕ0 )
640	639	nn0cnd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ∈ ℂ )
641	237	a1i	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ ℂ )
642	640 641	npcand	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) = ( 𝑢 ‘ 𝑖 ) )
643	610 635 619 617 638 637 642	offveq	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = 𝑢 )
644	643	fveq2d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝐹 ‘ ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝐹 ‘ 𝑢 ) )
645	634 644	oveq12d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ 𝑢 ) ) )
646	616 645	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ 𝑢 ) ) )
647	28	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑑 : 𝐼 ⟶ ℕ0 )
648	647	ffvelcdmda	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℕ0 )
649	648	nn0cnd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℂ )
650	649 640 641	subsub3d	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑑 ‘ 𝑖 ) − ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) = ( ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) − ( 𝑢 ‘ 𝑖 ) ) )
651	650	mpteq2dva	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑖 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑖 ) − ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) − ( 𝑢 ‘ 𝑖 ) ) ) )
652	65	adantr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑑 Fn 𝐼 )
653		eqidd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) = ( 𝑑 ‘ 𝑖 ) )
654	652 635 610 610 72 653 638	offval	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑑 ∘f − ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑖 ) − ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) ) )
655	652 619 610 610 72	offn	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) Fn 𝐼 )
656	652 619 610 610 72 653 637	ofval	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
657	655 617 610 610 72 656 636	offval	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) = ( 𝑖 ∈ 𝐼 ↦ ( ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) − ( 𝑢 ‘ 𝑖 ) ) ) )
658	651 654 657	3eqtr4d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑑 ∘f − ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) )
659	658	fveq2d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝐺 ‘ ( 𝑑 ∘f − ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) )
660	646 659	oveq12d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) = ( ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ 𝑢 ) ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) )
661	12	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑅 ∈ Ring )
662	572 612	ffvelcdmd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑢 ‘ 𝑋 ) ∈ ℕ0 )
663	662	nn0zd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑢 ‘ 𝑋 ) ∈ ℤ )
664	37	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝐹 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
665		simpllr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
666	18	ad3antrrr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
667		simprl	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
668		eqid	⊢ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } = { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) }
669	16 243 668	psrbagleadd1	⊢ ( ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
670	665 666 667 669	syl3anc	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
671		eleq1	⊢ ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↔ ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) )
672	670 671	syl5ibrcom	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → 𝑢 ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) )
673	483 672	sylbid	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑚 ∘f + 𝑛 ) → 𝑢 ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) )
674	479 673	sylbird	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) → 𝑢 ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) )
675	674	rexlimdvva	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ∃ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) → 𝑢 ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) )
676	455 675	mpd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑢 ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
677		elrabi	⊢ ( 𝑢 ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } → 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
678	676 677	syl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
679	664 678	ffvelcdmd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝐹 ‘ 𝑢 ) ∈ ( Base ‘ 𝑅 ) )
680	42	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝐺 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
681	21	adantr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
682	16 668	psrbagconcl	⊢ ( ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ 𝑢 ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
683	681 676 682	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
684		elrabi	⊢ ( ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
685	683 684	syl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
686	680 685	ffvelcdmd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ∈ ( Base ‘ 𝑅 ) )
687	10 24 35	mulgass2	⊢ ( ( 𝑅 ∈ Ring ∧ ( ( 𝑢 ‘ 𝑋 ) ∈ ℤ ∧ ( 𝐹 ‘ 𝑢 ) ∈ ( Base ‘ 𝑅 ) ∧ ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ∈ ( Base ‘ 𝑅 ) ) ) → ( ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ 𝑢 ) ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) = ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
688	661 663 679 686 687	syl13anc	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ 𝑢 ) ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) = ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
689	660 688	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) = ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
690	689	mpteq2dva	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) = ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) )
691	609 690	eqtrd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) ∘ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) )
692	691	oveq2d	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) ∘ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) = ( 𝑅 Σg ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) )
693		snex	⊢ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ∈ V
694	355 693	xpex	⊢ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ∈ V
695	694	funimaex	⊢ ( Fun ∘f + → ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∈ V )
696	528 695	mp1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∈ V )
697	26	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑅 ∈ Mnd )
698	10 35 661 679 686	ringcld	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ∈ ( Base ‘ 𝑅 ) )
699	10 24 697 662 698	mulgnn0cld	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
700		eqid	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) = ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
701	359 700	fnmpti	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) Fn { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) }
702	701	a1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) Fn { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
703	702 23 113	fndmfifsupp	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) finSupp ( 0g ‘ 𝑅 ) )
704	460	ad2antlr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) → 𝑚 Fn dom 𝑚 )
705	467	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) → 𝑛 Fn dom 𝑛 )
706	470	a1i	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) → dom 𝑚 ∈ V )
707	473	a1i	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) → dom 𝑛 ∈ V )
708		eqidd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ∧ 𝑜 ∈ dom 𝑚 ) → ( 𝑚 ‘ 𝑜 ) = ( 𝑚 ‘ 𝑜 ) )
709		eqidd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ∧ 𝑜 ∈ dom 𝑛 ) → ( 𝑛 ‘ 𝑜 ) = ( 𝑛 ‘ 𝑜 ) )
710	704 705 706 707 475 708 709	offval	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) → ( 𝑚 ∘f + 𝑛 ) = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) )
711	710	eqeq2d	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) → ( 𝑢 = ( 𝑚 ∘f + 𝑛 ) ↔ 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) ) )
712	711	rexbidva	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑚 ∘f + 𝑛 ) ↔ ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) ) )
713	18	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
714		oveq2	⊢ ( 𝑛 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) → ( 𝑚 ∘f + 𝑛 ) = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
715	714	eqeq2d	⊢ ( 𝑛 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) → ( 𝑢 = ( 𝑚 ∘f + 𝑛 ) ↔ 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
716	715	rexsng	⊢ ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → ( ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑚 ∘f + 𝑛 ) ↔ 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
717	713 716	syl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑚 ∘f + 𝑛 ) ↔ 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
718	712 717	bitr3d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) ↔ 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
719	718	rexbidva	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ∃ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) ↔ ∃ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
720		breq1	⊢ ( 𝑘 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
721		breq1	⊢ ( 𝑘 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑘 ∘r ≤ 𝑑 ↔ ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑑 ) )
722		fveq1	⊢ ( 𝑘 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑘 ‘ 𝑋 ) = ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) )
723	722	eqeq1d	⊢ ( 𝑘 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( ( 𝑘 ‘ 𝑋 ) = 0 ↔ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = 0 ) )
724	721 723	anbi12d	⊢ ( 𝑘 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ↔ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑑 ∧ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = 0 ) ) )
725	724	notbid	⊢ ( 𝑘 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ↔ ¬ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑑 ∧ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = 0 ) ) )
726	720 725	anbi12d	⊢ ( 𝑘 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) ↔ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑑 ∧ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = 0 ) ) ) )
727	456 713 562	syl2an2	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
728		simplr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
729		simpr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
730	16 243 44	psrbagleadd1	⊢ ( ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
731	728 713 729 730	syl3anc	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
732	720	elrab	⊢ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↔ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
733	732	simprbi	⊢ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
734	731 733	syl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
735	7	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑋 ∈ 𝐼 )
736	485	adantl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑚 : 𝐼 ⟶ ℕ0 )
737	736	ffnd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑚 Fn 𝐼 )
738	133	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) Fn 𝐼 )
739	5	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝐼 ∈ 𝑉 )
740		eqidd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑋 ∈ 𝐼 ) → ( 𝑚 ‘ 𝑋 ) = ( 𝑚 ‘ 𝑋 ) )
741	623	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑋 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑋 ) = 1 )
742	737 738 739 739 72 740 741	ofval	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑋 ∈ 𝐼 ) → ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = ( ( 𝑚 ‘ 𝑋 ) + 1 ) )
743	735 742	mpdan	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = ( ( 𝑚 ‘ 𝑋 ) + 1 ) )
744	736 735	ffvelcdmd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ‘ 𝑋 ) ∈ ℕ0 )
745		nn0p1nn	⊢ ( ( 𝑚 ‘ 𝑋 ) ∈ ℕ0 → ( ( 𝑚 ‘ 𝑋 ) + 1 ) ∈ ℕ )
746	744 745	syl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑚 ‘ 𝑋 ) + 1 ) ∈ ℕ )
747	743 746	eqeltrd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) ∈ ℕ )
748	747	nnne0d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) ≠ 0 )
749	748	neneqd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ¬ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = 0 )
750	749	intnand	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ¬ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑑 ∧ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = 0 ) )
751	734 750	jca	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑑 ∧ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = 0 ) ) )
752	726 727 751	elrabd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } )
753		eleq1	⊢ ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ↔ ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) )
754	752 753	syl5ibrcom	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) )
755		breq1	⊢ ( 𝑘 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑘 ∘r ≤ 𝑑 ↔ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑑 ) )
756		elrabi	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } → 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
757	756	adantl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
758	131	a1i	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) : 𝐼 ⟶ ℕ0 )
759	756 123	syl	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } → 𝑢 : 𝐼 ⟶ ℕ0 )
760	759	adantl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 𝑢 : 𝐼 ⟶ ℕ0 )
761	7	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 𝑋 ∈ 𝐼 )
762	760 761	ffvelcdmd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑢 ‘ 𝑋 ) ∈ ℕ0 )
763	339	notbid	⊢ ( 𝑘 = 𝑢 → ( ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ↔ ¬ ( 𝑢 ∘r ≤ 𝑑 ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ) )
764	118 763	anbi12d	⊢ ( 𝑘 = 𝑢 → ( ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) ↔ ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑢 ∘r ≤ 𝑑 ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ) ) )
765	764	elrab	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ↔ ( 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑢 ∘r ≤ 𝑑 ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ) ) )
766	765	simprbi	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } → ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑢 ∘r ≤ 𝑑 ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ) )
767	766	simpld	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } → 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
768	767	adantl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
769	768	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
770	756 124	syl	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } → 𝑢 Fn 𝐼 )
771	770	adantl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 𝑢 Fn 𝐼 )
772	771	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → 𝑢 Fn 𝐼 )
773	21	adantr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
774	88	ffnd	⊢ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) Fn 𝐼 )
775	773 774	syl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) Fn 𝐼 )
776	775	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) Fn 𝐼 )
777	5	ad3antrrr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → 𝐼 ∈ 𝑉 )
778		eqidd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) = ( 𝑢 ‘ 𝑖 ) )
779		eqidd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) )
780	772 776 777 777 72 778 779	ofrfval	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) ) )
781	769 780	mpbid	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) )
782	781	r19.21bi	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) )
783	782	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 ≠ 𝑋 ) → ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) )
784	65	ad3antrrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ≠ 𝑋 ) → 𝑑 Fn 𝐼 )
785	69	a1i	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ≠ 𝑋 ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) Fn 𝐼 )
786	5	ad4antr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ≠ 𝑋 ) → 𝐼 ∈ 𝑉 )
787		eqidd	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ≠ 𝑋 ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) = ( 𝑑 ‘ 𝑖 ) )
788	78	adantl	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ≠ 𝑋 ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑖 ) = if ( 𝑖 = 𝑋 , 1 , 0 ) )
789	784 785 786 786 72 787 788	ofval	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ≠ 𝑋 ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
790	789	an32s	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 ≠ 𝑋 ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
791	158	adantl	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 ≠ 𝑋 ) → if ( 𝑖 = 𝑋 , 1 , 0 ) = 0 )
792	791	oveq2d	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 ≠ 𝑋 ) → ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) = ( ( 𝑑 ‘ 𝑖 ) + 0 ) )
793	29	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → 𝑑 : 𝐼 ⟶ ℕ0 )
794	793	ffvelcdmda	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℕ0 )
795	794	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 ≠ 𝑋 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℕ0 )
796	795	nn0cnd	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 ≠ 𝑋 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℂ )
797	796	addridd	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 ≠ 𝑋 ) → ( ( 𝑑 ‘ 𝑖 ) + 0 ) = ( 𝑑 ‘ 𝑖 ) )
798	790 792 797	3eqtrd	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 ≠ 𝑋 ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( 𝑑 ‘ 𝑖 ) )
799	783 798	breqtrd	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 ≠ 𝑋 ) → ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) )
800		simpr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → ( 𝑢 ‘ 𝑋 ) = 0 )
801	29	adantr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 𝑑 : 𝐼 ⟶ ℕ0 )
802	801 761	ffvelcdmd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 )
803	802	nn0ge0d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 0 ≤ ( 𝑑 ‘ 𝑋 ) )
804	803	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → 0 ≤ ( 𝑑 ‘ 𝑋 ) )
805	800 804	eqbrtrd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → ( 𝑢 ‘ 𝑋 ) ≤ ( 𝑑 ‘ 𝑋 ) )
806	805	adantr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑋 ) ≤ ( 𝑑 ‘ 𝑋 ) )
807	175 799 806	pm2.61ne	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) )
808	807	ralrimiva	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) )
809	65	adantr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 𝑑 Fn 𝐼 )
810	809	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → 𝑑 Fn 𝐼 )
811		eqidd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) = ( 𝑑 ‘ 𝑖 ) )
812	772 810 777 777 72 778 811	ofrfval	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → ( 𝑢 ∘r ≤ 𝑑 ↔ ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
813	808 812	mpbird	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → 𝑢 ∘r ≤ 𝑑 )
814	813	ex	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( ( 𝑢 ‘ 𝑋 ) = 0 → 𝑢 ∘r ≤ 𝑑 ) )
815	766	simprd	⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } → ¬ ( 𝑢 ∘r ≤ 𝑑 ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) )
816	815	adantl	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ¬ ( 𝑢 ∘r ≤ 𝑑 ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) )
817		imnan	⊢ ( ( 𝑢 ∘r ≤ 𝑑 → ¬ ( 𝑢 ‘ 𝑋 ) = 0 ) ↔ ¬ ( 𝑢 ∘r ≤ 𝑑 ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) )
818	816 817	sylibr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑢 ∘r ≤ 𝑑 → ¬ ( 𝑢 ‘ 𝑋 ) = 0 ) )
819	818	con2d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( ( 𝑢 ‘ 𝑋 ) = 0 → ¬ 𝑢 ∘r ≤ 𝑑 ) )
820	814 819	pm2.65d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ¬ ( 𝑢 ‘ 𝑋 ) = 0 )
821	820	neqned	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑢 ‘ 𝑋 ) ≠ 0 )
822	762 821 191	sylanbrc	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑢 ‘ 𝑋 ) ∈ ℕ )
823	822	nnge1d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 1 ≤ ( 𝑢 ‘ 𝑋 ) )
824	823	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → 1 ≤ ( 𝑢 ‘ 𝑋 ) )
825	173	breq2d	⊢ ( 𝑖 = 𝑋 → ( 1 ≤ ( 𝑢 ‘ 𝑖 ) ↔ 1 ≤ ( 𝑢 ‘ 𝑋 ) ) )
826	824 825	syl5ibrcom	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑖 = 𝑋 → 1 ≤ ( 𝑢 ‘ 𝑖 ) ) )
827	826	imp	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 = 𝑋 ) → 1 ≤ ( 𝑢 ‘ 𝑖 ) )
828	760	ffvelcdmda	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ∈ ℕ0 )
829	828	nn0ge0d	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → 0 ≤ ( 𝑢 ‘ 𝑖 ) )
830	829	adantr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) ∧ ¬ 𝑖 = 𝑋 ) → 0 ≤ ( 𝑢 ‘ 𝑖 ) )
831	827 830	ifpimpda	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → if- ( 𝑖 = 𝑋 , 1 ≤ ( 𝑢 ‘ 𝑖 ) , 0 ≤ ( 𝑢 ‘ 𝑖 ) ) )
832		brif1	⊢ ( if ( 𝑖 = 𝑋 , 1 , 0 ) ≤ ( 𝑢 ‘ 𝑖 ) ↔ if- ( 𝑖 = 𝑋 , 1 ≤ ( 𝑢 ‘ 𝑖 ) , 0 ≤ ( 𝑢 ‘ 𝑖 ) ) )
833	831 832	sylibr	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → if ( 𝑖 = 𝑋 , 1 , 0 ) ≤ ( 𝑢 ‘ 𝑖 ) )
834	833	ralrimiva	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ∀ 𝑖 ∈ 𝐼 if ( 𝑖 = 𝑋 , 1 , 0 ) ≤ ( 𝑢 ‘ 𝑖 ) )
835	69	a1i	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) Fn 𝐼 )
836	5	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 𝐼 ∈ 𝑉 )
837	78	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑖 ) = if ( 𝑖 = 𝑋 , 1 , 0 ) )
838		eqidd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) = ( 𝑢 ‘ 𝑖 ) )
839	835 771 836 836 72 837 838	ofrfval	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∘r ≤ 𝑢 ↔ ∀ 𝑖 ∈ 𝐼 if ( 𝑖 = 𝑋 , 1 , 0 ) ≤ ( 𝑢 ‘ 𝑖 ) ) )
840	834 839	mpbird	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∘r ≤ 𝑢 )
841	16	psrbagcon	⊢ ( ( 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) : 𝐼 ⟶ ℕ0 ∧ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∘r ≤ 𝑢 ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑢 ) )
842	757 758 840 841	syl3anc	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑢 ) )
843	842	simpld	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
844		eqidd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) = ( 𝑑 ‘ 𝑖 ) )
845	809 835 836 836 72 844 837	ofval	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
846	771 775 836 836 72 838 845	ofrfval	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) )
847	768 846	mpbid	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
848	847	r19.21bi	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
849	828	nn0red	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ∈ ℝ )
850	60	a1i	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ ℝ )
851	801	ffvelcdmda	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℕ0 )
852	851	nn0red	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℝ )
853	849 850 852	lesubaddd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ≤ ( 𝑑 ‘ 𝑖 ) ↔ ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) )
854	848 853	mpbird	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ≤ ( 𝑑 ‘ 𝑖 ) )
855	854	ralrimiva	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ∀ 𝑖 ∈ 𝐼 ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ≤ ( 𝑑 ‘ 𝑖 ) )
856	771 835 836 836 72	offn	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) Fn 𝐼 )
857	771 835 836 836 72 838 837	ofval	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
858	856 809 836 836 72 857 844	ofrfval	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑑 ↔ ∀ 𝑖 ∈ 𝐼 ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
859	855 858	mpbird	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑑 )
860	755 843 859	elrabd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
861	828	nn0cnd	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ∈ ℂ )
862	237	a1i	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ ℂ )
863	861 862	npcand	⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) = ( 𝑢 ‘ 𝑖 ) )
864	863	mpteq2dva	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑖 ∈ 𝐼 ↦ ( ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( 𝑢 ‘ 𝑖 ) ) )
865	856 835 836 836 72 857 837	offval	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) )
866	760	feqmptd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 𝑢 = ( 𝑖 ∈ 𝐼 ↦ ( 𝑢 ‘ 𝑖 ) ) )
867	864 865 866	3eqtr4rd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 𝑢 = ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
868		oveq1	⊢ ( 𝑚 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
869	868	eqeq2d	⊢ ( 𝑚 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ 𝑢 = ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
870	754 860 867 869	rspceb2dv	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ∃ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) )
871	454 719 870	3bitrd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↔ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) )
872	871	eqrdv	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) = { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } )
873		difrab	⊢ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) = { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) }
874	872 873	eqtr4di	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) = ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) )
875		difssd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
876	874 875	eqsstrd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
877	703 876 113	fmptssfisupp	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) finSupp ( 0g ‘ 𝑅 ) )
878		difss	⊢ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 }
879		disjdif	⊢ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∩ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) = ∅
880		ssdisj	⊢ ( ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∩ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) = ∅ ) → ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∩ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) = ∅ )
881	878 879 880	mp2an	⊢ ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∩ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) = ∅
882	881	ineqcomi	⊢ ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∩ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) = ∅
883	882	a1i	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∩ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) = ∅ )
884	279 99	psdmullem	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∪ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) = ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) )
885	874 884	eqtr4d	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) = ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∪ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) )
886	10 104 11 14 696 699 877 883 885	gsumsplit2	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
887	692 886	eqtrd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) ∘ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
888	425 591 887	3eqtrd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) ‘ 𝑑 ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
889	415	adantr	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ∈ 𝐵 )
890	1 2 35 4 16 385 889 15	psrmulval	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) ‘ 𝑑 ) = ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ‘ ( 𝑑 ∘f − 𝑢 ) ) ) ) ) )
891	6	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑅 ∈ CRing )
892	9	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝐺 ∈ 𝐵 )
893	1 2 16 252 891 285 892 247	psdcoef	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ‘ ( 𝑑 ∘f − 𝑢 ) ) = ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f − 𝑢 ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) )
894	267	fveq2d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝐺 ‘ ( ( 𝑑 ∘f − 𝑢 ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) )
895	894	oveq2d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f − 𝑢 ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) )
896	893 895	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ‘ ( 𝑑 ∘f − 𝑢 ) ) = ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) )
897	896	oveq2d	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ‘ ( 𝑑 ∘f − 𝑢 ) ) ) = ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
898	309	nn0zd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ∈ ℤ )
899	10 24 35	mulgass3	⊢ ( ( 𝑅 ∈ Ring ∧ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ∈ ℤ ∧ ( 𝐹 ‘ 𝑢 ) ∈ ( Base ‘ 𝑅 ) ∧ ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ∈ ( Base ‘ 𝑅 ) ) ) → ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) = ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
900	224 898 228 271 899	syl13anc	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) = ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
901	897 900	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ‘ ( 𝑑 ∘f − 𝑢 ) ) ) = ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
902	901	mpteq2dva	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ‘ ( 𝑑 ∘f − 𝑢 ) ) ) ) = ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) )
903	902	oveq2d	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ‘ ( 𝑑 ∘f − 𝑢 ) ) ) ) ) = ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) )
904	10 11 14 221 321 275 282	gsummptfidmsplit	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
905	890 903 904	3eqtrd	⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) ‘ 𝑑 ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
906	419 421 422 422 423 888 905	offval	⊢ ( 𝜑 → ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) ∘f ( +g ‘ 𝑅 ) ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) ) = ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ) ) )
907	417 906	eqtrd	⊢ ( 𝜑 → ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) + ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) ) = ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ) ) )
908	409 411 907	3eqtr4d	⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐹 · 𝐺 ) ) = ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) + ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) ) )