Theorem hoidmvlelem3

Description: This is the contradiction proven in step (d) in the proof of Lemma 115B of Fremlin1 p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020)

	Ref	Expression
Hypotheses	hoidmvlelem3.l	⊢ 𝐿 = ( 𝑥 ∈ Fin ↦ ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) , 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) ) )
	hoidmvlelem3.x	⊢ ( 𝜑 → 𝑋 ∈ Fin )
	hoidmvlelem3.y	⊢ ( 𝜑 → 𝑌 ⊆ 𝑋 )
	hoidmvlelem3.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑋 ∖ 𝑌 ) )
	hoidmvlelem3.w	⊢ 𝑊 = ( 𝑌 ∪ { 𝑍 } )
	hoidmvlelem3.a	⊢ ( 𝜑 → 𝐴 : 𝑊 ⟶ ℝ )
	hoidmvlelem3.b	⊢ ( 𝜑 → 𝐵 : 𝑊 ⟶ ℝ )
	hoidmvlelem3.lt	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( 𝐴 ‘ 𝑘 ) < ( 𝐵 ‘ 𝑘 ) )
	hoidmvlelem3.f	⊢ 𝐹 = ( 𝑦 ∈ 𝑌 ↦ 0 )
	hoidmvlelem3.c	⊢ ( 𝜑 → 𝐶 : ℕ ⟶ ( ℝ ↑m 𝑊 ) )
	hoidmvlelem3.j	⊢ 𝐽 = ( 𝑗 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) )
	hoidmvlelem3.d	⊢ ( 𝜑 → 𝐷 : ℕ ⟶ ( ℝ ↑m 𝑊 ) )
	hoidmvlelem3.k	⊢ 𝐾 = ( 𝑗 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) )
	hoidmvlelem3.r	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ∈ ℝ )
	hoidmvlelem3.h	⊢ 𝐻 = ( 𝑥 ∈ ℝ ↦ ( 𝑐 ∈ ( ℝ ↑m 𝑊 ) ↦ ( 𝑗 ∈ 𝑊 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑐 ‘ 𝑗 ) , if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) ) ) ) )
	hoidmvlelem3.g	⊢ 𝐺 = ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) )
	hoidmvlelem3.e	⊢ ( 𝜑 → 𝐸 ∈ ℝ+ )
	hoidmvlelem3.u	⊢ 𝑈 = { 𝑧 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) }
	hoidmvlelem3.s	⊢ ( 𝜑 → 𝑆 ∈ 𝑈 )
	hoidmvlelem3.sb	⊢ ( 𝜑 → 𝑆 < ( 𝐵 ‘ 𝑍 ) )
	hoidmvlelem3.p	⊢ 𝑃 = ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) )
	hoidmvlelem3.i	⊢ ( 𝜑 → ∀ 𝑒 ∈ ( ℝ ↑m 𝑌 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑌 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) )
	hoidmvlelem3.i2	⊢ ( 𝜑 → X 𝑘 ∈ 𝑊 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
	hoidmvlelem3.o	⊢ 𝑂 = ( 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ↦ ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) )
Assertion	hoidmvlelem3	⊢ ( 𝜑 → ∃ 𝑢 ∈ 𝑈 𝑆 < 𝑢 )

Proof

Step	Hyp	Ref	Expression
1		hoidmvlelem3.l	⊢ 𝐿 = ( 𝑥 ∈ Fin ↦ ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) , 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) ) )
2		hoidmvlelem3.x	⊢ ( 𝜑 → 𝑋 ∈ Fin )
3		hoidmvlelem3.y	⊢ ( 𝜑 → 𝑌 ⊆ 𝑋 )
4		hoidmvlelem3.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑋 ∖ 𝑌 ) )
5		hoidmvlelem3.w	⊢ 𝑊 = ( 𝑌 ∪ { 𝑍 } )
6		hoidmvlelem3.a	⊢ ( 𝜑 → 𝐴 : 𝑊 ⟶ ℝ )
7		hoidmvlelem3.b	⊢ ( 𝜑 → 𝐵 : 𝑊 ⟶ ℝ )
8		hoidmvlelem3.lt	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( 𝐴 ‘ 𝑘 ) < ( 𝐵 ‘ 𝑘 ) )
9		hoidmvlelem3.f	⊢ 𝐹 = ( 𝑦 ∈ 𝑌 ↦ 0 )
10		hoidmvlelem3.c	⊢ ( 𝜑 → 𝐶 : ℕ ⟶ ( ℝ ↑m 𝑊 ) )
11		hoidmvlelem3.j	⊢ 𝐽 = ( 𝑗 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) )
12		hoidmvlelem3.d	⊢ ( 𝜑 → 𝐷 : ℕ ⟶ ( ℝ ↑m 𝑊 ) )
13		hoidmvlelem3.k	⊢ 𝐾 = ( 𝑗 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) )
14		hoidmvlelem3.r	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ∈ ℝ )
15		hoidmvlelem3.h	⊢ 𝐻 = ( 𝑥 ∈ ℝ ↦ ( 𝑐 ∈ ( ℝ ↑m 𝑊 ) ↦ ( 𝑗 ∈ 𝑊 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑐 ‘ 𝑗 ) , if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) ) ) ) )
16		hoidmvlelem3.g	⊢ 𝐺 = ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) )
17		hoidmvlelem3.e	⊢ ( 𝜑 → 𝐸 ∈ ℝ+ )
18		hoidmvlelem3.u	⊢ 𝑈 = { 𝑧 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) }
19		hoidmvlelem3.s	⊢ ( 𝜑 → 𝑆 ∈ 𝑈 )
20		hoidmvlelem3.sb	⊢ ( 𝜑 → 𝑆 < ( 𝐵 ‘ 𝑍 ) )
21		hoidmvlelem3.p	⊢ 𝑃 = ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) )
22		hoidmvlelem3.i	⊢ ( 𝜑 → ∀ 𝑒 ∈ ( ℝ ↑m 𝑌 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑌 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) )
23		hoidmvlelem3.i2	⊢ ( 𝜑 → X 𝑘 ∈ 𝑊 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
24		hoidmvlelem3.o	⊢ 𝑂 = ( 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ↦ ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) )
25		1nn	⊢ 1 ∈ ℕ
26	25	a1i	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → 1 ∈ ℕ )
27		0le0	⊢ 0 ≤ 0
28	27	a1i	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → 0 ≤ 0 )
29	16	a1i	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → 𝐺 = ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) )
30		fveq2	⊢ ( 𝑌 = ∅ → ( 𝐿 ‘ 𝑌 ) = ( 𝐿 ‘ ∅ ) )
31		reseq2	⊢ ( 𝑌 = ∅ → ( 𝐴 ↾ 𝑌 ) = ( 𝐴 ↾ ∅ ) )
32		res0	⊢ ( 𝐴 ↾ ∅ ) = ∅
33	32	a1i	⊢ ( 𝑌 = ∅ → ( 𝐴 ↾ ∅ ) = ∅ )
34	31 33	eqtrd	⊢ ( 𝑌 = ∅ → ( 𝐴 ↾ 𝑌 ) = ∅ )
35		reseq2	⊢ ( 𝑌 = ∅ → ( 𝐵 ↾ 𝑌 ) = ( 𝐵 ↾ ∅ ) )
36		res0	⊢ ( 𝐵 ↾ ∅ ) = ∅
37	36	a1i	⊢ ( 𝑌 = ∅ → ( 𝐵 ↾ ∅ ) = ∅ )
38	35 37	eqtrd	⊢ ( 𝑌 = ∅ → ( 𝐵 ↾ 𝑌 ) = ∅ )
39	30 34 38	oveq123d	⊢ ( 𝑌 = ∅ → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) = ( ∅ ( 𝐿 ‘ ∅ ) ∅ ) )
40	39	adantl	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) = ( ∅ ( 𝐿 ‘ ∅ ) ∅ ) )
41		f0	⊢ ∅ : ∅ ⟶ ℝ
42	41	a1i	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ∅ : ∅ ⟶ ℝ )
43	1 42 42	hoidmv0val	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( ∅ ( 𝐿 ‘ ∅ ) ∅ ) = 0 )
44	29 40 43	3eqtrd	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → 𝐺 = 0 )
45		nfcvd	⊢ ( 𝜑 → Ⅎ 𝑗 ( 𝑃 ‘ 1 ) )
46		nfv	⊢ Ⅎ 𝑗 𝜑
47		simpr	⊢ ( ( 𝜑 ∧ 𝑗 = 1 ) → 𝑗 = 1 )
48	47	fveq2d	⊢ ( ( 𝜑 ∧ 𝑗 = 1 ) → ( 𝑃 ‘ 𝑗 ) = ( 𝑃 ‘ 1 ) )
49		1red	⊢ ( 𝜑 → 1 ∈ ℝ )
50		rge0ssre	⊢ ( 0 [,) +∞ ) ⊆ ℝ
51		id	⊢ ( 𝜑 → 𝜑 )
52	25	a1i	⊢ ( 𝜑 → 1 ∈ ℕ )
53	25	elexi	⊢ 1 ∈ V
54		eleq1	⊢ ( 𝑗 = 1 → ( 𝑗 ∈ ℕ ↔ 1 ∈ ℕ ) )
55	54	anbi2d	⊢ ( 𝑗 = 1 → ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ↔ ( 𝜑 ∧ 1 ∈ ℕ ) ) )
56		fveq2	⊢ ( 𝑗 = 1 → ( 𝑃 ‘ 𝑗 ) = ( 𝑃 ‘ 1 ) )
57	56	eleq1d	⊢ ( 𝑗 = 1 → ( ( 𝑃 ‘ 𝑗 ) ∈ ( 0 [,) +∞ ) ↔ ( 𝑃 ‘ 1 ) ∈ ( 0 [,) +∞ ) ) )
58	55 57	imbi12d	⊢ ( 𝑗 = 1 → ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) ∈ ( 0 [,) +∞ ) ) ↔ ( ( 𝜑 ∧ 1 ∈ ℕ ) → ( 𝑃 ‘ 1 ) ∈ ( 0 [,) +∞ ) ) ) )
59		id	⊢ ( 𝑗 ∈ ℕ → 𝑗 ∈ ℕ )
60		ovexd	⊢ ( 𝑗 ∈ ℕ → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ∈ V )
61	21	fvmpt2	⊢ ( ( 𝑗 ∈ ℕ ∧ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ∈ V ) → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) )
62	59 60 61	syl2anc	⊢ ( 𝑗 ∈ ℕ → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) )
63	62	adantl	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) )
64	5	a1i	⊢ ( 𝜑 → 𝑊 = ( 𝑌 ∪ { 𝑍 } ) )
65	4	eldifad	⊢ ( 𝜑 → 𝑍 ∈ 𝑋 )
66		snssi	⊢ ( 𝑍 ∈ 𝑋 → { 𝑍 } ⊆ 𝑋 )
67	65 66	syl	⊢ ( 𝜑 → { 𝑍 } ⊆ 𝑋 )
68	3 67	unssd	⊢ ( 𝜑 → ( 𝑌 ∪ { 𝑍 } ) ⊆ 𝑋 )
69	64 68	eqsstrd	⊢ ( 𝜑 → 𝑊 ⊆ 𝑋 )
70	2 69	ssfid	⊢ ( 𝜑 → 𝑊 ∈ Fin )
71		ssun1	⊢ 𝑌 ⊆ ( 𝑌 ∪ { 𝑍 } )
72	5	eqcomi	⊢ ( 𝑌 ∪ { 𝑍 } ) = 𝑊
73	71 72	sseqtri	⊢ 𝑌 ⊆ 𝑊
74	73	a1i	⊢ ( 𝜑 → 𝑌 ⊆ 𝑊 )
75	70 74	ssfid	⊢ ( 𝜑 → 𝑌 ∈ Fin )
76	75	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑌 ∈ Fin )
77		iftrue	⊢ ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) )
78	77	adantl	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) )
79	10	ffvelrnda	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐶 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑊 ) )
80		elmapi	⊢ ( ( 𝐶 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑊 ) → ( 𝐶 ‘ 𝑗 ) : 𝑊 ⟶ ℝ )
81	79 80	syl	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐶 ‘ 𝑗 ) : 𝑊 ⟶ ℝ )
82	71 5	sseqtrri	⊢ 𝑌 ⊆ 𝑊
83	82	a1i	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑌 ⊆ 𝑊 )
84	81 83	fssresd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ )
85		reex	⊢ ℝ ∈ V
86	85	a1i	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ℝ ∈ V )
87	70 74	ssexd	⊢ ( 𝜑 → 𝑌 ∈ V )
88	87	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑌 ∈ V )
89	86 88	elmapd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) ↔ ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) )
90	84 89	mpbird	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) )
91	90	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) )
92	78 91	eqeltrd	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ ( ℝ ↑m 𝑌 ) )
93		iffalse	⊢ ( ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = 𝐹 )
94	93	adantl	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = 𝐹 )
95		0red	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑌 ) → 0 ∈ ℝ )
96	95 9	fmptd	⊢ ( 𝜑 → 𝐹 : 𝑌 ⟶ ℝ )
97	85	a1i	⊢ ( 𝜑 → ℝ ∈ V )
98	97 75	elmapd	⊢ ( 𝜑 → ( 𝐹 ∈ ( ℝ ↑m 𝑌 ) ↔ 𝐹 : 𝑌 ⟶ ℝ ) )
99	96 98	mpbird	⊢ ( 𝜑 → 𝐹 ∈ ( ℝ ↑m 𝑌 ) )
100	99	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → 𝐹 ∈ ( ℝ ↑m 𝑌 ) )
101	94 100	eqeltrd	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ ( ℝ ↑m 𝑌 ) )
102	92 101	pm2.61dan	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ ( ℝ ↑m 𝑌 ) )
103	102 11	fmptd	⊢ ( 𝜑 → 𝐽 : ℕ ⟶ ( ℝ ↑m 𝑌 ) )
104	103	ffvelrnda	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐽 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑌 ) )
105		elmapi	⊢ ( ( 𝐽 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑌 ) → ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ )
106	104 105	syl	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ )
107		iftrue	⊢ ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) )
108	107	adantl	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) )
109	12	ffvelrnda	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐷 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑊 ) )
110		elmapi	⊢ ( ( 𝐷 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑊 ) → ( 𝐷 ‘ 𝑗 ) : 𝑊 ⟶ ℝ )
111	109 110	syl	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐷 ‘ 𝑗 ) : 𝑊 ⟶ ℝ )
112	111 83	fssresd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ )
113	86 88	elmapd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) ↔ ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) )
114	112 113	mpbird	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) )
115	114	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) )
116	108 115	eqeltrd	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ ( ℝ ↑m 𝑌 ) )
117		iffalse	⊢ ( ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = 𝐹 )
118	117	adantl	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = 𝐹 )
119	118 100	eqeltrd	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ ( ℝ ↑m 𝑌 ) )
120	116 119	pm2.61dan	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ ( ℝ ↑m 𝑌 ) )
121	120 13	fmptd	⊢ ( 𝜑 → 𝐾 : ℕ ⟶ ( ℝ ↑m 𝑌 ) )
122	121	ffvelrnda	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐾 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑌 ) )
123		elmapi	⊢ ( ( 𝐾 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑌 ) → ( 𝐾 ‘ 𝑗 ) : 𝑌 ⟶ ℝ )
124	122 123	syl	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐾 ‘ 𝑗 ) : 𝑌 ⟶ ℝ )
125	1 76 106 124	hoidmvcl	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ∈ ( 0 [,) +∞ ) )
126	63 125	eqeltrd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) ∈ ( 0 [,) +∞ ) )
127	53 58 126	vtocl	⊢ ( ( 𝜑 ∧ 1 ∈ ℕ ) → ( 𝑃 ‘ 1 ) ∈ ( 0 [,) +∞ ) )
128	51 52 127	syl2anc	⊢ ( 𝜑 → ( 𝑃 ‘ 1 ) ∈ ( 0 [,) +∞ ) )
129	50 128	sselid	⊢ ( 𝜑 → ( 𝑃 ‘ 1 ) ∈ ℝ )
130	129	recnd	⊢ ( 𝜑 → ( 𝑃 ‘ 1 ) ∈ ℂ )
131	45 46 48 49 130	sumsnd	⊢ ( 𝜑 → Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) = ( 𝑃 ‘ 1 ) )
132	131	adantr	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) = ( 𝑃 ‘ 1 ) )
133		fveq2	⊢ ( 𝑗 = 1 → ( 𝐽 ‘ 𝑗 ) = ( 𝐽 ‘ 1 ) )
134		fveq2	⊢ ( 𝑗 = 1 → ( 𝐾 ‘ 𝑗 ) = ( 𝐾 ‘ 1 ) )
135	133 134	oveq12d	⊢ ( 𝑗 = 1 → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) = ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 1 ) ) )
136		ovex	⊢ ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 1 ) ) ∈ V
137	135 21 136	fvmpt	⊢ ( 1 ∈ ℕ → ( 𝑃 ‘ 1 ) = ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 1 ) ) )
138	25 137	ax-mp	⊢ ( 𝑃 ‘ 1 ) = ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 1 ) )
139	138	a1i	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( 𝑃 ‘ 1 ) = ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 1 ) ) )
140	30	oveqd	⊢ ( 𝑌 = ∅ → ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 1 ) ) = ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ ∅ ) ( 𝐾 ‘ 1 ) ) )
141	140	adantl	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 1 ) ) = ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ ∅ ) ( 𝐾 ‘ 1 ) ) )
142	133	feq1d	⊢ ( 𝑗 = 1 → ( ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ↔ ( 𝐽 ‘ 1 ) : 𝑌 ⟶ ℝ ) )
143	55 142	imbi12d	⊢ ( 𝑗 = 1 → ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ) ↔ ( ( 𝜑 ∧ 1 ∈ ℕ ) → ( 𝐽 ‘ 1 ) : 𝑌 ⟶ ℝ ) ) )
144	84	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ )
145	78	feq1d	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ ↔ ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) )
146	144 145	mpbird	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ )
147	96	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → 𝐹 : 𝑌 ⟶ ℝ )
148	94	feq1d	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ ↔ 𝐹 : 𝑌 ⟶ ℝ ) )
149	147 148	mpbird	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ )
150	146 149	pm2.61dan	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ )
151		simpr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑗 ∈ ℕ )
152		fvex	⊢ ( 𝐶 ‘ 𝑗 ) ∈ V
153	152	resex	⊢ ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ∈ V
154	78 153	eqeltrdi	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V )
155	99	elexd	⊢ ( 𝜑 → 𝐹 ∈ V )
156	155	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝐹 ∈ V )
157	156	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → 𝐹 ∈ V )
158	94 157	eqeltrd	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V )
159	154 158	pm2.61dan	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V )
160	11	fvmpt2	⊢ ( ( 𝑗 ∈ ℕ ∧ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V ) → ( 𝐽 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) )
161	151 159 160	syl2anc	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐽 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) )
162	161	feq1d	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ↔ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ ) )
163	150 162	mpbird	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ )
164	53 143 163	vtocl	⊢ ( ( 𝜑 ∧ 1 ∈ ℕ ) → ( 𝐽 ‘ 1 ) : 𝑌 ⟶ ℝ )
165	51 52 164	syl2anc	⊢ ( 𝜑 → ( 𝐽 ‘ 1 ) : 𝑌 ⟶ ℝ )
166	165	adantr	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( 𝐽 ‘ 1 ) : 𝑌 ⟶ ℝ )
167		id	⊢ ( 𝑌 = ∅ → 𝑌 = ∅ )
168	167	eqcomd	⊢ ( 𝑌 = ∅ → ∅ = 𝑌 )
169	168	feq2d	⊢ ( 𝑌 = ∅ → ( ( 𝐽 ‘ 1 ) : ∅ ⟶ ℝ ↔ ( 𝐽 ‘ 1 ) : 𝑌 ⟶ ℝ ) )
170	169	adantl	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( ( 𝐽 ‘ 1 ) : ∅ ⟶ ℝ ↔ ( 𝐽 ‘ 1 ) : 𝑌 ⟶ ℝ ) )
171	166 170	mpbird	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( 𝐽 ‘ 1 ) : ∅ ⟶ ℝ )
172	134	feq1d	⊢ ( 𝑗 = 1 → ( ( 𝐾 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ↔ ( 𝐾 ‘ 1 ) : 𝑌 ⟶ ℝ ) )
173	55 172	imbi12d	⊢ ( 𝑗 = 1 → ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐾 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ) ↔ ( ( 𝜑 ∧ 1 ∈ ℕ ) → ( 𝐾 ‘ 1 ) : 𝑌 ⟶ ℝ ) ) )
174	53 173 124	vtocl	⊢ ( ( 𝜑 ∧ 1 ∈ ℕ ) → ( 𝐾 ‘ 1 ) : 𝑌 ⟶ ℝ )
175	51 52 174	syl2anc	⊢ ( 𝜑 → ( 𝐾 ‘ 1 ) : 𝑌 ⟶ ℝ )
176	175	adantr	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( 𝐾 ‘ 1 ) : 𝑌 ⟶ ℝ )
177	168	feq2d	⊢ ( 𝑌 = ∅ → ( ( 𝐾 ‘ 1 ) : ∅ ⟶ ℝ ↔ ( 𝐾 ‘ 1 ) : 𝑌 ⟶ ℝ ) )
178	177	adantl	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( ( 𝐾 ‘ 1 ) : ∅ ⟶ ℝ ↔ ( 𝐾 ‘ 1 ) : 𝑌 ⟶ ℝ ) )
179	176 178	mpbird	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( 𝐾 ‘ 1 ) : ∅ ⟶ ℝ )
180	1 171 179	hoidmv0val	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ ∅ ) ( 𝐾 ‘ 1 ) ) = 0 )
181	141 180	eqtrd	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 1 ) ) = 0 )
182	132 139 181	3eqtrd	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) = 0 )
183	182	oveq2d	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) ) = ( ( 1 + 𝐸 ) · 0 ) )
184	17	rpred	⊢ ( 𝜑 → 𝐸 ∈ ℝ )
185	49 184	readdcld	⊢ ( 𝜑 → ( 1 + 𝐸 ) ∈ ℝ )
186	185	recnd	⊢ ( 𝜑 → ( 1 + 𝐸 ) ∈ ℂ )
187	186	mul01d	⊢ ( 𝜑 → ( ( 1 + 𝐸 ) · 0 ) = 0 )
188	187	adantr	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( ( 1 + 𝐸 ) · 0 ) = 0 )
189		eqidd	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → 0 = 0 )
190	183 188 189	3eqtrd	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) ) = 0 )
191	44 190	breq12d	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) ) ↔ 0 ≤ 0 ) )
192	28 191	mpbird	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) ) )
193		oveq2	⊢ ( 𝑚 = 1 → ( 1 ... 𝑚 ) = ( 1 ... 1 ) )
194	25	nnzi	⊢ 1 ∈ ℤ
195		fzsn	⊢ ( 1 ∈ ℤ → ( 1 ... 1 ) = { 1 } )
196	194 195	ax-mp	⊢ ( 1 ... 1 ) = { 1 }
197	196	a1i	⊢ ( 𝑚 = 1 → ( 1 ... 1 ) = { 1 } )
198	193 197	eqtrd	⊢ ( 𝑚 = 1 → ( 1 ... 𝑚 ) = { 1 } )
199	198	sumeq1d	⊢ ( 𝑚 = 1 → Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) = Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) )
200	199	oveq2d	⊢ ( 𝑚 = 1 → ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) = ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) ) )
201	200	breq2d	⊢ ( 𝑚 = 1 → ( 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ↔ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) ) ) )
202	201	rspcev	⊢ ( ( 1 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) ) ) → ∃ 𝑚 ∈ ℕ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) )
203	26 192 202	syl2anc	⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ∃ 𝑚 ∈ ℕ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) )
204		simpl	⊢ ( ( 𝜑 ∧ ¬ 𝑌 = ∅ ) → 𝜑 )
205		neqne	⊢ ( ¬ 𝑌 = ∅ → 𝑌 ≠ ∅ )
206	205	adantl	⊢ ( ( 𝜑 ∧ ¬ 𝑌 = ∅ ) → 𝑌 ≠ ∅ )
207		nfv	⊢ Ⅎ 𝑗 ( 𝜑 ∧ 𝑌 ≠ ∅ )
208	194	a1i	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → 1 ∈ ℤ )
209		nnuz	⊢ ℕ = ( ℤ≥ ‘ 1 )
210	126	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) ∈ ( 0 [,) +∞ ) )
211	82	a1i	⊢ ( 𝜑 → 𝑌 ⊆ 𝑊 )
212	6 211	fssresd	⊢ ( 𝜑 → ( 𝐴 ↾ 𝑌 ) : 𝑌 ⟶ ℝ )
213	7 211	fssresd	⊢ ( 𝜑 → ( 𝐵 ↾ 𝑌 ) : 𝑌 ⟶ ℝ )
214	1 75 212 213	hoidmvcl	⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ∈ ( 0 [,) +∞ ) )
215	50 214	sselid	⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ∈ ℝ )
216	16 215	eqeltrid	⊢ ( 𝜑 → 𝐺 ∈ ℝ )
217		0red	⊢ ( 𝜑 → 0 ∈ ℝ )
218		1rp	⊢ 1 ∈ ℝ+
219	218	a1i	⊢ ( 𝜑 → 1 ∈ ℝ+ )
220	219 17	jca	⊢ ( 𝜑 → ( 1 ∈ ℝ+ ∧ 𝐸 ∈ ℝ+ ) )
221		rpaddcl	⊢ ( ( 1 ∈ ℝ+ ∧ 𝐸 ∈ ℝ+ ) → ( 1 + 𝐸 ) ∈ ℝ+ )
222	220 221	syl	⊢ ( 𝜑 → ( 1 + 𝐸 ) ∈ ℝ+ )
223		rpgt0	⊢ ( ( 1 + 𝐸 ) ∈ ℝ+ → 0 < ( 1 + 𝐸 ) )
224	222 223	syl	⊢ ( 𝜑 → 0 < ( 1 + 𝐸 ) )
225	217 224	gtned	⊢ ( 𝜑 → ( 1 + 𝐸 ) ≠ 0 )
226	216 185 225	redivcld	⊢ ( 𝜑 → ( 𝐺 / ( 1 + 𝐸 ) ) ∈ ℝ )
227	226	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 𝐺 / ( 1 + 𝐸 ) ) ∈ ℝ )
228	226	ltpnfd	⊢ ( 𝜑 → ( 𝐺 / ( 1 + 𝐸 ) ) < +∞ )
229	228	adantr	⊢ ( ( 𝜑 ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ( 𝐺 / ( 1 + 𝐸 ) ) < +∞ )
230		id	⊢ ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ )
231	230	eqcomd	⊢ ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ → +∞ = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) )
232	231	adantl	⊢ ( ( 𝜑 ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → +∞ = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) )
233	229 232	breqtrd	⊢ ( ( 𝜑 ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ( 𝐺 / ( 1 + 𝐸 ) ) < ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) )
234	233	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ( 𝐺 / ( 1 + 𝐸 ) ) < ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) )
235		simpl	⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ( 𝜑 ∧ 𝑌 ≠ ∅ ) )
236		simpr	⊢ ( ( 𝜑 ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ )
237		nnex	⊢ ℕ ∈ V
238	237	a1i	⊢ ( ( 𝜑 ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ℕ ∈ V )
239		icossicc	⊢ ( 0 [,) +∞ ) ⊆ ( 0 [,] +∞ )
240	239 126	sselid	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) ∈ ( 0 [,] +∞ ) )
241		eqid	⊢ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) = ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) )
242	240 241	fmptd	⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) : ℕ ⟶ ( 0 [,] +∞ ) )
243	242	adantr	⊢ ( ( 𝜑 ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) : ℕ ⟶ ( 0 [,] +∞ ) )
244	238 243	sge0repnf	⊢ ( ( 𝜑 ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ↔ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) )
245	236 244	mpbird	⊢ ( ( 𝜑 ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ )
246	245	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ )
247	227	adantr	⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) → ( 𝐺 / ( 1 + 𝐸 ) ) ∈ ℝ )
248	216	adantr	⊢ ( ( 𝜑 ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) → 𝐺 ∈ ℝ )
249	248	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) → 𝐺 ∈ ℝ )
250		simpr	⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ )
251	49 17	ltaddrpd	⊢ ( 𝜑 → 1 < ( 1 + 𝐸 ) )
252	251	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → 1 < ( 1 + 𝐸 ) )
253	75	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → 𝑌 ∈ Fin )
254		simpr	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → 𝑌 ≠ ∅ )
255	212	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 𝐴 ↾ 𝑌 ) : 𝑌 ⟶ ℝ )
256	213	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 𝐵 ↾ 𝑌 ) : 𝑌 ⟶ ℝ )
257	1 253 254 255 256	hoidmvn0val	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) = ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) [,) ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) ) ) )
258	16	a1i	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → 𝐺 = ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) )
259		fvres	⊢ ( 𝑘 ∈ 𝑌 → ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) = ( 𝐴 ‘ 𝑘 ) )
260		fvres	⊢ ( 𝑘 ∈ 𝑌 → ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) = ( 𝐵 ‘ 𝑘 ) )
261	259 260	oveq12d	⊢ ( 𝑘 ∈ 𝑌 → ( ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) [,) ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
262	261	fveq2d	⊢ ( 𝑘 ∈ 𝑌 → ( vol ‘ ( ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) [,) ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) ) ) = ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) )
263	262	adantl	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( vol ‘ ( ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) [,) ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) ) ) = ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) )
264	6	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → 𝐴 : 𝑊 ⟶ ℝ )
265		elun1	⊢ ( 𝑘 ∈ 𝑌 → 𝑘 ∈ ( 𝑌 ∪ { 𝑍 } ) )
266	265 5	eleqtrrdi	⊢ ( 𝑘 ∈ 𝑌 → 𝑘 ∈ 𝑊 )
267	266	adantl	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → 𝑘 ∈ 𝑊 )
268	264 267	ffvelrnd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( 𝐴 ‘ 𝑘 ) ∈ ℝ )
269	7	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → 𝐵 : 𝑊 ⟶ ℝ )
270	269 267	ffvelrnd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( 𝐵 ‘ 𝑘 ) ∈ ℝ )
271		volico	⊢ ( ( ( 𝐴 ‘ 𝑘 ) ∈ ℝ ∧ ( 𝐵 ‘ 𝑘 ) ∈ ℝ ) → ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = if ( ( 𝐴 ‘ 𝑘 ) < ( 𝐵 ‘ 𝑘 ) , ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) , 0 ) )
272	268 270 271	syl2anc	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = if ( ( 𝐴 ‘ 𝑘 ) < ( 𝐵 ‘ 𝑘 ) , ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) , 0 ) )
273	267 8	syldan	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( 𝐴 ‘ 𝑘 ) < ( 𝐵 ‘ 𝑘 ) )
274	273	iftrued	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → if ( ( 𝐴 ‘ 𝑘 ) < ( 𝐵 ‘ 𝑘 ) , ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) , 0 ) = ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) )
275	263 272 274	3eqtrd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( vol ‘ ( ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) [,) ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) ) ) = ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) )
276	275	prodeq2dv	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) [,) ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑌 ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) )
277	276	eqcomd	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝑌 ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) = ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) [,) ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) ) ) )
278	277	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ∏ 𝑘 ∈ 𝑌 ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) = ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) [,) ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) ) ) )
279	257 258 278	3eqtr4d	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → 𝐺 = ∏ 𝑘 ∈ 𝑌 ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) )
280		difrp	⊢ ( ( ( 𝐴 ‘ 𝑘 ) ∈ ℝ ∧ ( 𝐵 ‘ 𝑘 ) ∈ ℝ ) → ( ( 𝐴 ‘ 𝑘 ) < ( 𝐵 ‘ 𝑘 ) ↔ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ ℝ+ ) )
281	268 270 280	syl2anc	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐴 ‘ 𝑘 ) < ( 𝐵 ‘ 𝑘 ) ↔ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ ℝ+ ) )
282	273 281	mpbid	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ ℝ+ )
283	75 282	fprodrpcl	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝑌 ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ ℝ+ )
284	283	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ∏ 𝑘 ∈ 𝑌 ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ ℝ+ )
285	279 284	eqeltrd	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → 𝐺 ∈ ℝ+ )
286	222	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 1 + 𝐸 ) ∈ ℝ+ )
287	285 286	ltdivgt1	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 1 < ( 1 + 𝐸 ) ↔ ( 𝐺 / ( 1 + 𝐸 ) ) < 𝐺 ) )
288	252 287	mpbid	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 𝐺 / ( 1 + 𝐸 ) ) < 𝐺 )
289	288	adantr	⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) → ( 𝐺 / ( 1 + 𝐸 ) ) < 𝐺 )
290	23	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → X 𝑘 ∈ 𝑊 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
291		fvexd	⊢ ( 𝜑 → ( 𝑥 ‘ 𝑘 ) ∈ V )
292	19	elexd	⊢ ( 𝜑 → 𝑆 ∈ V )
293	291 292	ifcld	⊢ ( 𝜑 → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ V )
294	293	ralrimivw	⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝑊 if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ V )
295	294	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ∀ 𝑘 ∈ 𝑊 if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ V )
296		eqid	⊢ ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) = ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) )
297	296	fnmpt	⊢ ( ∀ 𝑘 ∈ 𝑊 if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ V → ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) Fn 𝑊 )
298	295 297	syl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) Fn 𝑊 )
299		simpr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
300		mptexg	⊢ ( 𝑊 ∈ Fin → ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ∈ V )
301	70 300	syl	⊢ ( 𝜑 → ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ∈ V )
302	301	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ∈ V )
303	24	fvmpt2	⊢ ( ( 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∧ ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ∈ V ) → ( 𝑂 ‘ 𝑥 ) = ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) )
304	299 302 303	syl2anc	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( 𝑂 ‘ 𝑥 ) = ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) )
305	304	fneq1d	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( ( 𝑂 ‘ 𝑥 ) Fn 𝑊 ↔ ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) Fn 𝑊 ) )
306	298 305	mpbird	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( 𝑂 ‘ 𝑥 ) Fn 𝑊 )
307		nfv	⊢ Ⅎ 𝑘 𝜑
308		nfcv	⊢ Ⅎ 𝑘 𝑥
309		nfixp1	⊢ Ⅎ 𝑘 X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) )
310	308 309	nfel	⊢ Ⅎ 𝑘 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) )
311	307 310	nfan	⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
312	304	fveq1d	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) = ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑘 ) )
313	312	adantr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) = ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑘 ) )
314		simpr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → 𝑘 ∈ 𝑊 )
315	293	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ V )
316	296	fvmpt2	⊢ ( ( 𝑘 ∈ 𝑊 ∧ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ V ) → ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑘 ) = if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) )
317	314 315 316	syl2anc	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑘 ) = if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) )
318	317	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) → ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑘 ) = if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) )
319	313 318	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) = if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) )
320		iftrue	⊢ ( 𝑘 ∈ 𝑌 → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) = ( 𝑥 ‘ 𝑘 ) )
321	320	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) ∧ 𝑘 ∈ 𝑌 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) = ( 𝑥 ‘ 𝑘 ) )
322		vex	⊢ 𝑥 ∈ V
323	322	elixp	⊢ ( 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ↔ ( 𝑥 Fn 𝑌 ∧ ∀ 𝑘 ∈ 𝑌 ( 𝑥 ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) )
324	323	simprbi	⊢ ( 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) → ∀ 𝑘 ∈ 𝑌 ( 𝑥 ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
325	324	adantr	⊢ ( ( 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ∀ 𝑘 ∈ 𝑌 ( 𝑥 ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
326		simpr	⊢ ( ( 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → 𝑘 ∈ 𝑌 )
327		rspa	⊢ ( ( ∀ 𝑘 ∈ 𝑌 ( 𝑥 ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑥 ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
328	325 326 327	syl2anc	⊢ ( ( 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑥 ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
329	328	ad4ant24	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑥 ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
330	321 329	eqeltrd	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) ∧ 𝑘 ∈ 𝑌 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
331		snidg	⊢ ( 𝑍 ∈ ( 𝑋 ∖ 𝑌 ) → 𝑍 ∈ { 𝑍 } )
332	4 331	syl	⊢ ( 𝜑 → 𝑍 ∈ { 𝑍 } )
333		elun2	⊢ ( 𝑍 ∈ { 𝑍 } → 𝑍 ∈ ( 𝑌 ∪ { 𝑍 } ) )
334	332 333	syl	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑌 ∪ { 𝑍 } ) )
335	72	a1i	⊢ ( 𝜑 → ( 𝑌 ∪ { 𝑍 } ) = 𝑊 )
336	334 335	eleqtrd	⊢ ( 𝜑 → 𝑍 ∈ 𝑊 )
337	6 336	ffvelrnd	⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) ∈ ℝ )
338	337	rexrd	⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) ∈ ℝ* )
339	7 336	ffvelrnd	⊢ ( 𝜑 → ( 𝐵 ‘ 𝑍 ) ∈ ℝ )
340	339	rexrd	⊢ ( 𝜑 → ( 𝐵 ‘ 𝑍 ) ∈ ℝ* )
341		iccssxr	⊢ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ⊆ ℝ*
342		ssrab2	⊢ { 𝑧 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) } ⊆ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) )
343	18 342	eqsstri	⊢ 𝑈 ⊆ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) )
344	343 19	sselid	⊢ ( 𝜑 → 𝑆 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) )
345	341 344	sselid	⊢ ( 𝜑 → 𝑆 ∈ ℝ* )
346		iccgelb	⊢ ( ( ( 𝐴 ‘ 𝑍 ) ∈ ℝ* ∧ ( 𝐵 ‘ 𝑍 ) ∈ ℝ* ∧ 𝑆 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ) → ( 𝐴 ‘ 𝑍 ) ≤ 𝑆 )
347	338 340 344 346	syl3anc	⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) ≤ 𝑆 )
348	338 340 345 347 20	elicod	⊢ ( 𝜑 → 𝑆 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) )
349	348	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ¬ 𝑘 ∈ 𝑌 ) → 𝑆 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) )
350		iffalse	⊢ ( ¬ 𝑘 ∈ 𝑌 → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) = 𝑆 )
351	350	adantl	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ¬ 𝑘 ∈ 𝑌 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) = 𝑆 )
352	5	eleq2i	⊢ ( 𝑘 ∈ 𝑊 ↔ 𝑘 ∈ ( 𝑌 ∪ { 𝑍 } ) )
353	352	biimpi	⊢ ( 𝑘 ∈ 𝑊 → 𝑘 ∈ ( 𝑌 ∪ { 𝑍 } ) )
354	353	adantr	⊢ ( ( 𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌 ) → 𝑘 ∈ ( 𝑌 ∪ { 𝑍 } ) )
355		simpr	⊢ ( ( 𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌 ) → ¬ 𝑘 ∈ 𝑌 )
356		elunnel1	⊢ ( ( 𝑘 ∈ ( 𝑌 ∪ { 𝑍 } ) ∧ ¬ 𝑘 ∈ 𝑌 ) → 𝑘 ∈ { 𝑍 } )
357	354 355 356	syl2anc	⊢ ( ( 𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌 ) → 𝑘 ∈ { 𝑍 } )
358		elsni	⊢ ( 𝑘 ∈ { 𝑍 } → 𝑘 = 𝑍 )
359	357 358	syl	⊢ ( ( 𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌 ) → 𝑘 = 𝑍 )
360		fveq2	⊢ ( 𝑘 = 𝑍 → ( 𝐴 ‘ 𝑘 ) = ( 𝐴 ‘ 𝑍 ) )
361		fveq2	⊢ ( 𝑘 = 𝑍 → ( 𝐵 ‘ 𝑘 ) = ( 𝐵 ‘ 𝑍 ) )
362	360 361	oveq12d	⊢ ( 𝑘 = 𝑍 → ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) )
363	359 362	syl	⊢ ( ( 𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌 ) → ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) )
364	363	adantll	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ¬ 𝑘 ∈ 𝑌 ) → ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) )
365	351 364	eleq12d	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ¬ 𝑘 ∈ 𝑌 ) → ( if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ↔ 𝑆 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) )
366	349 365	mpbird	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ¬ 𝑘 ∈ 𝑌 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
367	366	adantllr	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) ∧ ¬ 𝑘 ∈ 𝑌 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
368	330 367	pm2.61dan	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
369	319 368	eqeltrd	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
370	369	ex	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( 𝑘 ∈ 𝑊 → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) )
371	311 370	ralrimi	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
372	306 371	jca	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( ( 𝑂 ‘ 𝑥 ) Fn 𝑊 ∧ ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) )
373		fvex	⊢ ( 𝑂 ‘ 𝑥 ) ∈ V
374	373	elixp	⊢ ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ↔ ( ( 𝑂 ‘ 𝑥 ) Fn 𝑊 ∧ ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) )
375	372 374	sylibr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
376	290 375	sseldd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( 𝑂 ‘ 𝑥 ) ∈ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
377		eliun	⊢ ( ( 𝑂 ‘ 𝑥 ) ∈ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ ∃ 𝑗 ∈ ℕ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
378	376 377	sylib	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ∃ 𝑗 ∈ ℕ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
379		ixpfn	⊢ ( 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) → 𝑥 Fn 𝑌 )
380	379	adantl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → 𝑥 Fn 𝑌 )
381	380	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → 𝑥 Fn 𝑌 )
382		nfv	⊢ Ⅎ 𝑘 𝑗 ∈ ℕ
383	311 382	nfan	⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ )
384		nfcv	⊢ Ⅎ 𝑘 ( 𝑂 ‘ 𝑥 )
385		nfixp1	⊢ Ⅎ 𝑘 X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) )
386	384 385	nfel	⊢ Ⅎ 𝑘 ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) )
387	383 386	nfan	⊢ Ⅎ 𝑘 ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
388	312	3adant3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) = ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑘 ) )
389	293	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ V )
390	267 389 316	syl2anc	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑘 ) = if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) )
391	390	3adant2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑘 ) = if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) )
392	320	3ad2ant3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) = ( 𝑥 ‘ 𝑘 ) )
393	388 391 392	3eqtrrd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑥 ‘ 𝑘 ) = ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) )
394	393	ad5ant125	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑥 ‘ 𝑘 ) = ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) )
395		simpl	⊢ ( ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
396	373	elixp	⊢ ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ ( ( 𝑂 ‘ 𝑥 ) Fn 𝑊 ∧ ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
397	395 396	sylib	⊢ ( ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝑂 ‘ 𝑥 ) Fn 𝑊 ∧ ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
398	397	simprd	⊢ ( ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
399	266	adantl	⊢ ( ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → 𝑘 ∈ 𝑊 )
400		rspa	⊢ ( ( ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑊 ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
401	398 399 400	syl2anc	⊢ ( ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
402	401	adantll	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
403	394 402	eqeltrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑥 ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
404	51	ad3antrrr	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → 𝜑 )
405	59	ad2antlr	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → 𝑗 ∈ ℕ )
406	304	fveq1d	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑍 ) = ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑍 ) )
407		eqidd	⊢ ( 𝜑 → ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) = ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) )
408		eleq1	⊢ ( 𝑘 = 𝑍 → ( 𝑘 ∈ 𝑌 ↔ 𝑍 ∈ 𝑌 ) )
409		fveq2	⊢ ( 𝑘 = 𝑍 → ( 𝑥 ‘ 𝑘 ) = ( 𝑥 ‘ 𝑍 ) )
410	408 409	ifbieq1d	⊢ ( 𝑘 = 𝑍 → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) = if ( 𝑍 ∈ 𝑌 , ( 𝑥 ‘ 𝑍 ) , 𝑆 ) )
411	410	adantl	⊢ ( ( 𝜑 ∧ 𝑘 = 𝑍 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) = if ( 𝑍 ∈ 𝑌 , ( 𝑥 ‘ 𝑍 ) , 𝑆 ) )
412		fvexd	⊢ ( 𝜑 → ( 𝑥 ‘ 𝑍 ) ∈ V )
413	412 292	ifcld	⊢ ( 𝜑 → if ( 𝑍 ∈ 𝑌 , ( 𝑥 ‘ 𝑍 ) , 𝑆 ) ∈ V )
414	407 411 336 413	fvmptd	⊢ ( 𝜑 → ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑍 ) = if ( 𝑍 ∈ 𝑌 , ( 𝑥 ‘ 𝑍 ) , 𝑆 ) )
415	414	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑍 ) = if ( 𝑍 ∈ 𝑌 , ( 𝑥 ‘ 𝑍 ) , 𝑆 ) )
416	4	eldifbd	⊢ ( 𝜑 → ¬ 𝑍 ∈ 𝑌 )
417	416	iffalsed	⊢ ( 𝜑 → if ( 𝑍 ∈ 𝑌 , ( 𝑥 ‘ 𝑍 ) , 𝑆 ) = 𝑆 )
418	417	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → if ( 𝑍 ∈ 𝑌 , ( 𝑥 ‘ 𝑍 ) , 𝑆 ) = 𝑆 )
419	406 415 418	3eqtrrd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → 𝑆 = ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑍 ) )
420	419	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → 𝑆 = ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑍 ) )
421	404 336	syl	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → 𝑍 ∈ 𝑊 )
422	396	simprbi	⊢ ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) → ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
423	422	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
424		fveq2	⊢ ( 𝑘 = 𝑍 → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) = ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑍 ) )
425		fveq2	⊢ ( 𝑘 = 𝑍 → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) )
426		fveq2	⊢ ( 𝑘 = 𝑍 → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) )
427	425 426	oveq12d	⊢ ( 𝑘 = 𝑍 → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) )
428	424 427	eleq12d	⊢ ( 𝑘 = 𝑍 → ( ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑍 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) )
429	428	rspcva	⊢ ( ( 𝑍 ∈ 𝑊 ∧ ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑍 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) )
430	421 423 429	syl2anc	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑍 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) )
431	420 430	eqeltrd	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) )
432	161	3adant3	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝐽 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) )
433	77	3ad2ant3	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) )
434	432 433	eqtrd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝐽 ‘ 𝑗 ) = ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) )
435	434	fveq1d	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) )
436	404 405 431 435	syl3anc	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) )
437	436	adantr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) )
438		fvres	⊢ ( 𝑘 ∈ 𝑌 → ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) )
439	438	adantl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) )
440	437 439	eqtrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) )
441	120	elexd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V )
442	13	fvmpt2	⊢ ( ( 𝑗 ∈ ℕ ∧ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V ) → ( 𝐾 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) )
443	151 441 442	syl2anc	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐾 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) )
444	443	3adant3	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝐾 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) )
445	107	3ad2ant3	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) )
446	444 445	eqtrd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝐾 ‘ 𝑗 ) = ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) )
447	446	fveq1d	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) )
448	404 405 431 447	syl3anc	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) )
449	448	adantr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) )
450		fvres	⊢ ( 𝑘 ∈ 𝑌 → ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) )
451	450	adantl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) )
452	449 451	eqtrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) )
453	440 452	oveq12d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
454	453	eqcomd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) )
455	403 454	eleqtrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑥 ‘ 𝑘 ) ∈ ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) )
456	455	ex	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → ( 𝑘 ∈ 𝑌 → ( 𝑥 ‘ 𝑘 ) ∈ ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
457	387 456	ralrimi	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → ∀ 𝑘 ∈ 𝑌 ( 𝑥 ‘ 𝑘 ) ∈ ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) )
458	381 457	jca	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → ( 𝑥 Fn 𝑌 ∧ ∀ 𝑘 ∈ 𝑌 ( 𝑥 ‘ 𝑘 ) ∈ ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
459	322	elixp	⊢ ( 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ ( 𝑥 Fn 𝑌 ∧ ∀ 𝑘 ∈ 𝑌 ( 𝑥 ‘ 𝑘 ) ∈ ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
460	458 459	sylibr	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) )
461	460	ex	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) → 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
462	461	reximdva	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( ∃ 𝑗 ∈ ℕ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) → ∃ 𝑗 ∈ ℕ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
463	378 462	mpd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ∃ 𝑗 ∈ ℕ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) )
464		eliun	⊢ ( 𝑥 ∈ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ ∃ 𝑗 ∈ ℕ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) )
465	463 464	sylibr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → 𝑥 ∈ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) )
466	465	ralrimiva	⊢ ( 𝜑 → ∀ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) 𝑥 ∈ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) )
467		dfss3	⊢ ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ ∀ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) 𝑥 ∈ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) )
468	466 467	sylibr	⊢ ( 𝜑 → X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) )
469		ovexd	⊢ ( 𝜑 → ( ℝ ↑m 𝑌 ) ∈ V )
470	237	a1i	⊢ ( 𝜑 → ℕ ∈ V )
471	469 470	elmapd	⊢ ( 𝜑 → ( 𝐾 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ↔ 𝐾 : ℕ ⟶ ( ℝ ↑m 𝑌 ) ) )
472	121 471	mpbird	⊢ ( 𝜑 → 𝐾 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) )
473	469 470	elmapd	⊢ ( 𝜑 → ( 𝐽 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ↔ 𝐽 : ℕ ⟶ ( ℝ ↑m 𝑌 ) ) )
474	103 473	mpbird	⊢ ( 𝜑 → 𝐽 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) )
475	97 87	elmapd	⊢ ( 𝜑 → ( ( 𝐵 ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) ↔ ( 𝐵 ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) )
476	213 475	mpbird	⊢ ( 𝜑 → ( 𝐵 ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) )
477	97 87	elmapd	⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) ↔ ( 𝐴 ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) )
478	212 477	mpbird	⊢ ( 𝜑 → ( 𝐴 ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) )
479		fveq1	⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → ( 𝑒 ‘ 𝑘 ) = ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) )
480	479	adantr	⊢ ( ( 𝑒 = ( 𝐴 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑒 ‘ 𝑘 ) = ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) )
481	259	adantl	⊢ ( ( 𝑒 = ( 𝐴 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) = ( 𝐴 ‘ 𝑘 ) )
482	480 481	eqtrd	⊢ ( ( 𝑒 = ( 𝐴 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑒 ‘ 𝑘 ) = ( 𝐴 ‘ 𝑘 ) )
483	482	oveq1d	⊢ ( ( 𝑒 = ( 𝐴 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) )
484	483	ixpeq2dva	⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → X 𝑘 ∈ 𝑌 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) )
485	484	sseq1d	⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → ( X 𝑘 ∈ 𝑌 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
486		oveq1	⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → ( 𝑒 ( 𝐿 ‘ 𝑌 ) 𝑓 ) = ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) )
487	486	breq1d	⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → ( ( 𝑒 ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ↔ ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) )
488	485 487	imbi12d	⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → ( ( X 𝑘 ∈ 𝑌 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) )
489	488	ralbidv	⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → ( ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) )
490	489	ralbidv	⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → ( ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) )
491	490	ralbidv	⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → ( ∀ 𝑓 ∈ ( ℝ ↑m 𝑌 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑓 ∈ ( ℝ ↑m 𝑌 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) )
492	491	rspcva	⊢ ( ( ( 𝐴 ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑌 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑌 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) → ∀ 𝑓 ∈ ( ℝ ↑m 𝑌 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) )
493	478 22 492	syl2anc	⊢ ( 𝜑 → ∀ 𝑓 ∈ ( ℝ ↑m 𝑌 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) )
494		fveq1	⊢ ( 𝑓 = ( 𝐵 ↾ 𝑌 ) → ( 𝑓 ‘ 𝑘 ) = ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) )
495	494	adantr	⊢ ( ( 𝑓 = ( 𝐵 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑓 ‘ 𝑘 ) = ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) )
496	260	adantl	⊢ ( ( 𝑓 = ( 𝐵 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) = ( 𝐵 ‘ 𝑘 ) )
497	495 496	eqtrd	⊢ ( ( 𝑓 = ( 𝐵 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑓 ‘ 𝑘 ) = ( 𝐵 ‘ 𝑘 ) )
498	497	oveq2d	⊢ ( ( 𝑓 = ( 𝐵 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
499	498	ixpeq2dva	⊢ ( 𝑓 = ( 𝐵 ↾ 𝑌 ) → X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
500	499	sseq1d	⊢ ( 𝑓 = ( 𝐵 ↾ 𝑌 ) → ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
501		oveq2	⊢ ( 𝑓 = ( 𝐵 ↾ 𝑌 ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) = ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) )
502	501	breq1d	⊢ ( 𝑓 = ( 𝐵 ↾ 𝑌 ) → ( ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ↔ ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) )
503	500 502	imbi12d	⊢ ( 𝑓 = ( 𝐵 ↾ 𝑌 ) → ( ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) )
504	503	ralbidv	⊢ ( 𝑓 = ( 𝐵 ↾ 𝑌 ) → ( ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) )
505	504	ralbidv	⊢ ( 𝑓 = ( 𝐵 ↾ 𝑌 ) → ( ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) )
506	505	rspcva	⊢ ( ( ( 𝐵 ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) ∧ ∀ 𝑓 ∈ ( ℝ ↑m 𝑌 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) → ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) )
507	476 493 506	syl2anc	⊢ ( 𝜑 → ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) )
508		fveq1	⊢ ( 𝑔 = 𝐽 → ( 𝑔 ‘ 𝑗 ) = ( 𝐽 ‘ 𝑗 ) )
509	508	fveq1d	⊢ ( 𝑔 = 𝐽 → ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) )
510	509	oveq1d	⊢ ( 𝑔 = 𝐽 → ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) )
511	510	ixpeq2dv	⊢ ( 𝑔 = 𝐽 → X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) )
512	511	iuneq2d	⊢ ( 𝑔 = 𝐽 → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) )
513	512	sseq2d	⊢ ( 𝑔 = 𝐽 → ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
514	508	oveq1d	⊢ ( 𝑔 = 𝐽 → ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) )
515	514	mpteq2dv	⊢ ( 𝑔 = 𝐽 → ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) )
516	515	fveq2d	⊢ ( 𝑔 = 𝐽 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) )
517	516	breq2d	⊢ ( 𝑔 = 𝐽 → ( ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ↔ ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) )
518	513 517	imbi12d	⊢ ( 𝑔 = 𝐽 → ( ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) )
519	518	ralbidv	⊢ ( 𝑔 = 𝐽 → ( ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) )
520	519	rspcva	⊢ ( ( 𝐽 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∧ ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) → ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) )
521	474 507 520	syl2anc	⊢ ( 𝜑 → ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) )
522		fveq1	⊢ ( ℎ = 𝐾 → ( ℎ ‘ 𝑗 ) = ( 𝐾 ‘ 𝑗 ) )
523	522	fveq1d	⊢ ( ℎ = 𝐾 → ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) )
524	523	oveq2d	⊢ ( ℎ = 𝐾 → ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) )
525	524	ixpeq2dv	⊢ ( ℎ = 𝐾 → X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) )
526	525	iuneq2d	⊢ ( ℎ = 𝐾 → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) )
527	526	sseq2d	⊢ ( ℎ = 𝐾 → ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
528	522	oveq2d	⊢ ( ℎ = 𝐾 → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) )
529	528	mpteq2dv	⊢ ( ℎ = 𝐾 → ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) )
530	529	fveq2d	⊢ ( ℎ = 𝐾 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) )
531	530	breq2d	⊢ ( ℎ = 𝐾 → ( ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ↔ ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) ) )
532	527 531	imbi12d	⊢ ( ℎ = 𝐾 → ( ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) ) ) )
533	532	rspcva	⊢ ( ( 𝐾 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∧ ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) → ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) ) )
534	472 521 533	syl2anc	⊢ ( 𝜑 → ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) ) )
535	468 534	mpd	⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) )
536		idd	⊢ ( 𝜑 → ( ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) ) )
537	535 536	mpd	⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) )
538	537	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) )
539	62	adantl	⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) )
540	539	mpteq2dva	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) )
541	540	fveq2d	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) )
542	258 541	breq12d	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 𝐺 ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ↔ ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) ) )
543	538 542	mpbird	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → 𝐺 ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) )
544	543	adantr	⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) → 𝐺 ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) )
545	247 249 250 289 544	ltletrd	⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) → ( 𝐺 / ( 1 + 𝐸 ) ) < ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) )
546	235 246 545	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ( 𝐺 / ( 1 + 𝐸 ) ) < ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) )
547	234 546	pm2.61dan	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 𝐺 / ( 1 + 𝐸 ) ) < ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) )
548	207 208 209 210 227 547	sge0uzfsumgt	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ∃ 𝑚 ∈ ℕ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) )
549	226	adantr	⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ( 𝐺 / ( 1 + 𝐸 ) ) ∈ ℝ )
550		fzfid	⊢ ( 𝜑 → ( 1 ... 𝑚 ) ∈ Fin )
551		simpl	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑚 ) ) → 𝜑 )
552		elfznn	⊢ ( 𝑗 ∈ ( 1 ... 𝑚 ) → 𝑗 ∈ ℕ )
553	552	adantl	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑚 ) ) → 𝑗 ∈ ℕ )
554	50 126	sselid	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) ∈ ℝ )
555	551 553 554	syl2anc	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑚 ) ) → ( 𝑃 ‘ 𝑗 ) ∈ ℝ )
556	550 555	fsumrecl	⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ∈ ℝ )
557	556	adantr	⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ∈ ℝ )
558		simpr	⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) )
559	549 557 558	ltled	⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ( 𝐺 / ( 1 + 𝐸 ) ) ≤ Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) )
560	216	adantr	⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → 𝐺 ∈ ℝ )
561	222	adantr	⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ( 1 + 𝐸 ) ∈ ℝ+ )
562	560 557 561	ledivmuld	⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ( ( 𝐺 / ( 1 + 𝐸 ) ) ≤ Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ↔ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) )
563	559 562	mpbid	⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) )
564	563	ex	⊢ ( 𝜑 → ( ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) )
565	564	adantr	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) )
566	565	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) )
567	566	reximdva	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( ∃ 𝑚 ∈ ℕ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) → ∃ 𝑚 ∈ ℕ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) )
568	548 567	mpd	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ∃ 𝑚 ∈ ℕ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) )
569	204 206 568	syl2anc	⊢ ( ( 𝜑 ∧ ¬ 𝑌 = ∅ ) → ∃ 𝑚 ∈ ℕ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) )
570	203 569	pm2.61dan	⊢ ( 𝜑 → ∃ 𝑚 ∈ ℕ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) )
571	2	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝑋 ∈ Fin )
572	3	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝑌 ⊆ 𝑋 )
573	4	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝑍 ∈ ( 𝑋 ∖ 𝑌 ) )
574	6	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝐴 : 𝑊 ⟶ ℝ )
575	7	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝐵 : 𝑊 ⟶ ℝ )
576	10	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝐶 : ℕ ⟶ ( ℝ ↑m 𝑊 ) )
577		eqid	⊢ ( 𝑦 ∈ 𝑌 ↦ 0 ) = ( 𝑦 ∈ 𝑌 ↦ 0 )
578		eqid	⊢ ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) = ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) )
579	12	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝐷 : ℕ ⟶ ( ℝ ↑m 𝑊 ) )
580		eqid	⊢ ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) = ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) )
581		fveq2	⊢ ( 𝑖 = 𝑗 → ( 𝐶 ‘ 𝑖 ) = ( 𝐶 ‘ 𝑗 ) )
582		fveq2	⊢ ( 𝑖 = 𝑗 → ( 𝐷 ‘ 𝑖 ) = ( 𝐷 ‘ 𝑗 ) )
583	581 582	oveq12d	⊢ ( 𝑖 = 𝑗 → ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑖 ) ) = ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑗 ) ) )
584	583	cbvmptv	⊢ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑖 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑗 ) ) )
585	584	fveq2i	⊢ ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑖 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑗 ) ) ) )
586	585 14	eqeltrid	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑖 ) ) ) ) ∈ ℝ )
587	586	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑖 ) ) ) ) ∈ ℝ )
588		eleq1w	⊢ ( 𝑗 = 𝑖 → ( 𝑗 ∈ 𝑌 ↔ 𝑖 ∈ 𝑌 ) )
589		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝑐 ‘ 𝑗 ) = ( 𝑐 ‘ 𝑖 ) )
590	589	breq1d	⊢ ( 𝑗 = 𝑖 → ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 ↔ ( 𝑐 ‘ 𝑖 ) ≤ 𝑥 ) )
591	590 589	ifbieq1d	⊢ ( 𝑗 = 𝑖 → if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) = if ( ( 𝑐 ‘ 𝑖 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑖 ) , 𝑥 ) )
592	588 589 591	ifbieq12d	⊢ ( 𝑗 = 𝑖 → if ( 𝑗 ∈ 𝑌 , ( 𝑐 ‘ 𝑗 ) , if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) ) = if ( 𝑖 ∈ 𝑌 , ( 𝑐 ‘ 𝑖 ) , if ( ( 𝑐 ‘ 𝑖 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑖 ) , 𝑥 ) ) )
593	592	cbvmptv	⊢ ( 𝑗 ∈ 𝑊 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑐 ‘ 𝑗 ) , if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) ) ) = ( 𝑖 ∈ 𝑊 ↦ if ( 𝑖 ∈ 𝑌 , ( 𝑐 ‘ 𝑖 ) , if ( ( 𝑐 ‘ 𝑖 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑖 ) , 𝑥 ) ) )
594	593	mpteq2i	⊢ ( 𝑐 ∈ ( ℝ ↑m 𝑊 ) ↦ ( 𝑗 ∈ 𝑊 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑐 ‘ 𝑗 ) , if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) ) ) ) = ( 𝑐 ∈ ( ℝ ↑m 𝑊 ) ↦ ( 𝑖 ∈ 𝑊 ↦ if ( 𝑖 ∈ 𝑌 , ( 𝑐 ‘ 𝑖 ) , if ( ( 𝑐 ‘ 𝑖 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑖 ) , 𝑥 ) ) ) )
595	594	mpteq2i	⊢ ( 𝑥 ∈ ℝ ↦ ( 𝑐 ∈ ( ℝ ↑m 𝑊 ) ↦ ( 𝑗 ∈ 𝑊 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑐 ‘ 𝑗 ) , if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) ) ) ) ) = ( 𝑥 ∈ ℝ ↦ ( 𝑐 ∈ ( ℝ ↑m 𝑊 ) ↦ ( 𝑖 ∈ 𝑊 ↦ if ( 𝑖 ∈ 𝑌 , ( 𝑐 ‘ 𝑖 ) , if ( ( 𝑐 ‘ 𝑖 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑖 ) , 𝑥 ) ) ) ) )
596	15 595	eqtri	⊢ 𝐻 = ( 𝑥 ∈ ℝ ↦ ( 𝑐 ∈ ( ℝ ↑m 𝑊 ) ↦ ( 𝑖 ∈ 𝑊 ↦ if ( 𝑖 ∈ 𝑌 , ( 𝑐 ‘ 𝑖 ) , if ( ( 𝑐 ‘ 𝑖 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑖 ) , 𝑥 ) ) ) ) )
597	17	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝐸 ∈ ℝ+ )
598		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝐶 ‘ 𝑗 ) = ( 𝐶 ‘ 𝑖 ) )
599		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝐷 ‘ 𝑗 ) = ( 𝐷 ‘ 𝑖 ) )
600	599	fveq2d	⊢ ( 𝑗 = 𝑖 → ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) = ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) )
601	598 600	oveq12d	⊢ ( 𝑗 = 𝑖 → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) = ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) ) )
602	601	cbvmptv	⊢ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) = ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) ) )
603	602	fveq2i	⊢ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) = ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) ) ) )
604	603	oveq2i	⊢ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) = ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) ) ) ) )
605	604	breq2i	⊢ ( ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ↔ ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) ) ) ) ) )
606	605	rabbii	⊢ { 𝑧 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) } = { 𝑧 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) ) ) ) ) }
607	18 606	eqtri	⊢ 𝑈 = { 𝑧 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) ) ) ) ) }
608	19	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝑆 ∈ 𝑈 )
609	20	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝑆 < ( 𝐵 ‘ 𝑍 ) )
610		eqid	⊢ ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) = ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) )
611		simp2	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝑚 ∈ ℕ )
612		id	⊢ ( 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) )
613		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝑃 ‘ 𝑗 ) = ( 𝑃 ‘ 𝑖 ) )
614	613	cbvsumv	⊢ Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) = Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑖 )
615	614	oveq2i	⊢ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) = ( ( 1 + 𝐸 ) · Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑖 ) )
616	615	a1i	⊢ ( 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) = ( ( 1 + 𝐸 ) · Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑖 ) ) )
617	612 616	breqtrd	⊢ ( 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑖 ) ) )
618	617	3ad2ant3	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑖 ) ) )
619		simpl	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑚 ) ) → 𝜑 )
620		elfznn	⊢ ( 𝑖 ∈ ( 1 ... 𝑚 ) → 𝑖 ∈ ℕ )
621	620	adantl	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑚 ) ) → 𝑖 ∈ ℕ )
622		eleq1w	⊢ ( 𝑗 = 𝑖 → ( 𝑗 ∈ ℕ ↔ 𝑖 ∈ ℕ ) )
623		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝐽 ‘ 𝑗 ) = ( 𝐽 ‘ 𝑖 ) )
624		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝐾 ‘ 𝑗 ) = ( 𝐾 ‘ 𝑖 ) )
625	623 624	oveq12d	⊢ ( 𝑗 = 𝑖 → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) = ( ( 𝐽 ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑖 ) ) )
626	613 625	eqeq12d	⊢ ( 𝑗 = 𝑖 → ( ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ↔ ( 𝑃 ‘ 𝑖 ) = ( ( 𝐽 ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑖 ) ) ) )
627	622 626	imbi12d	⊢ ( 𝑗 = 𝑖 → ( ( 𝑗 ∈ ℕ → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ↔ ( 𝑖 ∈ ℕ → ( 𝑃 ‘ 𝑖 ) = ( ( 𝐽 ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑖 ) ) ) ) )
628	627 62	chvarvv	⊢ ( 𝑖 ∈ ℕ → ( 𝑃 ‘ 𝑖 ) = ( ( 𝐽 ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑖 ) ) )
629	628	adantl	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( 𝑃 ‘ 𝑖 ) = ( ( 𝐽 ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑖 ) ) )
630	622	anbi2d	⊢ ( 𝑗 = 𝑖 → ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ↔ ( 𝜑 ∧ 𝑖 ∈ ℕ ) ) )
631	598	fveq1d	⊢ ( 𝑗 = 𝑖 → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) )
632	599	fveq1d	⊢ ( 𝑗 = 𝑖 → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) )
633	631 632	oveq12d	⊢ ( 𝑗 = 𝑖 → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) )
634	633	eleq2d	⊢ ( 𝑗 = 𝑖 → ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ↔ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) )
635	598	reseq1d	⊢ ( 𝑗 = 𝑖 → ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) = ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) )
636	634 635	ifbieq1d	⊢ ( 𝑗 = 𝑖 → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) )
637	623 636	eqeq12d	⊢ ( 𝑗 = 𝑖 → ( ( 𝐽 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ↔ ( 𝐽 ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) )
638	630 637	imbi12d	⊢ ( 𝑗 = 𝑖 → ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐽 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ) ↔ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( 𝐽 ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) ) )
639	638 161	chvarvv	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( 𝐽 ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) )
640	599	reseq1d	⊢ ( 𝑗 = 𝑖 → ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) = ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) )
641	634 640	ifbieq1d	⊢ ( 𝑗 = 𝑖 → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) )
642	624 641	eqeq12d	⊢ ( 𝑗 = 𝑖 → ( ( 𝐾 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ↔ ( 𝐾 ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) )
643	630 642	imbi12d	⊢ ( 𝑗 = 𝑖 → ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐾 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ) ↔ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( 𝐾 ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) ) )
644	643 443	chvarvv	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( 𝐾 ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) )
645	639 644	oveq12d	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝐽 ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑖 ) ) = ( if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ( 𝐿 ‘ 𝑌 ) if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) )
646	629 645	eqtrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( 𝑃 ‘ 𝑖 ) = ( if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ( 𝐿 ‘ 𝑌 ) if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) )
647		simpr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → 𝑖 ∈ ℕ )
648		ovexd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ∈ V )
649	610	fvmpt2	⊢ ( ( 𝑖 ∈ ℕ ∧ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ∈ V ) → ( ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) ‘ 𝑖 ) = ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) )
650	647 648 649	syl2anc	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) ‘ 𝑖 ) = ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) )
651		fvex	⊢ ( 𝐶 ‘ 𝑖 ) ∈ V
652	651	resex	⊢ ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) ∈ V
653	652	a1i	⊢ ( 𝜑 → ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) ∈ V )
654	9 155	eqeltrrid	⊢ ( 𝜑 → ( 𝑦 ∈ 𝑌 ↦ 0 ) ∈ V )
655	653 654	ifcld	⊢ ( 𝜑 → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ∈ V )
656	655	adantr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ∈ V )
657	578	fvmpt2	⊢ ( ( 𝑖 ∈ ℕ ∧ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ∈ V ) → ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) )
658	647 656 657	syl2anc	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) )
659	9	eqcomi	⊢ ( 𝑦 ∈ 𝑌 ↦ 0 ) = 𝐹
660		ifeq2	⊢ ( ( 𝑦 ∈ 𝑌 ↦ 0 ) = 𝐹 → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) )
661	659 660	ax-mp	⊢ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 )
662	661	a1i	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) )
663	658 662	eqtrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) )
664		fvex	⊢ ( 𝐷 ‘ 𝑖 ) ∈ V
665	664	resex	⊢ ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) ∈ V
666	665	a1i	⊢ ( 𝜑 → ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) ∈ V )
667	666 654	ifcld	⊢ ( 𝜑 → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ∈ V )
668	667	adantr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ∈ V )
669	580	fvmpt2	⊢ ( ( 𝑖 ∈ ℕ ∧ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ∈ V ) → ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) )
670	647 668 669	syl2anc	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) )
671		biid	⊢ ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ↔ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) )
672	671 659	ifbieq2i	⊢ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 )
673	672	a1i	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) )
674	670 673	eqtrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) )
675	663 674	oveq12d	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) = ( if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ( 𝐿 ‘ 𝑌 ) if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) )
676	650 675	eqtrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) ‘ 𝑖 ) = ( if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ( 𝐿 ‘ 𝑌 ) if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) )
677	646 676	eqtr4d	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( 𝑃 ‘ 𝑖 ) = ( ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) ‘ 𝑖 ) )
678	619 621 677	syl2anc	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑚 ) ) → ( 𝑃 ‘ 𝑖 ) = ( ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) ‘ 𝑖 ) )
679	678	3ad2antl1	⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑚 ) ) → ( 𝑃 ‘ 𝑖 ) = ( ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) ‘ 𝑖 ) )
680	679	sumeq2dv	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑖 ) = Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) ‘ 𝑖 ) )
681	680	oveq2d	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → ( ( 1 + 𝐸 ) · Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑖 ) ) = ( ( 1 + 𝐸 ) · Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) ‘ 𝑖 ) ) )
682	618 681	breqtrd	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) ‘ 𝑖 ) ) )
683		fveq2	⊢ ( 𝑗 = ℎ → ( 𝐷 ‘ 𝑗 ) = ( 𝐷 ‘ ℎ ) )
684	683	fveq1d	⊢ ( 𝑗 = ℎ → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) )
685	684	cbvmptv	⊢ ( 𝑗 ∈ { 𝑖 ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( ℎ ∈ { 𝑖 ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) )
686	685	rneqi	⊢ ran ( 𝑗 ∈ { 𝑖 ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ran ( ℎ ∈ { 𝑖 ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) )
687		fveq2	⊢ ( ℎ = 𝑖 → ( 𝐶 ‘ ℎ ) = ( 𝐶 ‘ 𝑖 ) )
688	687	fveq1d	⊢ ( ℎ = 𝑖 → ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) = ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) )
689		fveq2	⊢ ( ℎ = 𝑖 → ( 𝐷 ‘ ℎ ) = ( 𝐷 ‘ 𝑖 ) )
690	689	fveq1d	⊢ ( ℎ = 𝑖 → ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) )
691	688 690	oveq12d	⊢ ( ℎ = 𝑖 → ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) )
692	691	eleq2d	⊢ ( ℎ = 𝑖 → ( 𝑆 ∈ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) ↔ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) )
693	692	cbvrabv	⊢ { ℎ ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) } = { 𝑖 ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) }
694	693	mpteq1i	⊢ ( 𝑗 ∈ { ℎ ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( 𝑗 ∈ { 𝑖 ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) )
695	694	rneqi	⊢ ran ( 𝑗 ∈ { ℎ ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ran ( 𝑗 ∈ { 𝑖 ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) )
696	695	uneq2i	⊢ ( { ( 𝐵 ‘ 𝑍 ) } ∪ ran ( 𝑗 ∈ { ℎ ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) = ( { ( 𝐵 ‘ 𝑍 ) } ∪ ran ( 𝑗 ∈ { 𝑖 ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) )
697		eqid	⊢ inf ( ( { ( 𝐵 ‘ 𝑍 ) } ∪ ran ( 𝑗 ∈ { ℎ ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) , ℝ , < ) = inf ( ( { ( 𝐵 ‘ 𝑍 ) } ∪ ran ( 𝑗 ∈ { ℎ ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) , ℝ , < )
698	1 571 572 573 5 574 575 576 577 578 579 580 587 596 16 597 607 608 609 610 611 682 686 696 697	hoidmvlelem2	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → ∃ 𝑢 ∈ 𝑈 𝑆 < 𝑢 )
699	698	3exp	⊢ ( 𝜑 → ( 𝑚 ∈ ℕ → ( 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ∃ 𝑢 ∈ 𝑈 𝑆 < 𝑢 ) ) )
700	699	rexlimdv	⊢ ( 𝜑 → ( ∃ 𝑚 ∈ ℕ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ∃ 𝑢 ∈ 𝑈 𝑆 < 𝑢 ) )
701	570 700	mpd	⊢ ( 𝜑 → ∃ 𝑢 ∈ 𝑈 𝑆 < 𝑢 )