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	ffvelcdmda	⊢ ( ( 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	ffvelcdmda	⊢ ( ( 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	ffvelcdmda	⊢ ( ( 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	ffvelcdmda	⊢ ( ( 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	ffvelcdmd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( 𝐴 ‘ 𝑘 ) ∈ ℝ )
269	7	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → 𝐵 : 𝑊 ⟶ ℝ )
270	269 267	ffvelcdmd	⊢ ( ( 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	ffvelcdmd	⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) ∈ ℝ )
338	337	rexrd	⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) ∈ ℝ* )
339	7 336	ffvelcdmd	⊢ ( 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	birani	⊢ ( ( 𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌 ) → 𝑘 ∈ ( 𝑌 ∪ { 𝑍 } ) )
354		simpr	⊢ ( ( 𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌 ) → ¬ 𝑘 ∈ 𝑌 )
355		elunnel1	⊢ ( ( 𝑘 ∈ ( 𝑌 ∪ { 𝑍 } ) ∧ ¬ 𝑘 ∈ 𝑌 ) → 𝑘 ∈ { 𝑍 } )
356	353 354 355	syl2anc	⊢ ( ( 𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌 ) → 𝑘 ∈ { 𝑍 } )
357		elsni	⊢ ( 𝑘 ∈ { 𝑍 } → 𝑘 = 𝑍 )
358	356 357	syl	⊢ ( ( 𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌 ) → 𝑘 = 𝑍 )
359		fveq2	⊢ ( 𝑘 = 𝑍 → ( 𝐴 ‘ 𝑘 ) = ( 𝐴 ‘ 𝑍 ) )
360		fveq2	⊢ ( 𝑘 = 𝑍 → ( 𝐵 ‘ 𝑘 ) = ( 𝐵 ‘ 𝑍 ) )
361	359 360	oveq12d	⊢ ( 𝑘 = 𝑍 → ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) )
362	358 361	syl	⊢ ( ( 𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌 ) → ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) )
363	362	adantll	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ¬ 𝑘 ∈ 𝑌 ) → ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) )
364	351 363	eleq12d	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ¬ 𝑘 ∈ 𝑌 ) → ( if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ↔ 𝑆 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) )
365	349 364	mpbird	⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ¬ 𝑘 ∈ 𝑌 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
366	365	adantllr	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) ∧ ¬ 𝑘 ∈ 𝑌 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
367	330 366	pm2.61dan	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
368	319 367	eqeltrd	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
369	368	ex	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( 𝑘 ∈ 𝑊 → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) )
370	311 369	ralrimi	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
371	306 370	jca	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( ( 𝑂 ‘ 𝑥 ) Fn 𝑊 ∧ ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) )
372		fvex	⊢ ( 𝑂 ‘ 𝑥 ) ∈ V
373	372	elixp	⊢ ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ↔ ( ( 𝑂 ‘ 𝑥 ) Fn 𝑊 ∧ ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) )
374	371 373	sylibr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
375	290 374	sseldd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( 𝑂 ‘ 𝑥 ) ∈ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
376		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 ) ) )
377	375 376	sylib	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ∃ 𝑗 ∈ ℕ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
378		ixpfn	⊢ ( 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) → 𝑥 Fn 𝑌 )
379	378	adantl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → 𝑥 Fn 𝑌 )
380	379	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 )
381		nfv	⊢ Ⅎ 𝑘 𝑗 ∈ ℕ
382	311 381	nfan	⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ )
383		nfcv	⊢ Ⅎ 𝑘 ( 𝑂 ‘ 𝑥 )
384		nfixp1	⊢ Ⅎ 𝑘 X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) )
385	383 384	nfel	⊢ Ⅎ 𝑘 ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) )
386	382 385	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 ) ) )
387	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 ) )
388	293	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ V )
389	267 388 316	syl2anc	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑘 ) = if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) )
390	389	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 ) )
391	320	3ad2ant3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) = ( 𝑥 ‘ 𝑘 ) )
392	387 390 391	3eqtrrd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑥 ‘ 𝑘 ) = ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) )
393	392	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 ) )
394	372	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 ) ) ) )
395	394	birani	⊢ ( ( ( 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 ) ) ) )
396	395	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 ) ) )
397	266	adantl	⊢ ( ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → 𝑘 ∈ 𝑊 )
398		rspa	⊢ ( ( ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑊 ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
399	396 397 398	syl2anc	⊢ ( ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
400	399	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 ) ) )
401	393 400	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 ) ) )
402	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 )
403	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 )
404	304	fveq1d	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑍 ) = ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑍 ) )
405		eqidd	⊢ ( 𝜑 → ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) = ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) )
406		eleq1	⊢ ( 𝑘 = 𝑍 → ( 𝑘 ∈ 𝑌 ↔ 𝑍 ∈ 𝑌 ) )
407		fveq2	⊢ ( 𝑘 = 𝑍 → ( 𝑥 ‘ 𝑘 ) = ( 𝑥 ‘ 𝑍 ) )
408	406 407	ifbieq1d	⊢ ( 𝑘 = 𝑍 → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) = if ( 𝑍 ∈ 𝑌 , ( 𝑥 ‘ 𝑍 ) , 𝑆 ) )
409	408	adantl	⊢ ( ( 𝜑 ∧ 𝑘 = 𝑍 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) = if ( 𝑍 ∈ 𝑌 , ( 𝑥 ‘ 𝑍 ) , 𝑆 ) )
410		fvexd	⊢ ( 𝜑 → ( 𝑥 ‘ 𝑍 ) ∈ V )
411	410 292	ifcld	⊢ ( 𝜑 → if ( 𝑍 ∈ 𝑌 , ( 𝑥 ‘ 𝑍 ) , 𝑆 ) ∈ V )
412	405 409 336 411	fvmptd	⊢ ( 𝜑 → ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑍 ) = if ( 𝑍 ∈ 𝑌 , ( 𝑥 ‘ 𝑍 ) , 𝑆 ) )
413	412	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 ) )
414	4	eldifbd	⊢ ( 𝜑 → ¬ 𝑍 ∈ 𝑌 )
415	414	iffalsed	⊢ ( 𝜑 → if ( 𝑍 ∈ 𝑌 , ( 𝑥 ‘ 𝑍 ) , 𝑆 ) = 𝑆 )
416	415	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → if ( 𝑍 ∈ 𝑌 , ( 𝑥 ‘ 𝑍 ) , 𝑆 ) = 𝑆 )
417	404 413 416	3eqtrrd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → 𝑆 = ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑍 ) )
418	417	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 ) )
419	402 336	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 )
420	394	simprbi	⊢ ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) → ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
421	420	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 ) ) )
422		fveq2	⊢ ( 𝑘 = 𝑍 → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) = ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑍 ) )
423		fveq2	⊢ ( 𝑘 = 𝑍 → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) )
424		fveq2	⊢ ( 𝑘 = 𝑍 → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) )
425	423 424	oveq12d	⊢ ( 𝑘 = 𝑍 → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) )
426	422 425	eleq12d	⊢ ( 𝑘 = 𝑍 → ( ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑍 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) )
427	426	rspcva	⊢ ( ( 𝑍 ∈ 𝑊 ∧ ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑍 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) )
428	419 421 427	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 ) ) )
429	418 428	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 ) ) )
430	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 ) )
431	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 ) )
432	430 431	eqtrd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝐽 ‘ 𝑗 ) = ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) )
433	432	fveq1d	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) )
434	402 403 429 433	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 ) )
435	434	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 ) )
436		fvres	⊢ ( 𝑘 ∈ 𝑌 → ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) )
437	436	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 ) )
438	435 437	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 ) )
439	120	elexd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V )
440	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 ) )
441	151 439 440	syl2anc	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐾 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) )
442	441	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 ) )
443	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 ) )
444	442 443	eqtrd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝐾 ‘ 𝑗 ) = ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) )
445	444	fveq1d	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) )
446	402 403 429 445	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 ) )
447	446	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 ) )
448		fvres	⊢ ( 𝑘 ∈ 𝑌 → ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) )
449	448	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 ) )
450	447 449	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 ) )
451	438 450	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 ) ) )
452	451	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 ) ) )
453	401 452	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 ) ) )
454	453	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 ) ) ) )
455	386 454	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 ) ) )
456	380 455	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 ) ) ) )
457	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 ) ) ) )
458	456 457	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 ) ) )
459	458	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 ) ) ) )
460	459	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 ) ) ) )
461	377 460	mpd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ∃ 𝑗 ∈ ℕ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) )
462		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 ) ) )
463	461 462	sylibr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → 𝑥 ∈ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) )
464	463	ralrimiva	⊢ ( 𝜑 → ∀ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) 𝑥 ∈ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) )
465		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 ) ) )
466	464 465	sylibr	⊢ ( 𝜑 → X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) )
467		ovexd	⊢ ( 𝜑 → ( ℝ ↑m 𝑌 ) ∈ V )
468	237	a1i	⊢ ( 𝜑 → ℕ ∈ V )
469	467 468	elmapd	⊢ ( 𝜑 → ( 𝐾 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ↔ 𝐾 : ℕ ⟶ ( ℝ ↑m 𝑌 ) ) )
470	121 469	mpbird	⊢ ( 𝜑 → 𝐾 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) )
471	467 468	elmapd	⊢ ( 𝜑 → ( 𝐽 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ↔ 𝐽 : ℕ ⟶ ( ℝ ↑m 𝑌 ) ) )
472	103 471	mpbird	⊢ ( 𝜑 → 𝐽 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) )
473	97 87	elmapd	⊢ ( 𝜑 → ( ( 𝐵 ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) ↔ ( 𝐵 ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) )
474	213 473	mpbird	⊢ ( 𝜑 → ( 𝐵 ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) )
475	97 87	elmapd	⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) ↔ ( 𝐴 ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) )
476	212 475	mpbird	⊢ ( 𝜑 → ( 𝐴 ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) )
477		fveq1	⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → ( 𝑒 ‘ 𝑘 ) = ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) )
478	477	adantr	⊢ ( ( 𝑒 = ( 𝐴 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑒 ‘ 𝑘 ) = ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) )
479	259	adantl	⊢ ( ( 𝑒 = ( 𝐴 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) = ( 𝐴 ‘ 𝑘 ) )
480	478 479	eqtrd	⊢ ( ( 𝑒 = ( 𝐴 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑒 ‘ 𝑘 ) = ( 𝐴 ‘ 𝑘 ) )
481	480	oveq1d	⊢ ( ( 𝑒 = ( 𝐴 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) )
482	481	ixpeq2dva	⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → X 𝑘 ∈ 𝑌 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) )
483	482	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 ) ) ) )
484		oveq1	⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → ( 𝑒 ( 𝐿 ‘ 𝑌 ) 𝑓 ) = ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) )
485	484	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 ) ) ) ) ) )
486	483 485	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 ) ) ) ) ) ) )
487	486	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 ) ) ) ) ) ) )
488	487	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 ) ) ) ) ) ) )
489	488	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 ) ) ) ) ) ) )
490	489	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 ) ) ) ) ) )
491	476 22 490	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 ) ) ) ) ) )
492		fveq1	⊢ ( 𝑓 = ( 𝐵 ↾ 𝑌 ) → ( 𝑓 ‘ 𝑘 ) = ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) )
493	492	adantr	⊢ ( ( 𝑓 = ( 𝐵 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑓 ‘ 𝑘 ) = ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) )
494	260	adantl	⊢ ( ( 𝑓 = ( 𝐵 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) = ( 𝐵 ‘ 𝑘 ) )
495	493 494	eqtrd	⊢ ( ( 𝑓 = ( 𝐵 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑓 ‘ 𝑘 ) = ( 𝐵 ‘ 𝑘 ) )
496	495	oveq2d	⊢ ( ( 𝑓 = ( 𝐵 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
497	496	ixpeq2dva	⊢ ( 𝑓 = ( 𝐵 ↾ 𝑌 ) → X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
498	497	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 ) ) ) )
499		oveq2	⊢ ( 𝑓 = ( 𝐵 ↾ 𝑌 ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) = ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) )
500	499	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 ) ) ) ) ) )
501	498 500	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 ) ) ) ) ) ) )
502	501	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 ) ) ) ) ) ) )
503	502	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 ) ) ) ) ) ) )
504	503	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 ) ) ) ) ) )
505	474 491 504	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 ) ) ) ) ) )
506		fveq1	⊢ ( 𝑔 = 𝐽 → ( 𝑔 ‘ 𝑗 ) = ( 𝐽 ‘ 𝑗 ) )
507	506	fveq1d	⊢ ( 𝑔 = 𝐽 → ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) )
508	507	oveq1d	⊢ ( 𝑔 = 𝐽 → ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) )
509	508	ixpeq2dv	⊢ ( 𝑔 = 𝐽 → X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) )
510	509	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 ) ) )
511	510	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 ) ) ) )
512	506	oveq1d	⊢ ( 𝑔 = 𝐽 → ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) )
513	512	mpteq2dv	⊢ ( 𝑔 = 𝐽 → ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) )
514	513	fveq2d	⊢ ( 𝑔 = 𝐽 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) )
515	514	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 ) ) ) ) ) )
516	511 515	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 ) ) ) ) ) ) )
517	516	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 ) ) ) ) ) ) )
518	517	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 ) ) ) ) ) )
519	472 505 518	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 ) ) ) ) ) )
520		fveq1	⊢ ( ℎ = 𝐾 → ( ℎ ‘ 𝑗 ) = ( 𝐾 ‘ 𝑗 ) )
521	520	fveq1d	⊢ ( ℎ = 𝐾 → ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) )
522	521	oveq2d	⊢ ( ℎ = 𝐾 → ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) )
523	522	ixpeq2dv	⊢ ( ℎ = 𝐾 → X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) )
524	523	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 ) ) )
525	524	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 ) ) ) )
526	520	oveq2d	⊢ ( ℎ = 𝐾 → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) )
527	526	mpteq2dv	⊢ ( ℎ = 𝐾 → ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) )
528	527	fveq2d	⊢ ( ℎ = 𝐾 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) )
529	528	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 ) ) ) ) ) )
530	525 529	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 ) ) ) ) ) ) )
531	530	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 ) ) ) ) ) )
532	470 519 531	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 ) ) ) ) ) )
533	466 532	mpd	⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) )
534		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 ) ) ) ) ) )
535	533 534	mpd	⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) )
536	535	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) )
537	62	adantl	⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) )
538	537	mpteq2dva	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) )
539	538	fveq2d	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) )
540	258 539	breq12d	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 𝐺 ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ↔ ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) ) )
541	536 540	mpbird	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → 𝐺 ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) )
542	541	adantr	⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) → 𝐺 ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) )
543	247 249 250 289 542	ltletrd	⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) → ( 𝐺 / ( 1 + 𝐸 ) ) < ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) )
544	235 246 543	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ( 𝐺 / ( 1 + 𝐸 ) ) < ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) )
545	234 544	pm2.61dan	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 𝐺 / ( 1 + 𝐸 ) ) < ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) )
546	207 208 209 210 227 545	sge0uzfsumgt	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ∃ 𝑚 ∈ ℕ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) )
547	226	adantr	⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ( 𝐺 / ( 1 + 𝐸 ) ) ∈ ℝ )
548		fzfid	⊢ ( 𝜑 → ( 1 ... 𝑚 ) ∈ Fin )
549		simpl	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑚 ) ) → 𝜑 )
550		elfznn	⊢ ( 𝑗 ∈ ( 1 ... 𝑚 ) → 𝑗 ∈ ℕ )
551	550	adantl	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑚 ) ) → 𝑗 ∈ ℕ )
552	50 126	sselid	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) ∈ ℝ )
553	549 551 552	syl2anc	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑚 ) ) → ( 𝑃 ‘ 𝑗 ) ∈ ℝ )
554	548 553	fsumrecl	⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ∈ ℝ )
555	554	adantr	⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ∈ ℝ )
556		simpr	⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) )
557	547 555 556	ltled	⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ( 𝐺 / ( 1 + 𝐸 ) ) ≤ Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) )
558	216	adantr	⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → 𝐺 ∈ ℝ )
559	222	adantr	⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ( 1 + 𝐸 ) ∈ ℝ+ )
560	558 555 559	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 ) ) ) )
561	557 560	mpbid	⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) )
562	561	ex	⊢ ( 𝜑 → ( ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) )
563	562	adantr	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) )
564	563	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) )
565	564	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 ) ) ) )
566	546 565	mpd	⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ∃ 𝑚 ∈ ℕ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) )
567	204 206 566	syl2anc	⊢ ( ( 𝜑 ∧ ¬ 𝑌 = ∅ ) → ∃ 𝑚 ∈ ℕ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) )
568	203 567	pm2.61dan	⊢ ( 𝜑 → ∃ 𝑚 ∈ ℕ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) )
569	2	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝑋 ∈ Fin )
570	3	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝑌 ⊆ 𝑋 )
571	4	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝑍 ∈ ( 𝑋 ∖ 𝑌 ) )
572	6	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝐴 : 𝑊 ⟶ ℝ )
573	7	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝐵 : 𝑊 ⟶ ℝ )
574	10	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝐶 : ℕ ⟶ ( ℝ ↑m 𝑊 ) )
575		eqid	⊢ ( 𝑦 ∈ 𝑌 ↦ 0 ) = ( 𝑦 ∈ 𝑌 ↦ 0 )
576		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 ) ) )
577	12	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝐷 : ℕ ⟶ ( ℝ ↑m 𝑊 ) )
578		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 ) ) )
579		fveq2	⊢ ( 𝑖 = 𝑗 → ( 𝐶 ‘ 𝑖 ) = ( 𝐶 ‘ 𝑗 ) )
580		fveq2	⊢ ( 𝑖 = 𝑗 → ( 𝐷 ‘ 𝑖 ) = ( 𝐷 ‘ 𝑗 ) )
581	579 580	oveq12d	⊢ ( 𝑖 = 𝑗 → ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑖 ) ) = ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑗 ) ) )
582	581	cbvmptv	⊢ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑖 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑗 ) ) )
583	582	fveq2i	⊢ ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑖 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑗 ) ) ) )
584	583 14	eqeltrid	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑖 ) ) ) ) ∈ ℝ )
585	584	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑖 ) ) ) ) ∈ ℝ )
586		eleq1w	⊢ ( 𝑗 = 𝑖 → ( 𝑗 ∈ 𝑌 ↔ 𝑖 ∈ 𝑌 ) )
587		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝑐 ‘ 𝑗 ) = ( 𝑐 ‘ 𝑖 ) )
588	587	breq1d	⊢ ( 𝑗 = 𝑖 → ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 ↔ ( 𝑐 ‘ 𝑖 ) ≤ 𝑥 ) )
589	588 587	ifbieq1d	⊢ ( 𝑗 = 𝑖 → if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) = if ( ( 𝑐 ‘ 𝑖 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑖 ) , 𝑥 ) )
590	586 587 589	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 ) ) )
591	590	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 ) ) )
592	591	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 ) ) ) )
593	592	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 ) ) ) ) )
594	15 593	eqtri	⊢ 𝐻 = ( 𝑥 ∈ ℝ ↦ ( 𝑐 ∈ ( ℝ ↑m 𝑊 ) ↦ ( 𝑖 ∈ 𝑊 ↦ if ( 𝑖 ∈ 𝑌 , ( 𝑐 ‘ 𝑖 ) , if ( ( 𝑐 ‘ 𝑖 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑖 ) , 𝑥 ) ) ) ) )
595	17	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝐸 ∈ ℝ+ )
596		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝐶 ‘ 𝑗 ) = ( 𝐶 ‘ 𝑖 ) )
597		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝐷 ‘ 𝑗 ) = ( 𝐷 ‘ 𝑖 ) )
598	597	fveq2d	⊢ ( 𝑗 = 𝑖 → ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) = ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) )
599	596 598	oveq12d	⊢ ( 𝑗 = 𝑖 → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) = ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) ) )
600	599	cbvmptv	⊢ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) = ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) ) )
601	600	fveq2i	⊢ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) = ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) ) ) )
602	601	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 ) ) ) ) ) )
603	602	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 ) ) ) ) ) ) )
604	603	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 ) ) ) ) ) ) }
605	18 604	eqtri	⊢ 𝑈 = { 𝑧 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) ) ) ) ) }
606	19	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝑆 ∈ 𝑈 )
607	20	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝑆 < ( 𝐵 ‘ 𝑍 ) )
608		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 ) ) )
609		simp2	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝑚 ∈ ℕ )
610		id	⊢ ( 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) )
611		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝑃 ‘ 𝑗 ) = ( 𝑃 ‘ 𝑖 ) )
612	611	cbvsumv	⊢ Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) = Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑖 )
613	612	oveq2i	⊢ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) = ( ( 1 + 𝐸 ) · Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑖 ) )
614	613	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 ) ) )
615	610 614	breqtrd	⊢ ( 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑖 ) ) )
616	615	3ad2ant3	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑖 ) ) )
617		simpl	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑚 ) ) → 𝜑 )
618		elfznn	⊢ ( 𝑖 ∈ ( 1 ... 𝑚 ) → 𝑖 ∈ ℕ )
619	618	adantl	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑚 ) ) → 𝑖 ∈ ℕ )
620		eleq1w	⊢ ( 𝑗 = 𝑖 → ( 𝑗 ∈ ℕ ↔ 𝑖 ∈ ℕ ) )
621		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝐽 ‘ 𝑗 ) = ( 𝐽 ‘ 𝑖 ) )
622		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝐾 ‘ 𝑗 ) = ( 𝐾 ‘ 𝑖 ) )
623	621 622	oveq12d	⊢ ( 𝑗 = 𝑖 → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) = ( ( 𝐽 ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑖 ) ) )
624	611 623	eqeq12d	⊢ ( 𝑗 = 𝑖 → ( ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ↔ ( 𝑃 ‘ 𝑖 ) = ( ( 𝐽 ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑖 ) ) ) )
625	620 624	imbi12d	⊢ ( 𝑗 = 𝑖 → ( ( 𝑗 ∈ ℕ → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ↔ ( 𝑖 ∈ ℕ → ( 𝑃 ‘ 𝑖 ) = ( ( 𝐽 ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑖 ) ) ) ) )
626	625 62	chvarvv	⊢ ( 𝑖 ∈ ℕ → ( 𝑃 ‘ 𝑖 ) = ( ( 𝐽 ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑖 ) ) )
627	626	adantl	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( 𝑃 ‘ 𝑖 ) = ( ( 𝐽 ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑖 ) ) )
628	620	anbi2d	⊢ ( 𝑗 = 𝑖 → ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ↔ ( 𝜑 ∧ 𝑖 ∈ ℕ ) ) )
629	596	fveq1d	⊢ ( 𝑗 = 𝑖 → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) )
630	597	fveq1d	⊢ ( 𝑗 = 𝑖 → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) )
631	629 630	oveq12d	⊢ ( 𝑗 = 𝑖 → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) )
632	631	eleq2d	⊢ ( 𝑗 = 𝑖 → ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ↔ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) )
633	596	reseq1d	⊢ ( 𝑗 = 𝑖 → ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) = ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) )
634	632 633	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 ) )
635	621 634	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 ) ) )
636	628 635	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 ) ) ) )
637	636 161	chvarvv	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( 𝐽 ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) )
638	597	reseq1d	⊢ ( 𝑗 = 𝑖 → ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) = ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) )
639	632 638	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 ) )
640	622 639	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 ) ) )
641	628 640	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 ) ) ) )
642	641 441	chvarvv	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( 𝐾 ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) )
643	637 642	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 ) ) )
644	627 643	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 ) ) )
645		simpr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → 𝑖 ∈ ℕ )
646		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 )
647	608	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 ) ) )
648	645 646 647	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 ) ) )
649		fvex	⊢ ( 𝐶 ‘ 𝑖 ) ∈ V
650	649	resex	⊢ ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) ∈ V
651	650	a1i	⊢ ( 𝜑 → ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) ∈ V )
652	9 155	eqeltrrid	⊢ ( 𝜑 → ( 𝑦 ∈ 𝑌 ↦ 0 ) ∈ V )
653	651 652	ifcld	⊢ ( 𝜑 → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ∈ V )
654	653	adantr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ∈ V )
655	576	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 ) ) )
656	645 654 655	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 ) ) )
657	9	eqcomi	⊢ ( 𝑦 ∈ 𝑌 ↦ 0 ) = 𝐹
658		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 ) )
659	657 658	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 )
660	659	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 ) )
661	656 660	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 ) )
662		fvex	⊢ ( 𝐷 ‘ 𝑖 ) ∈ V
663	662	resex	⊢ ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) ∈ V
664	663	a1i	⊢ ( 𝜑 → ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) ∈ V )
665	664 652	ifcld	⊢ ( 𝜑 → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ∈ V )
666	665	adantr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ∈ V )
667	578	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 ) ) )
668	645 666 667	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 ) ) )
669		biid	⊢ ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ↔ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) )
670	669 657	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 )
671	670	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 ) )
672	668 671	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 ) )
673	661 672	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 ) ) )
674	648 673	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 ) ) )
675	644 674	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 ) )
676	617 619 675	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 ) )
677	676	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 ) )
678	677	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 ) )
679	678	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 ) ) )
680	616 679	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 ) ) )
681		fveq2	⊢ ( 𝑗 = ℎ → ( 𝐷 ‘ 𝑗 ) = ( 𝐷 ‘ ℎ ) )
682	681	fveq1d	⊢ ( 𝑗 = ℎ → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) )
683	682	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 ) )
684	683	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 ) )
685		fveq2	⊢ ( ℎ = 𝑖 → ( 𝐶 ‘ ℎ ) = ( 𝐶 ‘ 𝑖 ) )
686	685	fveq1d	⊢ ( ℎ = 𝑖 → ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) = ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) )
687		fveq2	⊢ ( ℎ = 𝑖 → ( 𝐷 ‘ ℎ ) = ( 𝐷 ‘ 𝑖 ) )
688	687	fveq1d	⊢ ( ℎ = 𝑖 → ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) )
689	686 688	oveq12d	⊢ ( ℎ = 𝑖 → ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) )
690	689	eleq2d	⊢ ( ℎ = 𝑖 → ( 𝑆 ∈ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) ↔ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) )
691	690	cbvrabv	⊢ { ℎ ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) } = { 𝑖 ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) }
692	691	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 ) )
693	692	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 ) )
694	693	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 ) ) )
695		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 , < )
696	1 569 570 571 5 572 573 574 575 576 577 578 585 594 16 595 605 606 607 608 609 680 684 694 695	hoidmvlelem2	⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → ∃ 𝑢 ∈ 𝑈 𝑆 < 𝑢 )
697	696	3exp	⊢ ( 𝜑 → ( 𝑚 ∈ ℕ → ( 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ∃ 𝑢 ∈ 𝑈 𝑆 < 𝑢 ) ) )
698	697	rexlimdv	⊢ ( 𝜑 → ( ∃ 𝑚 ∈ ℕ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ∃ 𝑢 ∈ 𝑈 𝑆 < 𝑢 ) )
699	568 698	mpd	⊢ ( 𝜑 → ∃ 𝑢 ∈ 𝑈 𝑆 < 𝑢 )