hoidmvlelem3 - Mathbox for Glauco Siliprandi

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Theorem hoidmvlelem3 44135

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.)

**Hypotheses**
Ref	Expression
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)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
hoidmvlelem3.i2	⊢ (𝜑 → X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
hoidmvlelem3.o	⊢ 𝑂 = (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↦ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))

**Assertion**
Ref	Expression
hoidmvlelem3	⊢ (𝜑 → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)

Distinct variable groups: 𝐴,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥 𝐴,𝑒,𝑓,𝑔,ℎ,𝑗,𝑘 𝑧,𝐴,ℎ,𝑗 𝐵,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦 𝐵,𝑐,ℎ,𝑗,𝑘,𝑥 𝐵,𝑓,𝑔 𝑢,𝐵,ℎ,𝑗 𝑧,𝐵 𝐶,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦 𝐶,𝑐 𝑢,𝐶 𝑧,𝐶 𝐷,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦 𝐷,𝑐 𝑢,𝐷 𝑧,𝐷 𝐸,𝑎,𝑏,ℎ,𝑘,𝑥,𝑦 𝐸,𝑐 𝑧,𝐸 𝑗,𝐹 𝐺,𝑎,𝑏,ℎ,𝑘,𝑥,𝑦 𝐺,𝑐 𝑧,𝐺 𝐻,𝑎,𝑏,𝑗,𝑘 𝑧,𝐻 𝐽,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥 𝑔,𝐽 𝐾,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥 𝐿,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥 𝑒,𝐿,𝑓,𝑔 𝑧,𝐿 𝑗,𝑂,𝑘 𝑃,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦 𝑃,𝑐 𝑆,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦 𝑆,𝑐 𝑢,𝑆 𝑧,𝑆 𝑢,𝑈 𝑊,𝑎,𝑏,𝑗,𝑘,𝑥 𝑊,𝑐 𝑧,𝑊 𝑌,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦 𝑌,𝑐 𝑒,𝑌,𝑓,𝑔 𝑍,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦 𝑍,𝑐 𝑢,𝑍 𝑧,𝑍 𝜑,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦 𝜑,𝑐 Allowed substitution hints: 𝜑(𝑧,𝑢,𝑒,𝑓,𝑔) 𝐴(𝑦,𝑢,𝑐) 𝐵(𝑒) 𝐶(𝑒,𝑓,𝑔) 𝐷(𝑒,𝑓,𝑔) 𝑃(𝑧,𝑢,𝑒,𝑓,𝑔) 𝑆(𝑒,𝑓,𝑔) 𝑈(𝑥,𝑦,𝑧,𝑒,𝑓,𝑔,ℎ,𝑗,𝑘,𝑎,𝑏,𝑐) 𝐸(𝑢,𝑒,𝑓,𝑔,𝑗) 𝐹(𝑥,𝑦,𝑧,𝑢,𝑒,𝑓,𝑔,ℎ,𝑘,𝑎,𝑏,𝑐) 𝐺(𝑢,𝑒,𝑓,𝑔,𝑗) 𝐻(𝑥,𝑦,𝑢,𝑒,𝑓,𝑔,ℎ,𝑐) 𝐽(𝑦,𝑧,𝑢,𝑒,𝑓,𝑐) 𝐾(𝑦,𝑧,𝑢,𝑒,𝑓,𝑔,𝑐) 𝐿(𝑦,𝑢,𝑐) 𝑂(𝑥,𝑦,𝑧,𝑢,𝑒,𝑓,𝑔,ℎ,𝑎,𝑏,𝑐) 𝑊(𝑦,𝑢,𝑒,𝑓,𝑔,ℎ) 𝑋(𝑥,𝑦,𝑧,𝑢,𝑒,𝑓,𝑔,ℎ,𝑗,𝑘,𝑎,𝑏,𝑐) 𝑌(𝑧,𝑢) 𝑍(𝑒,𝑓,𝑔)

Proof of Theorem hoidmvlelem3 Dummy variables 𝑖 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1nn 11984	. . . . 5 ⊢ 1 ∈ ℕ
2	1	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 1 ∈ ℕ)
3		0le0 12074	. . . . . 6 ⊢ 0 ≤ 0
4	3	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 0 ≤ 0)
5		hoidmvlelem3.g	. . . . . . . 8 ⊢ 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))
6	5	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)))
7		fveq2 6774	. . . . . . . . 9 ⊢ (𝑌 = ∅ → (𝐿‘𝑌) = (𝐿‘∅))
8		reseq2 5886	. . . . . . . . . 10 ⊢ (𝑌 = ∅ → (𝐴 ↾ 𝑌) = (𝐴 ↾ ∅))
9		res0 5895	. . . . . . . . . . 11 ⊢ (𝐴 ↾ ∅) = ∅
10	9	a1i 11	. . . . . . . . . 10 ⊢ (𝑌 = ∅ → (𝐴 ↾ ∅) = ∅)
11	8, 10	eqtrd 2778	. . . . . . . . 9 ⊢ (𝑌 = ∅ → (𝐴 ↾ 𝑌) = ∅)
12		reseq2 5886	. . . . . . . . . 10 ⊢ (𝑌 = ∅ → (𝐵 ↾ 𝑌) = (𝐵 ↾ ∅))
13		res0 5895	. . . . . . . . . . 11 ⊢ (𝐵 ↾ ∅) = ∅
14	13	a1i 11	. . . . . . . . . 10 ⊢ (𝑌 = ∅ → (𝐵 ↾ ∅) = ∅)
15	12, 14	eqtrd 2778	. . . . . . . . 9 ⊢ (𝑌 = ∅ → (𝐵 ↾ 𝑌) = ∅)
16	7, 11, 15	oveq123d 7296	. . . . . . . 8 ⊢ (𝑌 = ∅ → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) = (∅(𝐿‘∅)∅))
17	16	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) = (∅(𝐿‘∅)∅))
18		hoidmvlelem3.l	. . . . . . . 8 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑_m 𝑥), 𝑏 ∈ (ℝ ↑_m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
19		f0 6655	. . . . . . . . 9 ⊢ ∅:∅⟶ℝ
20	19	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ∅:∅⟶ℝ)
21	18, 20, 20	hoidmv0val 44121	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (∅(𝐿‘∅)∅) = 0)
22	6, 17, 21	3eqtrd 2782	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 𝐺 = 0)
23		nfcvd 2908	. . . . . . . . . . 11 ⊢ (𝜑 → Ⅎ𝑗(𝑃‘1))
24		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑗𝜑
25		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 = 1) → 𝑗 = 1)
26	25	fveq2d 6778	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 = 1) → (𝑃‘𝑗) = (𝑃‘1))
27		1red 10976	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℝ)
28		rge0ssre 13188	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ⊆ ℝ
29		id 22	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝜑)
30	1	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℕ)
31	1	elexi 3451	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
32		eleq1 2826	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 1 → (𝑗 ∈ ℕ ↔ 1 ∈ ℕ))
33	32	anbi2d 629	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 1 → ((𝜑 ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ 1 ∈ ℕ)))
34		fveq2 6774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 1 → (𝑃‘𝑗) = (𝑃‘1))
35	34	eleq1d 2823	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 1 → ((𝑃‘𝑗) ∈ (0[,)+∞) ↔ (𝑃‘1) ∈ (0[,)+∞)))
36	33, 35	imbi12d 345	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 1 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,)+∞)) ↔ ((𝜑 ∧ 1 ∈ ℕ) → (𝑃‘1) ∈ (0[,)+∞))))
37		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
38		ovexd 7310	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ℕ → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ V)
39		hoidmvlelem3.p	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑃 = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
40	39	fvmpt2 6886	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ℕ ∧ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ V) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
41	37, 38, 40	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
42	41	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
43		hoidmvlelem3.x	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑋 ∈ Fin)
44		hoidmvlelem3.w	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑊 = (𝑌 ∪ {𝑍})
45	44	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑊 = (𝑌 ∪ {𝑍}))
46		hoidmvlelem3.y	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
47		hoidmvlelem3.z	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))
48	47	eldifad 3899	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑍 ∈ 𝑋)
49		snssi 4741	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑍 ∈ 𝑋 → {𝑍} ⊆ 𝑋)
50	48, 49	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {𝑍} ⊆ 𝑋)
51	46, 50	unssd 4120	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑌 ∪ {𝑍}) ⊆ 𝑋)
52	45, 51	eqsstrd 3959	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑊 ⊆ 𝑋)
53	43, 52	ssfid 9042	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑊 ∈ Fin)
54		ssun1 4106	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑌 ⊆ (𝑌 ∪ {𝑍})
55	44	eqcomi 2747	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑌 ∪ {𝑍}) = 𝑊
56	54, 55	sseqtri 3957	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑌 ⊆ 𝑊
57	56	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑌 ⊆ 𝑊)
58	53, 57	ssfid 9042	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑌 ∈ Fin)
59	58	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ∈ Fin)
60		iftrue 4465	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = ((𝐶‘𝑗) ↾ 𝑌))
61	60	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((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))
62		hoidmvlelem3.c	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑_m 𝑊))
63	62	ffvelrnda 6961	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑_m 𝑊))
64		elmapi 8637	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐶‘𝑗) ∈ (ℝ ↑_m 𝑊) → (𝐶‘𝑗):𝑊⟶ℝ)
65	63, 64	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑊⟶ℝ)
66	54, 44	sseqtrri 3958	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑌 ⊆ 𝑊
67	66	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ⊆ 𝑊)
68	65, 67	fssresd 6641	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
69		reex 10962	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ℝ ∈ V
70	69	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ℝ ∈ V)
71	53, 57	ssexd 5248	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑌 ∈ V)
72	71	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ∈ V)
73	70, 72	elmapd 8629	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐶‘𝑗) ↾ 𝑌) ∈ (ℝ ↑_m 𝑌) ↔ ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ))
74	68, 73	mpbird 256	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗) ↾ 𝑌) ∈ (ℝ ↑_m 𝑌))
75	74	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐶‘𝑗) ↾ 𝑌) ∈ (ℝ ↑_m 𝑌))
76	61, 75	eqeltrd 2839	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((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))
77		iffalse 4468	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
78	77	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
79		0red 10978	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ∈ ℝ)
80		hoidmvlelem3.f	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝐹 = (𝑦 ∈ 𝑌 ↦ 0)
81	79, 80	fmptd 6988	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐹:𝑌⟶ℝ)
82	69	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ℝ ∈ V)
83	82, 58	elmapd 8629	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝐹 ∈ (ℝ ↑_m 𝑌) ↔ 𝐹:𝑌⟶ℝ))
84	81, 83	mpbird 256	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐹 ∈ (ℝ ↑_m 𝑌))
85	84	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝐹 ∈ (ℝ ↑_m 𝑌))
86	78, 85	eqeltrd 2839	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((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))
87	76, 86	pm2.61dan 810	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ (ℝ ↑_m 𝑌))
88		hoidmvlelem3.j	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐽 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
89	87, 88	fmptd 6988	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐽:ℕ⟶(ℝ ↑_m 𝑌))
90	89	ffvelrnda 6961	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗) ∈ (ℝ ↑_m 𝑌))
91		elmapi 8637	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽‘𝑗) ∈ (ℝ ↑_m 𝑌) → (𝐽‘𝑗):𝑌⟶ℝ)
92	90, 91	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗):𝑌⟶ℝ)
93		iftrue 4465	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = ((𝐷‘𝑗) ↾ 𝑌))
94	93	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((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))
95		hoidmvlelem3.d	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑_m 𝑊))
96	95	ffvelrnda 6961	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑_m 𝑊))
97		elmapi 8637	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐷‘𝑗) ∈ (ℝ ↑_m 𝑊) → (𝐷‘𝑗):𝑊⟶ℝ)
98	96, 97	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑊⟶ℝ)
99	98, 67	fssresd 6641	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
100	70, 72	elmapd 8629	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐷‘𝑗) ↾ 𝑌) ∈ (ℝ ↑_m 𝑌) ↔ ((𝐷‘𝑗) ↾ 𝑌):𝑌⟶ℝ))
101	99, 100	mpbird 256	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗) ↾ 𝑌) ∈ (ℝ ↑_m 𝑌))
102	101	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐷‘𝑗) ↾ 𝑌) ∈ (ℝ ↑_m 𝑌))
103	94, 102	eqeltrd 2839	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((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))
104		iffalse 4468	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
105	104	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
106	105, 85	eqeltrd 2839	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((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))
107	103, 106	pm2.61dan 810	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ (ℝ ↑_m 𝑌))
108		hoidmvlelem3.k	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐾 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
109	107, 108	fmptd 6988	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾:ℕ⟶(ℝ ↑_m 𝑌))
110	109	ffvelrnda 6961	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗) ∈ (ℝ ↑_m 𝑌))
111		elmapi 8637	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾‘𝑗) ∈ (ℝ ↑_m 𝑌) → (𝐾‘𝑗):𝑌⟶ℝ)
112	110, 111	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗):𝑌⟶ℝ)
113	18, 59, 92, 112	hoidmvcl 44120	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ (0[,)+∞))
114	42, 113	eqeltrd 2839	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,)+∞))
115	31, 36, 114	vtocl 3498	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 ∈ ℕ) → (𝑃‘1) ∈ (0[,)+∞))
116	29, 30, 115	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃‘1) ∈ (0[,)+∞))
117	28, 116	sselid 3919	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃‘1) ∈ ℝ)
118	117	recnd 11003	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃‘1) ∈ ℂ)
119	23, 24, 26, 27, 118	sumsnd 42569	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑗 ∈ {1} (𝑃‘𝑗) = (𝑃‘1))
120	119	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = ∅) → Σ𝑗 ∈ {1} (𝑃‘𝑗) = (𝑃‘1))
121		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑗 = 1 → (𝐽‘𝑗) = (𝐽‘1))
122		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑗 = 1 → (𝐾‘𝑗) = (𝐾‘1))
123	121, 122	oveq12d 7293	. . . . . . . . . . . 12 ⊢ (𝑗 = 1 → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)))
124		ovex 7308	. . . . . . . . . . . 12 ⊢ ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) ∈ V
125	123, 39, 124	fvmpt 6875	. . . . . . . . . . 11 ⊢ (1 ∈ ℕ → (𝑃‘1) = ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)))
126	1, 125	ax-mp 5	. . . . . . . . . 10 ⊢ (𝑃‘1) = ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1))
127	126	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝑃‘1) = ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)))
128	7	oveqd 7292	. . . . . . . . . . 11 ⊢ (𝑌 = ∅ → ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) = ((𝐽‘1)(𝐿‘∅)(𝐾‘1)))
129	128	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) = ((𝐽‘1)(𝐿‘∅)(𝐾‘1)))
130	121	feq1d 6585	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 1 → ((𝐽‘𝑗):𝑌⟶ℝ ↔ (𝐽‘1):𝑌⟶ℝ))
131	33, 130	imbi12d 345	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 1 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗):𝑌⟶ℝ) ↔ ((𝜑 ∧ 1 ∈ ℕ) → (𝐽‘1):𝑌⟶ℝ)))
132	68	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
133	61	feq1d 6585	. . . . . . . . . . . . . . . . . 18 ⊢ (((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))
134	132, 133	mpbird 256	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
135	81	ad2antrr 723	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝐹:𝑌⟶ℝ)
136	78	feq1d 6585	. . . . . . . . . . . . . . . . . 18 ⊢ (((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))
137	135, 136	mpbird 256	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
138	134, 137	pm2.61dan 810	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
139		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
140		fvex 6787	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐶‘𝑗) ∈ V
141	140	resex 5939	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶‘𝑗) ↾ 𝑌) ∈ V
142	61, 141	eqeltrdi 2847	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
143	84	elexd 3452	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹 ∈ V)
144	143	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐹 ∈ V)
145	144	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝐹 ∈ V)
146	78, 145	eqeltrd 2839	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
147	142, 146	pm2.61dan 810	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
148	88	fvmpt2 6886	. . . . . . . . . . . . . . . . . 18 ⊢ ((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))
149	139, 147, 148	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
150	149	feq1d 6585	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐽‘𝑗):𝑌⟶ℝ ↔ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ))
151	138, 150	mpbird 256	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗):𝑌⟶ℝ)
152	31, 131, 151	vtocl 3498	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 ∈ ℕ) → (𝐽‘1):𝑌⟶ℝ)
153	29, 30, 152	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐽‘1):𝑌⟶ℝ)
154	153	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐽‘1):𝑌⟶ℝ)
155		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑌 = ∅ → 𝑌 = ∅)
156	155	eqcomd 2744	. . . . . . . . . . . . . 14 ⊢ (𝑌 = ∅ → ∅ = 𝑌)
157	156	feq2d 6586	. . . . . . . . . . . . 13 ⊢ (𝑌 = ∅ → ((𝐽‘1):∅⟶ℝ ↔ (𝐽‘1):𝑌⟶ℝ))
158	157	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1):∅⟶ℝ ↔ (𝐽‘1):𝑌⟶ℝ))
159	154, 158	mpbird 256	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐽‘1):∅⟶ℝ)
160	122	feq1d 6585	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 1 → ((𝐾‘𝑗):𝑌⟶ℝ ↔ (𝐾‘1):𝑌⟶ℝ))
161	33, 160	imbi12d 345	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 1 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗):𝑌⟶ℝ) ↔ ((𝜑 ∧ 1 ∈ ℕ) → (𝐾‘1):𝑌⟶ℝ)))
162	31, 161, 112	vtocl 3498	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 ∈ ℕ) → (𝐾‘1):𝑌⟶ℝ)
163	29, 30, 162	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾‘1):𝑌⟶ℝ)
164	163	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐾‘1):𝑌⟶ℝ)
165	156	feq2d 6586	. . . . . . . . . . . . 13 ⊢ (𝑌 = ∅ → ((𝐾‘1):∅⟶ℝ ↔ (𝐾‘1):𝑌⟶ℝ))
166	165	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐾‘1):∅⟶ℝ ↔ (𝐾‘1):𝑌⟶ℝ))
167	164, 166	mpbird 256	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐾‘1):∅⟶ℝ)
168	18, 159, 167	hoidmv0val 44121	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1)(𝐿‘∅)(𝐾‘1)) = 0)
169	129, 168	eqtrd 2778	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) = 0)
170	120, 127, 169	3eqtrd 2782	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = ∅) → Σ𝑗 ∈ {1} (𝑃‘𝑗) = 0)
171	170	oveq2d 7291	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)) = ((1 + 𝐸) · 0))
172		hoidmvlelem3.e	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
173	172	rpred 12772	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ)
174	27, 173	readdcld 11004	. . . . . . . . . 10 ⊢ (𝜑 → (1 + 𝐸) ∈ ℝ)
175	174	recnd 11003	. . . . . . . . 9 ⊢ (𝜑 → (1 + 𝐸) ∈ ℂ)
176	175	mul01d 11174	. . . . . . . 8 ⊢ (𝜑 → ((1 + 𝐸) · 0) = 0)
177	176	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((1 + 𝐸) · 0) = 0)
178		eqidd 2739	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 0 = 0)
179	171, 177, 178	3eqtrd 2782	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)) = 0)
180	22, 179	breq12d 5087	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)) ↔ 0 ≤ 0))
181	4, 180	mpbird 256	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)))
182		oveq2 7283	. . . . . . . . 9 ⊢ (𝑚 = 1 → (1...𝑚) = (1...1))
183	1	nnzi 12344	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
184		fzsn 13298	. . . . . . . . . . 11 ⊢ (1 ∈ ℤ → (1...1) = {1})
185	183, 184	ax-mp 5	. . . . . . . . . 10 ⊢ (1...1) = {1}
186	185	a1i 11	. . . . . . . . 9 ⊢ (𝑚 = 1 → (1...1) = {1})
187	182, 186	eqtrd 2778	. . . . . . . 8 ⊢ (𝑚 = 1 → (1...𝑚) = {1})
188	187	sumeq1d 15413	. . . . . . 7 ⊢ (𝑚 = 1 → Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) = Σ𝑗 ∈ {1} (𝑃‘𝑗))
189	188	oveq2d 7291	. . . . . 6 ⊢ (𝑚 = 1 → ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) = ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)))
190	189	breq2d 5086	. . . . 5 ⊢ (𝑚 = 1 → (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) ↔ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗))))
191	190	rspcev 3561	. . . 4 ⊢ ((1 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗))) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
192	2, 181, 191	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
193		simpl 483	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 = ∅) → 𝜑)
194		neqne 2951	. . . . 5 ⊢ (¬ 𝑌 = ∅ → 𝑌 ≠ ∅)
195	194	adantl 482	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 = ∅) → 𝑌 ≠ ∅)
196		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑌 ≠ ∅)
197	183	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 1 ∈ ℤ)
198		nnuz 12621	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
199	114	adantlr 712	. . . . . 6 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,)+∞))
200		hoidmvlelem3.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴:𝑊⟶ℝ)
201	66	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ⊆ 𝑊)
202	200, 201	fssresd 6641	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ↾ 𝑌):𝑌⟶ℝ)
203		hoidmvlelem3.b	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵:𝑊⟶ℝ)
204	203, 201	fssresd 6641	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ↾ 𝑌):𝑌⟶ℝ)
205	18, 58, 202, 204	hoidmvcl 44120	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ∈ (0[,)+∞))
206	28, 205	sselid 3919	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ∈ ℝ)
207	5, 206	eqeltrid 2843	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ℝ)
208		0red 10978	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ)
209		1rp 12734	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ⁺
210	209	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ⁺)
211	210, 172	jca 512	. . . . . . . . . . 11 ⊢ (𝜑 → (1 ∈ ℝ⁺ ∧ 𝐸 ∈ ℝ⁺))
212		rpaddcl 12752	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ⁺ ∧ 𝐸 ∈ ℝ⁺) → (1 + 𝐸) ∈ ℝ⁺)
213	211, 212	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (1 + 𝐸) ∈ ℝ⁺)
214		rpgt0 12742	. . . . . . . . . 10 ⊢ ((1 + 𝐸) ∈ ℝ⁺ → 0 < (1 + 𝐸))
215	213, 214	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 0 < (1 + 𝐸))
216	208, 215	gtned 11110	. . . . . . . 8 ⊢ (𝜑 → (1 + 𝐸) ≠ 0)
217	207, 174, 216	redivcld 11803	. . . . . . 7 ⊢ (𝜑 → (𝐺 / (1 + 𝐸)) ∈ ℝ)
218	217	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 / (1 + 𝐸)) ∈ ℝ)
219	217	ltpnfd 12857	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 / (1 + 𝐸)) < +∞)
220	219	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) < +∞)
221		id 22	. . . . . . . . . . 11 ⊢ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞ → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞)
222	221	eqcomd 2744	. . . . . . . . . 10 ⊢ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞ → +∞ = (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
223	222	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → +∞ = (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
224	220, 223	breqtrd 5100	. . . . . . . 8 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) < (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
225	224	adantlr 712	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) < (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
226		simpl 483	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝜑 ∧ 𝑌 ≠ ∅))
227		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞)
228		nnex 11979	. . . . . . . . . . . 12 ⊢ ℕ ∈ V
229	228	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → ℕ ∈ V)
230		icossicc 13168	. . . . . . . . . . . . . 14 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
231	230, 114	sselid 3919	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,]+∞))
232		eqid 2738	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)) = (𝑗 ∈ ℕ ↦ (𝑃‘𝑗))
233	231, 232	fmptd 6988	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)):ℕ⟶(0[,]+∞))
234	233	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)):ℕ⟶(0[,]+∞))
235	229, 234	sge0repnf 43924	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ ↔ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞))
236	227, 235	mpbird 256	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ)
237	236	adantlr 712	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ)
238	218	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → (𝐺 / (1 + 𝐸)) ∈ ℝ)
239	207	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → 𝐺 ∈ ℝ)
240	239	adantlr 712	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → 𝐺 ∈ ℝ)
241		simpr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ)
242	27, 172	ltaddrpd 12805	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 < (1 + 𝐸))
243	242	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 1 < (1 + 𝐸))
244	58	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝑌 ∈ Fin)
245		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝑌 ≠ ∅)
246	202	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐴 ↾ 𝑌):𝑌⟶ℝ)
247	204	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐵 ↾ 𝑌):𝑌⟶ℝ)
248	18, 244, 245, 246, 247	hoidmvn0val 44122	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))))
249	5	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)))
250		fvres 6793	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑌 → ((𝐴 ↾ 𝑌)‘𝑘) = (𝐴‘𝑘))
251		fvres 6793	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑌 → ((𝐵 ↾ 𝑌)‘𝑘) = (𝐵‘𝑘))
252	250, 251	oveq12d 7293	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ 𝑌 → (((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
253	252	fveq2d 6778	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ 𝑌 → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
254	253	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
255	200	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐴:𝑊⟶ℝ)
256		elun1 4110	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑌 → 𝑘 ∈ (𝑌 ∪ {𝑍}))
257	256, 44	eleqtrrdi 2850	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑌 → 𝑘 ∈ 𝑊)
258	257	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑊)
259	255, 258	ffvelrnd 6962	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐴‘𝑘) ∈ ℝ)
260	203	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐵:𝑊⟶ℝ)
261	260, 258	ffvelrnd 6962	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐵‘𝑘) ∈ ℝ)
262		volico 43524	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0))
263	259, 261, 262	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0))
264		hoidmvlelem3.lt	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴‘𝑘) < (𝐵‘𝑘))
265	258, 264	syldan 591	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐴‘𝑘) < (𝐵‘𝑘))
266	265	iftrued 4467	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
267	254, 263, 266	3eqtrd 2782	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
268	267	prodeq2dv 15633	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)))
269	268	eqcomd 2744	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))))
270	269	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))))
271	248, 249, 270	3eqtr4d 2788	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 = ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)))
272		difrp 12768	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ⁺))
273	259, 261, 272	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ⁺))
274	265, 273	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ⁺)
275	58, 274	fprodrpcl 15666	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ⁺)
276	275	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ⁺)
277	271, 276	eqeltrd 2839	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 ∈ ℝ⁺)
278	213	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (1 + 𝐸) ∈ ℝ⁺)
279	277, 278	ltdivgt1 42895	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (1 < (1 + 𝐸) ↔ (𝐺 / (1 + 𝐸)) < 𝐺))
280	243, 279	mpbid 231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 / (1 + 𝐸)) < 𝐺)
281	280	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → (𝐺 / (1 + 𝐸)) < 𝐺)
282		hoidmvlelem3.i2	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
283	282	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
284		fvexd 6789	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑥‘𝑘) ∈ V)
285		hoidmvlelem3.s	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑆 ∈ 𝑈)
286	285	elexd 3452	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑆 ∈ V)
287	284, 286	ifcld 4505	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
288	287	ralrimivw 3104	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ∀𝑘 ∈ 𝑊 if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
289	288	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∀𝑘 ∈ 𝑊 if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
290		eqid 2738	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
291	290	fnmpt 6573	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑘 ∈ 𝑊 if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) Fn 𝑊)
292	289, 291	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) Fn 𝑊)
293		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
294		mptexg 7097	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑊 ∈ Fin → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) ∈ V)
295	53, 294	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) ∈ V)
296	295	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) ∈ V)
297		hoidmvlelem3.o	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑂 = (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↦ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))
298	297	fvmpt2 6886	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) ∈ V) → (𝑂‘𝑥) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))
299	293, 296, 298	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))
300	299	fneq1d 6526	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥) Fn 𝑊 ↔ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) Fn 𝑊))
301	292, 300	mpbird 256	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) Fn 𝑊)
302		nfv 1917	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘𝜑
303		nfcv 2907	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑘𝑥
304		nfixp1 8706	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑘X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
305	303, 304	nfel 2921	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
306	302, 305	nfan 1902	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
307	299	fveq1d 6776	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥)‘𝑘) = ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘))
308	307	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) = ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘))
309		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑊)
310	287	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
311	290	fvmpt2 6886	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑘 ∈ 𝑊 ∧ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
312	309, 310, 311	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
313	312	adantlr 712	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
314	308, 313	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
315		iftrue 4465	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ 𝑌 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = (𝑥‘𝑘))
316	315	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = (𝑥‘𝑘))
317		vex 3436	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 𝑥 ∈ V
318	317	elixp 8692	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↔ (𝑥 Fn 𝑌 ∧ ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))))
319	318	simprbi 497	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) → ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
320	319	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
321		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑌)
322		rspa 3132	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
323	320, 321, 322	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
324	323	ad4ant24 751	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
325	316, 324	eqeltrd 2839	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
326		snidg 4595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑍 ∈ (𝑋 ∖ 𝑌) → 𝑍 ∈ {𝑍})
327	47, 326	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → 𝑍 ∈ {𝑍})
328		elun2 4111	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑌 ∪ {𝑍}))
329	327, 328	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → 𝑍 ∈ (𝑌 ∪ {𝑍}))
330	55	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → (𝑌 ∪ {𝑍}) = 𝑊)
331	329, 330	eleqtrd 2841	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
332	200, 331	ffvelrnd 6962	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ)
333	332	rexrd 11025	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ^*)
334	203, 331	ffvelrnd 6962	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ)
335	334	rexrd 11025	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ^*)
336		iccssxr 13162	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ⊆ ℝ^*
337		hoidmvlelem3.u	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))}
338		ssrab2 4013	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ {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))
339	337, 338	eqsstri 3955	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍))
340	339, 285	sselid 3919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
341	336, 340	sselid 3919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝑆 ∈ ℝ^*)
342		iccgelb 13135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐴‘𝑍) ∈ ℝ^* ∧ (𝐵‘𝑍) ∈ ℝ^* ∧ 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → (𝐴‘𝑍) ≤ 𝑆)
343	333, 335, 340, 342	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → (𝐴‘𝑍) ≤ 𝑆)
344		hoidmvlelem3.sb	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝑆 < (𝐵‘𝑍))
345	333, 335, 341, 343, 344	elicod 13129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
346	345	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → 𝑆 ∈ ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
347		iffalse 4468	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ 𝑘 ∈ 𝑌 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = 𝑆)
348	347	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = 𝑆)
349	44	eleq2i 2830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑘 ∈ 𝑊 ↔ 𝑘 ∈ (𝑌 ∪ {𝑍}))
350	349	biimpi 215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑘 ∈ 𝑊 → 𝑘 ∈ (𝑌 ∪ {𝑍}))
351	350	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 ∈ (𝑌 ∪ {𝑍}))
352		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → ¬ 𝑘 ∈ 𝑌)
353		elunnel1 4084	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑘 ∈ (𝑌 ∪ {𝑍}) ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 ∈ {𝑍})
354	351, 352, 353	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 ∈ {𝑍})
355		elsni 4578	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 ∈ {𝑍} → 𝑘 = 𝑍)
356	354, 355	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 = 𝑍)
357		fveq2 6774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 = 𝑍 → (𝐴‘𝑘) = (𝐴‘𝑍))
358		fveq2 6774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 = 𝑍 → (𝐵‘𝑘) = (𝐵‘𝑍))
359	357, 358	oveq12d 7293	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 = 𝑍 → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
360	356, 359	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
361	360	adantll 711	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
362	348, 361	eleq12d 2833	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → (if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↔ 𝑆 ∈ ((𝐴‘𝑍)[,)(𝐵‘𝑍))))
363	346, 362	mpbird 256	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
364	363	adantllr 716	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
365	325, 364	pm2.61dan 810	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
366	314, 365	eqeltrd 2839	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
367	366	ex 413	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑘 ∈ 𝑊 → ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))))
368	306, 367	ralrimi 3141	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
369	301, 368	jca 512	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥) Fn 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))))
370		fvex 6787	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑂‘𝑥) ∈ V
371	370	elixp 8692	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↔ ((𝑂‘𝑥) Fn 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))))
372	369, 371	sylibr 233	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
373	283, 372	sseldd 3922	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
374		eliun 4928	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑂‘𝑥) ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ↔ ∃𝑗 ∈ ℕ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
375	373, 374	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∃𝑗 ∈ ℕ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
376		ixpfn 8691	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) → 𝑥 Fn 𝑌)
377	376	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑥 Fn 𝑌)
378	377	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑥 Fn 𝑌)
379		nfv 1917	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑘 𝑗 ∈ ℕ
380	306, 379	nfan 1902	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ)
381		nfcv 2907	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑘(𝑂‘𝑥)
382		nfixp1 8706	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑘X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))
383	381, 382	nfel 2921	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘(𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))
384	380, 383	nfan 1902	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
385	307	3adant3 1131	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ((𝑂‘𝑥)‘𝑘) = ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘))
386	287	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
387	258, 386, 311	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
388	387	3adant2 1130	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
389	315	3ad2ant3 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = (𝑥‘𝑘))
390	385, 388, 389	3eqtrrd 2783	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) = ((𝑂‘𝑥)‘𝑘))
391	390	ad5ant125 1365	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) = ((𝑂‘𝑥)‘𝑘))
392		simpl 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
393	370	elixp 8692	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ↔ ((𝑂‘𝑥) Fn 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
394	392, 393	sylib 217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ((𝑂‘𝑥) Fn 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
395	394	simprd 496	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
396	257	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑊)
397		rspa 3132	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
398	395, 396, 397	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
399	398	adantll 711	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
400	391, 399	eqeltrd 2839	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
401	29	ad3antrrr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝜑)
402	37	ad2antlr 724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑗 ∈ ℕ)
403	299	fveq1d 6776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥)‘𝑍) = ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑍))
404		eqidd 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))
405		eleq1 2826	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑘 = 𝑍 → (𝑘 ∈ 𝑌 ↔ 𝑍 ∈ 𝑌))
406		fveq2 6774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑘 = 𝑍 → (𝑥‘𝑘) = (𝑥‘𝑍))
407	405, 406	ifbieq1d 4483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑘 = 𝑍 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆))
408	407	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ 𝑘 = 𝑍) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆))
409		fvexd 6789	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → (𝑥‘𝑍) ∈ V)
410	409, 286	ifcld 4505	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆) ∈ V)
411	404, 408, 331, 410	fvmptd 6882	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑍) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆))
412	411	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑍) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆))
413	47	eldifbd 3900	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
414	413	iffalsed 4470	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆) = 𝑆)
415	414	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆) = 𝑆)
416	403, 412, 415	3eqtrrd 2783	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑆 = ((𝑂‘𝑥)‘𝑍))
417	416	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑆 = ((𝑂‘𝑥)‘𝑍))
418	401, 331	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑍 ∈ 𝑊)
419	393	simprbi 497	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
420	419	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
421		fveq2 6774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑘 = 𝑍 → ((𝑂‘𝑥)‘𝑘) = ((𝑂‘𝑥)‘𝑍))
422		fveq2 6774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑘 = 𝑍 → ((𝐶‘𝑗)‘𝑘) = ((𝐶‘𝑗)‘𝑍))
423		fveq2 6774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑘 = 𝑍 → ((𝐷‘𝑗)‘𝑘) = ((𝐷‘𝑗)‘𝑍))
424	422, 423	oveq12d 7293	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑘 = 𝑍 → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
425	421, 424	eleq12d 2833	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑘 = 𝑍 → (((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ↔ ((𝑂‘𝑥)‘𝑍) ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))))
426	425	rspcva 3559	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑍 ∈ 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ((𝑂‘𝑥)‘𝑍) ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
427	418, 420, 426	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ((𝑂‘𝑥)‘𝑍) ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
428	417, 427	eqeltrd 2839	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
429	149	3adant3 1131	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
430	60	3ad2ant3 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((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))
431	429, 430	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐽‘𝑗) = ((𝐶‘𝑗) ↾ 𝑌))
432	431	fveq1d 6776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐽‘𝑗)‘𝑘) = (((𝐶‘𝑗) ↾ 𝑌)‘𝑘))
433	401, 402, 428, 432	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ((𝐽‘𝑗)‘𝑘) = (((𝐶‘𝑗) ↾ 𝑌)‘𝑘))
434	433	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → ((𝐽‘𝑗)‘𝑘) = (((𝐶‘𝑗) ↾ 𝑌)‘𝑘))
435		fvres 6793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ 𝑌 → (((𝐶‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
436	435	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (((𝐶‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
437	434, 436	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → ((𝐽‘𝑗)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
438	107	elexd 3452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
439	108	fvmpt2 6886	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((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))
440	139, 438, 439	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
441	440	3adant3 1131	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
442	93	3ad2ant3 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((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))
443	441, 442	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐾‘𝑗) = ((𝐷‘𝑗) ↾ 𝑌))
444	443	fveq1d 6776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐾‘𝑗)‘𝑘) = (((𝐷‘𝑗) ↾ 𝑌)‘𝑘))
445	401, 402, 428, 444	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ((𝐾‘𝑗)‘𝑘) = (((𝐷‘𝑗) ↾ 𝑌)‘𝑘))
446	445	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → ((𝐾‘𝑗)‘𝑘) = (((𝐷‘𝑗) ↾ 𝑌)‘𝑘))
447		fvres 6793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ 𝑌 → (((𝐷‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
448	447	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (((𝐷‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
449	446, 448	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → ((𝐾‘𝑗)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
450	437, 449	oveq12d 7293	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
451	450	eqcomd 2744	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
452	400, 451	eleqtrd 2841	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
453	452	ex 413	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → (𝑘 ∈ 𝑌 → (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
454	384, 453	ralrimi 3141	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
455	378, 454	jca 512	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → (𝑥 Fn 𝑌 ∧ ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
456	317	elixp 8692	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) ↔ (𝑥 Fn 𝑌 ∧ ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
457	455, 456	sylibr 233	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
458	457	ex 413	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) → ((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
459	458	reximdva 3203	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (∃𝑗 ∈ ℕ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → ∃𝑗 ∈ ℕ 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
460	375, 459	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∃𝑗 ∈ ℕ 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
461		eliun 4928	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) ↔ ∃𝑗 ∈ ℕ 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
462	460, 461	sylibr 233	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑥 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
463	462	ralrimiva 3103	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ X 𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))𝑥 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
464		dfss3 3909	. . . . . . . . . . . . . . 15 ⊢ (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) ↔ ∀𝑥 ∈ X 𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))𝑥 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
465	463, 464	sylibr 233	. . . . . . . . . . . . . 14 ⊢ (𝜑 → X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
466		ovexd 7310	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℝ ↑_m 𝑌) ∈ V)
467	228	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ℕ ∈ V)
468	466, 467	elmapd 8629	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐾 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ) ↔ 𝐾:ℕ⟶(ℝ ↑_m 𝑌)))
469	109, 468	mpbird 256	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ))
470	466, 467	elmapd 8629	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐽 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ) ↔ 𝐽:ℕ⟶(ℝ ↑_m 𝑌)))
471	89, 470	mpbird 256	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐽 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ))
472	82, 71	elmapd 8629	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝐵 ↾ 𝑌) ∈ (ℝ ↑_m 𝑌) ↔ (𝐵 ↾ 𝑌):𝑌⟶ℝ))
473	204, 472	mpbird 256	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 ↾ 𝑌) ∈ (ℝ ↑_m 𝑌))
474	82, 71	elmapd 8629	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐴 ↾ 𝑌) ∈ (ℝ ↑_m 𝑌) ↔ (𝐴 ↾ 𝑌):𝑌⟶ℝ))
475	202, 474	mpbird 256	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 ↾ 𝑌) ∈ (ℝ ↑_m 𝑌))
476		hoidmvlelem3.i	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑒 ∈ (ℝ ↑_m 𝑌)∀𝑓 ∈ (ℝ ↑_m 𝑌)∀𝑔 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)∀ℎ ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
477		fveq1 6773	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑒 = (𝐴 ↾ 𝑌) → (𝑒‘𝑘) = ((𝐴 ↾ 𝑌)‘𝑘))
478	477	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑒‘𝑘) = ((𝐴 ↾ 𝑌)‘𝑘))
479	250	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝐴 ↾ 𝑌)‘𝑘) = (𝐴‘𝑘))
480	478, 479	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑒‘𝑘) = (𝐴‘𝑘))
481	480	oveq1d 7290	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = ((𝐴‘𝑘)[,)(𝑓‘𝑘)))
482	481	ixpeq2dva 8700	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒 = (𝐴 ↾ 𝑌) → X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)))
483	482	sseq1d 3952	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 = (𝐴 ↾ 𝑌) → (X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘))))
484		oveq1 7282	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒 = (𝐴 ↾ 𝑌) → (𝑒(𝐿‘𝑌)𝑓) = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓))
485	484	breq1d 5084	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (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 345	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑒 = (𝐴 ↾ 𝑌) → ((X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
487	486	ralbidv 3112	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑒 = (𝐴 ↾ 𝑌) → (∀ℎ ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
488	487	ralbidv 3112	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑒 = (𝐴 ↾ 𝑌) → (∀𝑔 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)∀ℎ ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
489	488	ralbidv 3112	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = (𝐴 ↾ 𝑌) → (∀𝑓 ∈ (ℝ ↑_m 𝑌)∀𝑔 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)∀ℎ ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
490	489	rspcva 3559	. . . . . . . . . . . . . . . . . 18 ⊢ (((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)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
491	475, 476, 490	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑓 ∈ (ℝ ↑_m 𝑌)∀𝑔 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)∀ℎ ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
492		fveq1 6773	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 = (𝐵 ↾ 𝑌) → (𝑓‘𝑘) = ((𝐵 ↾ 𝑌)‘𝑘))
493	492	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑓‘𝑘) = ((𝐵 ↾ 𝑌)‘𝑘))
494	251	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝐵 ↾ 𝑌)‘𝑘) = (𝐵‘𝑘))
495	493, 494	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑓‘𝑘) = (𝐵‘𝑘))
496	495	oveq2d 7291	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘)[,)(𝑓‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
497	496	ixpeq2dva 8700	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = (𝐵 ↾ 𝑌) → X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) = X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
498	497	sseq1d 3952	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = (𝐵 ↾ 𝑌) → (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘))))
499		oveq2 7283	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = (𝐵 ↾ 𝑌) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)))
500	499	breq1d 5084	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (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 345	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = (𝐵 ↾ 𝑌) → ((X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 3112	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝐵 ↾ 𝑌) → (∀ℎ ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 3112	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝐵 ↾ 𝑌) → (∀𝑔 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)∀ℎ ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 3559	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ↾ 𝑌) ∈ (ℝ ↑_m 𝑌) ∧ ∀𝑓 ∈ (ℝ ↑_m 𝑌)∀𝑔 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)∀ℎ ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
505	473, 491, 504	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑔 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)∀ℎ ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
506		fveq1 6773	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑔 = 𝐽 → (𝑔‘𝑗) = (𝐽‘𝑗))
507	506	fveq1d 6776	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑔 = 𝐽 → ((𝑔‘𝑗)‘𝑘) = ((𝐽‘𝑗)‘𝑘))
508	507	oveq1d 7290	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔 = 𝐽 → (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))
509	508	ixpeq2dv 8701	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = 𝐽 → X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))
510	509	iuneq2d 4953	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = 𝐽 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))
511	510	sseq2d 3953	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝐽 → (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘))))
512	506	oveq1d 7290	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔 = 𝐽 → ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)) = ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))
513	512	mpteq2dv 5176	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = 𝐽 → (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))
514	513	fveq2d 6778	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = 𝐽 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))
515	514	breq2d 5086	. . . . . . . . . . . . . . . . . . 19 ⊢ (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 345	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = 𝐽 → ((X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 3112	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝐽 → (∀ℎ ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 3559	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ) ∧ ∀𝑔 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)∀ℎ ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
519	471, 505, 518	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀ℎ ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
520		fveq1 6773	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ = 𝐾 → (ℎ‘𝑗) = (𝐾‘𝑗))
521	520	fveq1d 6776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ = 𝐾 → ((ℎ‘𝑗)‘𝑘) = ((𝐾‘𝑗)‘𝑘))
522	521	oveq2d 7291	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = 𝐾 → (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
523	522	ixpeq2dv 8701	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = 𝐾 → X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
524	523	iuneq2d 4953	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = 𝐾 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
525	524	sseq2d 3953	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = 𝐾 → (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
526	520	oveq2d 7291	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = 𝐾 → ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
527	526	mpteq2dv 5176	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = 𝐾 → (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))
528	527	fveq2d 6778	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = 𝐾 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
529	528	breq2d 5086	. . . . . . . . . . . . . . . . 17 ⊢ (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 345	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = 𝐾 → ((X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 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 3559	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ) ∧ ∀ℎ ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) → (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))))
532	469, 519, 531	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))))
533	465, 532	mpd 15	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
534		idd 24	. . . . . . . . . . . . 13 ⊢ (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 15	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
536	535	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
537	41	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
538	537	mpteq2dva 5174	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)) = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))
539	538	fveq2d 6778	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
540	249, 539	breq12d 5087	. . . . . . . . . . 11 ⊢ ((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 256	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
542	541	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → 𝐺 ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
543	238, 240, 241, 281, 542	ltletrd 11135	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → (𝐺 / (1 + 𝐸)) < (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
544	226, 237, 543	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) < (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
545	225, 544	pm2.61dan 810	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 / (1 + 𝐸)) < (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
546	196, 197, 198, 199, 218, 545	sge0uzfsumgt 43982	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ∃𝑚 ∈ ℕ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))
547	217	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (𝐺 / (1 + 𝐸)) ∈ ℝ)
548		fzfid 13693	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...𝑚) ∈ Fin)
549		simpl 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → 𝜑)
550		elfznn 13285	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (1...𝑚) → 𝑗 ∈ ℕ)
551	550	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → 𝑗 ∈ ℕ)
552	28, 114	sselid 3919	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ ℝ)
553	549, 551, 552	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → (𝑃‘𝑗) ∈ ℝ)
554	548, 553	fsumrecl 15446	. . . . . . . . . . . 12 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) ∈ ℝ)
555	554	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) ∈ ℝ)
556		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))
557	547, 555, 556	ltled 11123	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (𝐺 / (1 + 𝐸)) ≤ Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))
558	207	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ∈ ℝ)
559	213	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (1 + 𝐸) ∈ ℝ⁺)
560	558, 555, 559	ledivmuld 12825	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → ((𝐺 / (1 + 𝐸)) ≤ Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) ↔ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
561	557, 560	mpbid 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
562	561	ex 413	. . . . . . . 8 ⊢ (𝜑 → ((𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
563	562	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
564	563	adantlr 712	. . . . . 6 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ 𝑚 ∈ ℕ) → ((𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
565	564	reximdva 3203	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (∃𝑚 ∈ ℕ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
566	546, 565	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
567	193, 195, 566	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 = ∅) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
568	192, 567	pm2.61dan 810	. 2 ⊢ (𝜑 → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
569	43	3ad2ant1 1132	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑋 ∈ Fin)
570	46	3ad2ant1 1132	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑌 ⊆ 𝑋)
571	47	3ad2ant1 1132	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑍 ∈ (𝑋 ∖ 𝑌))
572	200	3ad2ant1 1132	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐴:𝑊⟶ℝ)
573	203	3ad2ant1 1132	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐵:𝑊⟶ℝ)
574	62	3ad2ant1 1132	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐶:ℕ⟶(ℝ ↑_m 𝑊))
575		eqid 2738	. . . . 5 ⊢ (𝑦 ∈ 𝑌 ↦ 0) = (𝑦 ∈ 𝑌 ↦ 0)
576		eqid 2738	. . . . 5 ⊢ (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	95	3ad2ant1 1132	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐷:ℕ⟶(ℝ ↑_m 𝑊))
578		eqid 2738	. . . . 5 ⊢ (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 6774	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝐶‘𝑖) = (𝐶‘𝑗))
580		fveq2 6774	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗))
581	579, 580	oveq12d 7293	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)) = ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))
582	581	cbvmptv 5187	. . . . . . . 8 ⊢ (𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))
583	582	fveq2i 6777	. . . . . . 7 ⊢ (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))
584		hoidmvlelem3.r	. . . . . . 7 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
585	583, 584	eqeltrid 2843	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ)
586	585	3ad2ant1 1132	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ)
587		hoidmvlelem3.h	. . . . . 6 ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))
588		eleq1w 2821	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ 𝑌 ↔ 𝑖 ∈ 𝑌))
589		fveq2 6774	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝑐‘𝑗) = (𝑐‘𝑖))
590	589	breq1d 5084	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → ((𝑐‘𝑗) ≤ 𝑥 ↔ (𝑐‘𝑖) ≤ 𝑥))
591	590, 589	ifbieq1d 4483	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥) = if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥))
592	588, 589, 591	ifbieq12d 4487	. . . . . . . . 9 ⊢ (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 5187	. . . . . . . 8 ⊢ (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 5179	. . . . . . 7 ⊢ (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 5179	. . . . . 6 ⊢ (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	587, 595	eqtri 2766	. . . . 5 ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥)))))
597	172	3ad2ant1 1132	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐸 ∈ ℝ⁺)
598		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → (𝐶‘𝑗) = (𝐶‘𝑖))
599		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑖 → (𝐷‘𝑗) = (𝐷‘𝑖))
600	599	fveq2d 6778	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘𝑧)‘(𝐷‘𝑖)))
601	598, 600	oveq12d 7293	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))
602	601	cbvmptv 5187	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))) = (𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))
603	602	fveq2i 6777	. . . . . . . . 9 ⊢ (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 7286	. . . . . . . 8 ⊢ ((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 5082	. . . . . . 7 ⊢ ((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 3408	. . . . . 6 ⊢ {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	337, 606	eqtri 2766	. . . . 5 ⊢ 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))))}
608	285	3ad2ant1 1132	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑆 ∈ 𝑈)
609	344	3ad2ant1 1132	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑆 < (𝐵‘𝑍))
610		eqid 2738	. . . . 5 ⊢ (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 1136	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑚 ∈ ℕ)
612		id 22	. . . . . . . 8 ⊢ (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
613		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → (𝑃‘𝑗) = (𝑃‘𝑖))
614	613	cbvsumv 15408	. . . . . . . . . 10 ⊢ Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) = Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)
615	614	oveq2i 7286	. . . . . . . . 9 ⊢ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) = ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖))
616	615	a1i 11	. . . . . . . 8 ⊢ (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) = ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)))
617	612, 616	breqtrd 5100	. . . . . . 7 ⊢ (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)))
618	617	3ad2ant3 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)))
619		simpl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑚)) → 𝜑)
620		elfznn 13285	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...𝑚) → 𝑖 ∈ ℕ)
621	620	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ∈ ℕ)
622		eleq1w 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ ℕ ↔ 𝑖 ∈ ℕ))
623		fveq2 6774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑖 → (𝐽‘𝑗) = (𝐽‘𝑖))
624		fveq2 6774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑖 → (𝐾‘𝑗) = (𝐾‘𝑖))
625	623, 624	oveq12d 7293	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑖 → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)))
626	613, 625	eqeq12d 2754	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → ((𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ↔ (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖))))
627	622, 626	imbi12d 345	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑖 → ((𝑗 ∈ ℕ → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))) ↔ (𝑖 ∈ ℕ → (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)))))
628	627, 41	chvarvv 2002	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ℕ → (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)))
629	628	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)))
630	622	anbi2d 629	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → ((𝜑 ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ 𝑖 ∈ ℕ)))
631	598	fveq1d 6776	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑖 → ((𝐶‘𝑗)‘𝑍) = ((𝐶‘𝑖)‘𝑍))
632	599	fveq1d 6776	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑖 → ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑖)‘𝑍))
633	631, 632	oveq12d 7293	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑖 → (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) = (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
634	633	eleq2d 2824	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑖 → (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))))
635	598	reseq1d 5890	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑖 → ((𝐶‘𝑗) ↾ 𝑌) = ((𝐶‘𝑖) ↾ 𝑌))
636	634, 635	ifbieq1d 4483	. . . . . . . . . . . . . . . 16 ⊢ (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 2754	. . . . . . . . . . . . . . 15 ⊢ (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 345	. . . . . . . . . . . . . 14 ⊢ (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, 149	chvarvv 2002	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐽‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹))
640	599	reseq1d 5890	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑖 → ((𝐷‘𝑗) ↾ 𝑌) = ((𝐷‘𝑖) ↾ 𝑌))
641	634, 640	ifbieq1d 4483	. . . . . . . . . . . . . . . 16 ⊢ (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 2754	. . . . . . . . . . . . . . 15 ⊢ (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 345	. . . . . . . . . . . . . 14 ⊢ (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, 440	chvarvv 2002	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐾‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹))
645	639, 644	oveq12d 7293	. . . . . . . . . . . 12 ⊢ ((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 2778	. . . . . . . . . . 11 ⊢ ((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 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
648		ovexd 7310	. . . . . . . . . . . . 13 ⊢ ((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 6886	. . . . . . . . . . . . 13 ⊢ ((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 584	. . . . . . . . . . . 12 ⊢ ((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 6787	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶‘𝑖) ∈ V
652	651	resex 5939	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶‘𝑖) ↾ 𝑌) ∈ V
653	652	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐶‘𝑖) ↾ 𝑌) ∈ V)
654	80, 143	eqeltrrid 2844	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ 0) ∈ V)
655	653, 654	ifcld 4505	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V)
656	655	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V)
657	576	fvmpt2 6886	. . . . . . . . . . . . . . 15 ⊢ ((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 584	. . . . . . . . . . . . . 14 ⊢ ((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	80	eqcomi 2747	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑌 ↦ 0) = 𝐹
660		ifeq2 4464	. . . . . . . . . . . . . . . 16 ⊢ ((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 5	. . . . . . . . . . . . . . 15 ⊢ 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 11	. . . . . . . . . . . . . 14 ⊢ ((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 2778	. . . . . . . . . . . . 13 ⊢ ((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 6787	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷‘𝑖) ∈ V
665	664	resex 5939	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷‘𝑖) ↾ 𝑌) ∈ V
666	665	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐷‘𝑖) ↾ 𝑌) ∈ V)
667	666, 654	ifcld 4505	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V)
668	667	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V)
669	578	fvmpt2 6886	. . . . . . . . . . . . . . 15 ⊢ ((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 584	. . . . . . . . . . . . . 14 ⊢ ((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 260	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
672	671, 659	ifbieq2i 4484	. . . . . . . . . . . . . . 15 ⊢ 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 11	. . . . . . . . . . . . . 14 ⊢ ((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 2778	. . . . . . . . . . . . 13 ⊢ ((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 7293	. . . . . . . . . . . 12 ⊢ ((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 2778	. . . . . . . . . . 11 ⊢ ((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 2781	. . . . . . . . . 10 ⊢ ((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 584	. . . . . . . . 9 ⊢ ((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 1184	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (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 15415	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖) = Σ𝑖 ∈ (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 7291	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)) = ((1 + 𝐸) · Σ𝑖 ∈ (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 5100	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑖 ∈ (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 6774	. . . . . . . 8 ⊢ (𝑗 = ℎ → (𝐷‘𝑗) = (𝐷‘ℎ))
684	683	fveq1d 6776	. . . . . . 7 ⊢ (𝑗 = ℎ → ((𝐷‘𝑗)‘𝑍) = ((𝐷‘ℎ)‘𝑍))
685	684	cbvmptv 5187	. . . . . 6 ⊢ (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 5846	. . . . 5 ⊢ 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 6774	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑖 → (𝐶‘ℎ) = (𝐶‘𝑖))
688	687	fveq1d 6776	. . . . . . . . . . 11 ⊢ (ℎ = 𝑖 → ((𝐶‘ℎ)‘𝑍) = ((𝐶‘𝑖)‘𝑍))
689		fveq2 6774	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑖 → (𝐷‘ℎ) = (𝐷‘𝑖))
690	689	fveq1d 6776	. . . . . . . . . . 11 ⊢ (ℎ = 𝑖 → ((𝐷‘ℎ)‘𝑍) = ((𝐷‘𝑖)‘𝑍))
691	688, 690	oveq12d 7293	. . . . . . . . . 10 ⊢ (ℎ = 𝑖 → (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍)) = (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
692	691	eleq2d 2824	. . . . . . . . 9 ⊢ (ℎ = 𝑖 → (𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))))
693	692	cbvrabv 3426	. . . . . . . 8 ⊢ {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} = {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))}
694	693	mpteq1i 5170	. . . . . . 7 ⊢ (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 5846	. . . . . 6 ⊢ 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 4094	. . . . 5 ⊢ ({(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 2738	. . . . 5 ⊢ 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	18, 569, 570, 571, 44, 572, 573, 574, 575, 576, 577, 578, 586, 596, 5, 597, 607, 608, 609, 610, 611, 682, 686, 696, 697	hoidmvlelem2 44134	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
699	698	3exp 1118	. . 3 ⊢ (𝜑 → (𝑚 ∈ ℕ → (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)))
700	699	rexlimdv 3212	. 2 ⊢ (𝜑 → (∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢))
701	568, 700	mpd 15	1 ⊢ (𝜑 → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 {crab 3068 Vcvv 3432 ∖ cdif 3884 ∪ cun 3885 ⊆ wss 3887 ∅c0 4256 ifcif 4459 {csn 4561 ∪ ciun 4924 class class class wbr 5074 ↦ cmpt 5157 ran crn 5590 ↾ cres 5591 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 ↑_m cmap 8615 Xcixp 8685 Fincfn 8733 infcinf 9200 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 +∞cpnf 11006 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 − cmin 11205 / cdiv 11632 ℕcn 11973 ℤcz 12319 ℝ⁺crp 12730 [,)cico 13081 [,]cicc 13082 ...cfz 13239 Σcsu 15397 ∏cprod 15615 volcvol 24627 Σ^{^}csumge0 43900

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sum 15398 df-prod 15616 df-rest 17133 df-topgen 17154 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-top 22043 df-topon 22060 df-bases 22096 df-cmp 22538 df-ovol 24628 df-vol 24629 df-sumge0 43901

This theorem is referenced by: hoidmvlelem4 44136

W3C validator