hoidmvlelem3 - Mathbox for Glauco Siliprandi

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Theorem hoidmvlelem3 47048

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 12177	. . . . 5 ⊢ 1 ∈ ℕ
2	1	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 1 ∈ ℕ)
3		0le0 12274	. . . . . 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 6828	. . . . . . . . 9 ⊢ (𝑌 = ∅ → (𝐿‘𝑌) = (𝐿‘∅))
8		reseq2 5927	. . . . . . . . . 10 ⊢ (𝑌 = ∅ → (𝐴 ↾ 𝑌) = (𝐴 ↾ ∅))
9		res0 5936	. . . . . . . . . . 11 ⊢ (𝐴 ↾ ∅) = ∅
10	9	a1i 11	. . . . . . . . . 10 ⊢ (𝑌 = ∅ → (𝐴 ↾ ∅) = ∅)
11	8, 10	eqtrd 2774	. . . . . . . . 9 ⊢ (𝑌 = ∅ → (𝐴 ↾ 𝑌) = ∅)
12		reseq2 5927	. . . . . . . . . 10 ⊢ (𝑌 = ∅ → (𝐵 ↾ 𝑌) = (𝐵 ↾ ∅))
13		res0 5936	. . . . . . . . . . 11 ⊢ (𝐵 ↾ ∅) = ∅
14	13	a1i 11	. . . . . . . . . 10 ⊢ (𝑌 = ∅ → (𝐵 ↾ ∅) = ∅)
15	12, 14	eqtrd 2774	. . . . . . . . 9 ⊢ (𝑌 = ∅ → (𝐵 ↾ 𝑌) = ∅)
16	7, 11, 15	oveq123d 7378	. . . . . . . 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 6709	. . . . . . . . 9 ⊢ ∅:∅⟶ℝ
20	19	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ∅:∅⟶ℝ)
21	18, 20, 20	hoidmv0val 47034	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (∅(𝐿‘∅)∅) = 0)
22	6, 17, 21	3eqtrd 2778	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 𝐺 = 0)
23		nfcvd 2902	. . . . . . . . . . 11 ⊢ (𝜑 → Ⅎ𝑗(𝑃‘1))
24		nfv 1921	. . . . . . . . . . 11 ⊢ Ⅎ𝑗𝜑
25		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 = 1) → 𝑗 = 1)
26	25	fveq2d 6832	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 = 1) → (𝑃‘𝑗) = (𝑃‘1))
27		1red 11137	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℝ)
28		rge0ssre 13401	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ⊆ ℝ
29		id 22	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝜑)
30	1	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℕ)
31	1	elexi 3453	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
32		eleq1 2827	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 1 → (𝑗 ∈ ℕ ↔ 1 ∈ ℕ))
33	32	anbi2d 636	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 1 → ((𝜑 ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ 1 ∈ ℕ)))
34		fveq2 6828	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 1 → (𝑃‘𝑗) = (𝑃‘1))
35	34	eleq1d 2824	. . . . . . . . . . . . . . . 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 7392	. . . . . . . . . . . . . . . . . 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 6948	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ℕ ∧ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ V) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
41	37, 38, 40	syl2anc 590	. . . . . . . . . . . . . . . . 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 3895	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑍 ∈ 𝑋)
49		snssi 4718	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑍 ∈ 𝑋 → {𝑍} ⊆ 𝑋)
50	48, 49	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {𝑍} ⊆ 𝑋)
51	46, 50	unssd 4122	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑌 ∪ {𝑍}) ⊆ 𝑋)
52	45, 51	eqsstrd 3949	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑊 ⊆ 𝑋)
53	43, 52	ssfid 9170	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑊 ∈ Fin)
54		ssun1 4108	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑌 ⊆ (𝑌 ∪ {𝑍})
55	44	eqcomi 2748	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑌 ∪ {𝑍}) = 𝑊
56	54, 55	sseqtri 3963	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑌 ⊆ 𝑊
57	56	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑌 ⊆ 𝑊)
58	53, 57	ssfid 9170	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑌 ∈ Fin)
59	58	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ∈ Fin)
60		iftrue 4461	. . . . . . . . . . . . . . . . . . . . . . 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	ffvelcdmda 7026	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑_m 𝑊))
64		elmapi 8787	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3964	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑌 ⊆ 𝑊
67	66	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ⊆ 𝑊)
68	65, 67	fssresd 6695	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
69		reex 11121	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ℝ ∈ V
70	69	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ℝ ∈ V)
71	53, 57	ssexd 5253	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑌 ∈ V)
72	71	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ∈ V)
73	70, 72	elmapd 8778	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐶‘𝑗) ↾ 𝑌) ∈ (ℝ ↑_m 𝑌) ↔ ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ))
74	68, 73	mpbird 258	. . . . . . . . . . . . . . . . . . . . . . 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 4464	. . . . . . . . . . . . . . . . . . . . . . 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 11139	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ∈ ℝ)
80		hoidmvlelem3.f	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝐹 = (𝑦 ∈ 𝑌 ↦ 0)
81	79, 80	fmptd 7056	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐹:𝑌⟶ℝ)
82	69	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ℝ ∈ V)
83	82, 58	elmapd 8778	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝐹 ∈ (ℝ ↑_m 𝑌) ↔ 𝐹:𝑌⟶ℝ))
84	81, 83	mpbird 258	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐹 ∈ (ℝ ↑_m 𝑌))
85	84	ad2antrr 732	. . . . . . . . . . . . . . . . . . . . . 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 818	. . . . . . . . . . . . . . . . . . . 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 7056	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐽:ℕ⟶(ℝ ↑_m 𝑌))
90	89	ffvelcdmda 7026	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗) ∈ (ℝ ↑_m 𝑌))
91		elmapi 8787	. . . . . . . . . . . . . . . . . 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 4461	. . . . . . . . . . . . . . . . . . . . . . 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	ffvelcdmda 7026	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑_m 𝑊))
97		elmapi 8787	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6695	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
100	70, 72	elmapd 8778	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐷‘𝑗) ↾ 𝑌) ∈ (ℝ ↑_m 𝑌) ↔ ((𝐷‘𝑗) ↾ 𝑌):𝑌⟶ℝ))
101	99, 100	mpbird 258	. . . . . . . . . . . . . . . . . . . . . . 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 4464	. . . . . . . . . . . . . . . . . . . . . . 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 818	. . . . . . . . . . . . . . . . . . . 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 7056	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾:ℕ⟶(ℝ ↑_m 𝑌))
110	109	ffvelcdmda 7026	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗) ∈ (ℝ ↑_m 𝑌))
111		elmapi 8787	. . . . . . . . . . . . . . . . . 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 47033	. . . . . . . . . . . . . . . 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 3503	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 ∈ ℕ) → (𝑃‘1) ∈ (0[,)+∞))
116	29, 30, 115	syl2anc 590	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃‘1) ∈ (0[,)+∞))
117	28, 116	sselid 3913	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃‘1) ∈ ℝ)
118	117	recnd 11165	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃‘1) ∈ ℂ)
119	23, 24, 26, 27, 118	sumsnd 45483	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑗 ∈ {1} (𝑃‘𝑗) = (𝑃‘1))
120	119	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = ∅) → Σ𝑗 ∈ {1} (𝑃‘𝑗) = (𝑃‘1))
121		fveq2 6828	. . . . . . . . . . . . 13 ⊢ (𝑗 = 1 → (𝐽‘𝑗) = (𝐽‘1))
122		fveq2 6828	. . . . . . . . . . . . 13 ⊢ (𝑗 = 1 → (𝐾‘𝑗) = (𝐾‘1))
123	121, 122	oveq12d 7375	. . . . . . . . . . . 12 ⊢ (𝑗 = 1 → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)))
124		ovex 7390	. . . . . . . . . . . 12 ⊢ ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) ∈ V
125	123, 39, 124	fvmpt 6936	. . . . . . . . . . 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 7374	. . . . . . . . . . 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 6638	. . . . . . . . . . . . . . . 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 6638	. . . . . . . . . . . . . . . . . 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 258	. . . . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝐹:𝑌⟶ℝ)
136	78	feq1d 6638	. . . . . . . . . . . . . . . . . 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 258	. . . . . . . . . . . . . . . . 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 818	. . . . . . . . . . . . . . . 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 6841	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐶‘𝑗) ∈ V
141	140	resex 5982	. . . . . . . . . . . . . . . . . . . 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 3454	. . . . . . . . . . . . . . . . . . . . . 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 818	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
148	88	fvmpt2 6948	. . . . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
150	149	feq1d 6638	. . . . . . . . . . . . . . . 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 258	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗):𝑌⟶ℝ)
152	31, 131, 151	vtocl 3503	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 ∈ ℕ) → (𝐽‘1):𝑌⟶ℝ)
153	29, 30, 152	syl2anc 590	. . . . . . . . . . . . 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 2745	. . . . . . . . . . . . . 14 ⊢ (𝑌 = ∅ → ∅ = 𝑌)
157	156	feq2d 6640	. . . . . . . . . . . . 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 258	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐽‘1):∅⟶ℝ)
160	122	feq1d 6638	. . . . . . . . . . . . . . . 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 3503	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 ∈ ℕ) → (𝐾‘1):𝑌⟶ℝ)
163	29, 30, 162	syl2anc 590	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾‘1):𝑌⟶ℝ)
164	163	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐾‘1):𝑌⟶ℝ)
165	156	feq2d 6640	. . . . . . . . . . . . 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 258	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐾‘1):∅⟶ℝ)
168	18, 159, 167	hoidmv0val 47034	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1)(𝐿‘∅)(𝐾‘1)) = 0)
169	129, 168	eqtrd 2774	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) = 0)
170	120, 127, 169	3eqtrd 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = ∅) → Σ𝑗 ∈ {1} (𝑃‘𝑗) = 0)
171	170	oveq2d 7373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)) = ((1 + 𝐸) · 0))
172		hoidmvlelem3.e	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
173	172	rpred 12978	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ)
174	27, 173	readdcld 11166	. . . . . . . . . 10 ⊢ (𝜑 → (1 + 𝐸) ∈ ℝ)
175	174	recnd 11165	. . . . . . . . 9 ⊢ (𝜑 → (1 + 𝐸) ∈ ℂ)
176	175	mul01d 11337	. . . . . . . 8 ⊢ (𝜑 → ((1 + 𝐸) · 0) = 0)
177	176	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((1 + 𝐸) · 0) = 0)
178		eqidd 2740	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 0 = 0)
179	171, 177, 178	3eqtrd 2778	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)) = 0)
180	22, 179	breq12d 5086	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)) ↔ 0 ≤ 0))
181	4, 180	mpbird 258	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)))
182		oveq2 7365	. . . . . . . . 9 ⊢ (𝑚 = 1 → (1...𝑚) = (1...1))
183	1	nnzi 12543	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
184		fzsn 13512	. . . . . . . . . . 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 2774	. . . . . . . 8 ⊢ (𝑚 = 1 → (1...𝑚) = {1})
188	187	sumeq1d 15654	. . . . . . 7 ⊢ (𝑚 = 1 → Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) = Σ𝑗 ∈ {1} (𝑃‘𝑗))
189	188	oveq2d 7373	. . . . . 6 ⊢ (𝑚 = 1 → ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) = ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)))
190	189	breq2d 5085	. . . . 5 ⊢ (𝑚 = 1 → (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) ↔ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗))))
191	190	rspcev 3560	. . . 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 590	. . 3 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
193		simpl 483	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 = ∅) → 𝜑)
194		neqne 2942	. . . . 5 ⊢ (¬ 𝑌 = ∅ → 𝑌 ≠ ∅)
195	194	adantl 482	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 = ∅) → 𝑌 ≠ ∅)
196		nfv 1921	. . . . . 6 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑌 ≠ ∅)
197	183	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 1 ∈ ℤ)
198		nnuz 12819	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
199	114	adantlr 721	. . . . . 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 6695	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ↾ 𝑌):𝑌⟶ℝ)
203		hoidmvlelem3.b	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵:𝑊⟶ℝ)
204	203, 201	fssresd 6695	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ↾ 𝑌):𝑌⟶ℝ)
205	18, 58, 202, 204	hoidmvcl 47033	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ∈ (0[,)+∞))
206	28, 205	sselid 3913	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ∈ ℝ)
207	5, 206	eqeltrid 2843	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ℝ)
208		0red 11139	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ)
209		1rp 12938	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ⁺
210	209	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ⁺)
211	210, 172	jca 516	. . . . . . . . . . 11 ⊢ (𝜑 → (1 ∈ ℝ⁺ ∧ 𝐸 ∈ ℝ⁺))
212		rpaddcl 12958	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ⁺ ∧ 𝐸 ∈ ℝ⁺) → (1 + 𝐸) ∈ ℝ⁺)
213	211, 212	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (1 + 𝐸) ∈ ℝ⁺)
214		rpgt0 12947	. . . . . . . . . 10 ⊢ ((1 + 𝐸) ∈ ℝ⁺ → 0 < (1 + 𝐸))
215	213, 214	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 0 < (1 + 𝐸))
216	208, 215	gtned 11273	. . . . . . . 8 ⊢ (𝜑 → (1 + 𝐸) ≠ 0)
217	207, 174, 216	redivcld 11975	. . . . . . 7 ⊢ (𝜑 → (𝐺 / (1 + 𝐸)) ∈ ℝ)
218	217	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 / (1 + 𝐸)) ∈ ℝ)
219	217	ltpnfd 13064	. . . . . . . . . 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 2745	. . . . . . . . . 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 5099	. . . . . . . 8 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) < (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
225	224	adantlr 721	. . . . . . 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 12172	. . . . . . . . . . . 12 ⊢ ℕ ∈ V
229	228	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → ℕ ∈ V)
230		icossicc 13381	. . . . . . . . . . . . . 14 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
231	230, 114	sselid 3913	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,]+∞))
232		eqid 2739	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)) = (𝑗 ∈ ℕ ↦ (𝑃‘𝑗))
233	231, 232	fmptd 7056	. . . . . . . . . . . 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 46837	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ ↔ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞))
236	227, 235	mpbird 258	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ)
237	236	adantlr 721	. . . . . . . 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 721	. . . . . . . . 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 13011	. . . . . . . . . . . 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 47035	. . . . . . . . . . . . . 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 6847	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑌 → ((𝐴 ↾ 𝑌)‘𝑘) = (𝐴‘𝑘))
251		fvres 6847	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑌 → ((𝐵 ↾ 𝑌)‘𝑘) = (𝐵‘𝑘))
252	250, 251	oveq12d 7375	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ 𝑌 → (((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
253	252	fveq2d 6832	. . . . . . . . . . . . . . . . . . 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 4112	. . . . . . . . . . . . . . . . . . . . . 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	ffvelcdmd 7027	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐴‘𝑘) ∈ ℝ)
260	203	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐵:𝑊⟶ℝ)
261	260, 258	ffvelcdmd 7027	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐵‘𝑘) ∈ ℝ)
262		volico 46434	. . . . . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . . . . 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 597	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐴‘𝑘) < (𝐵‘𝑘))
266	265	iftrued 4463	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
267	254, 263, 266	3eqtrd 2778	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
268	267	prodeq2dv 15879	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)))
269	268	eqcomd 2745	. . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 = ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)))
272		difrp 12974	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ⁺))
273	259, 261, 272	syl2anc 590	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ⁺))
274	265, 273	mpbid 233	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ⁺)
275	58, 274	fprodrpcl 15913	. . . . . . . . . . . . . 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 45809	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (1 < (1 + 𝐸) ↔ (𝐺 / (1 + 𝐸)) < 𝐺))
280	243, 279	mpbid 233	. . . . . . . . . 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 6843	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑥‘𝑘) ∈ V)
285		hoidmvlelem3.s	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑆 ∈ 𝑈)
286	285	elexd 3454	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑆 ∈ V)
287	284, 286	ifcld 4502	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
288	287	ralrimivw 3135	. . . . . . . . . . . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
291	290	fnmpt 6626	. . . . . . . . . . . . . . . . . . . . . . . 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 7166	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6948	. . . . . . . . . . . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))
300	299	fneq1d 6579	. . . . . . . . . . . . . . . . . . . . . . 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 258	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) Fn 𝑊)
302		nfv 1921	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘𝜑
303		nfcv 2901	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑘𝑥
304		nfixp1 8857	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑘X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
305	303, 304	nfel 2915	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
306	302, 305	nfan 1906	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
307	299	fveq1d 6830	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6948	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
313	312	adantlr 721	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
315		iftrue 4461	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 𝑥 ∈ V
318	317	elixp 8843	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↔ (𝑥 Fn 𝑌 ∧ ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))))
319	318	simprbi 498	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3228	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
323	320, 321, 322	syl2anc 590	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
324	323	ad4ant24 760	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 4593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑍 ∈ (𝑋 ∖ 𝑌) → 𝑍 ∈ {𝑍})
327	47, 326	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → 𝑍 ∈ {𝑍})
328		elun2 4113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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	ffvelcdmd 7027	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ)
333	332	rexrd 11187	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ^*)
334	203, 331	ffvelcdmd 7027	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ)
335	334	rexrd 11187	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ^*)
336		iccssxr 13375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4012	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3961	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍))
340	339, 285	sselid 3913	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
341	336, 340	sselid 3913	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝑆 ∈ ℝ^*)
342		iccgelb 13347	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐴‘𝑍) ∈ ℝ^* ∧ (𝐵‘𝑍) ∈ ℝ^* ∧ 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → (𝐴‘𝑍) ≤ 𝑆)
343	333, 335, 340, 342	syl3anc 1379	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → (𝐴‘𝑍) ≤ 𝑆)
344		hoidmvlelem3.sb	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝑆 < (𝐵‘𝑍))
345	333, 335, 341, 343, 344	elicod 13340	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
346	345	ad2antrr 732	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → 𝑆 ∈ ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
347		iffalse 4464	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑘 ∈ 𝑊 ↔ 𝑘 ∈ (𝑌 ∪ {𝑍}))
350	349	birani 504	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 ∈ (𝑌 ∪ {𝑍}))
351		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → ¬ 𝑘 ∈ 𝑌)
352		elunnel1 4085	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑘 ∈ (𝑌 ∪ {𝑍}) ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 ∈ {𝑍})
353	350, 351, 352	syl2anc 590	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 ∈ {𝑍})
354		elsni 4573	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 ∈ {𝑍} → 𝑘 = 𝑍)
355	353, 354	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 = 𝑍)
356		fveq2 6828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 = 𝑍 → (𝐴‘𝑘) = (𝐴‘𝑍))
357		fveq2 6828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 = 𝑍 → (𝐵‘𝑘) = (𝐵‘𝑍))
358	356, 357	oveq12d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 = 𝑍 → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
359	355, 358	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
360	359	adantll 720	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
361	348, 360	eleq12d 2833	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → (if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↔ 𝑆 ∈ ((𝐴‘𝑍)[,)(𝐵‘𝑍))))
362	346, 361	mpbird 258	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
363	362	adantllr 725	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
364	325, 363	pm2.61dan 818	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
365	314, 364	eqeltrd 2839	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
366	365	ex 413	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑘 ∈ 𝑊 → ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))))
367	306, 366	ralrimi 3237	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
368	301, 367	jca 516	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥) Fn 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))))
369		fvex 6841	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑂‘𝑥) ∈ V
370	369	elixp 8843	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↔ ((𝑂‘𝑥) Fn 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))))
371	368, 370	sylibr 235	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
372	283, 371	sseldd 3916	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
373		eliun 4926	. . . . . . . . . . . . . . . . . . 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)))
374	372, 373	sylib 219	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∃𝑗 ∈ ℕ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
375		ixpfn 8842	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) → 𝑥 Fn 𝑌)
376	375	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑥 Fn 𝑌)
377	376	ad2antrr 732	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑥 Fn 𝑌)
378		nfv 1921	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑘 𝑗 ∈ ℕ
379	306, 378	nfan 1906	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ)
380		nfcv 2901	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑘(𝑂‘𝑥)
381		nfixp1 8857	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑘X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))
382	380, 381	nfel 2915	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘(𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))
383	379, 382	nfan 1906	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
384	307	3adant3 1138	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ((𝑂‘𝑥)‘𝑘) = ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘))
385	287	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
386	258, 385, 311	syl2anc 590	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
387	386	3adant2 1137	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
388	315	3ad2ant3 1141	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = (𝑥‘𝑘))
389	384, 387, 388	3eqtrrd 2779	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) = ((𝑂‘𝑥)‘𝑘))
390	389	ad5ant125 1374	. . . . . . . . . . . . . . . . . . . . . . . . . 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))
391	369	elixp 8843	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
392	391	birani 504	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
393	392	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)))
394	257	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑊)
395		rspa 3228	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
396	393, 394, 395	syl2anc 590	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
397	396	adantll 720	. . . . . . . . . . . . . . . . . . . . . . . . . 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)))
398	390, 397	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)))
399	29	ad3antrrr 736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝜑)
400	37	ad2antlr 733	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑗 ∈ ℕ)
401	299	fveq1d 6830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥)‘𝑍) = ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑍))
402		eqidd 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))
403		eleq1 2827	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑘 = 𝑍 → (𝑘 ∈ 𝑌 ↔ 𝑍 ∈ 𝑌))
404		fveq2 6828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑘 = 𝑍 → (𝑥‘𝑘) = (𝑥‘𝑍))
405	403, 404	ifbieq1d 4480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑘 = 𝑍 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆))
406	405	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ 𝑘 = 𝑍) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆))
407		fvexd 6843	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → (𝑥‘𝑍) ∈ V)
408	407, 286	ifcld 4502	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆) ∈ V)
409	402, 406, 331, 408	fvmptd 6944	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑍) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆))
410	409	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))
411	47	eldifbd 3896	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
412	411	iffalsed 4466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆) = 𝑆)
413	412	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆) = 𝑆)
414	401, 410, 413	3eqtrrd 2779	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑆 = ((𝑂‘𝑥)‘𝑍))
415	414	ad2antrr 732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
416	399, 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)
417	391	simprbi 498	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
418	417	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)))
419		fveq2 6828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑘 = 𝑍 → ((𝑂‘𝑥)‘𝑘) = ((𝑂‘𝑥)‘𝑍))
420		fveq2 6828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑘 = 𝑍 → ((𝐶‘𝑗)‘𝑘) = ((𝐶‘𝑗)‘𝑍))
421		fveq2 6828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑘 = 𝑍 → ((𝐷‘𝑗)‘𝑘) = ((𝐷‘𝑗)‘𝑍))
422	420, 421	oveq12d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑘 = 𝑍 → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
423	419, 422	eleq12d 2833	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑘 = 𝑍 → (((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ↔ ((𝑂‘𝑥)‘𝑍) ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))))
424	423	rspcva 3558	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑍 ∈ 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ((𝑂‘𝑥)‘𝑍) ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
425	416, 418, 424	syl2anc 590	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
426	415, 425	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)))
427	149	3adant3 1138	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
428	60	3ad2ant3 1141	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
429	427, 428	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐽‘𝑗) = ((𝐶‘𝑗) ↾ 𝑌))
430	429	fveq1d 6830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐽‘𝑗)‘𝑘) = (((𝐶‘𝑗) ↾ 𝑌)‘𝑘))
431	399, 400, 426, 430	syl3anc 1379	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
432	431	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))
433		fvres 6847	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ 𝑌 → (((𝐶‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
434	433	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))
435	432, 434	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . 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))
436	107	elexd 3454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
437	108	fvmpt2 6948	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
438	139, 436, 437	syl2anc 590	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
439	438	3adant3 1138	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
440	93	3ad2ant3 1141	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
441	439, 440	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐾‘𝑗) = ((𝐷‘𝑗) ↾ 𝑌))
442	441	fveq1d 6830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐾‘𝑗)‘𝑘) = (((𝐷‘𝑗) ↾ 𝑌)‘𝑘))
443	399, 400, 426, 442	syl3anc 1379	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
444	443	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))
445		fvres 6847	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ 𝑌 → (((𝐷‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
446	445	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))
447	444, 446	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . 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))
448	435, 447	oveq12d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . 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)))
449	448	eqcomd 2745	. . . . . . . . . . . . . . . . . . . . . . . . 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)))
450	398, 449	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)))
451	450	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))))
452	383, 451	ralrimi 3237	. . . . . . . . . . . . . . . . . . . . . 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)))
453	377, 452	jca 516	. . . . . . . . . . . . . . . . . . . . 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))))
454	317	elixp 8843	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) ↔ (𝑥 Fn 𝑌 ∧ ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
455	453, 454	sylibr 235	. . . . . . . . . . . . . . . . . . . 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)))
456	455	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))))
457	456	reximdva 3152	. . . . . . . . . . . . . . . . . 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))))
458	374, 457	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∃𝑗 ∈ ℕ 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
459		eliun 4926	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) ↔ ∃𝑗 ∈ ℕ 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
460	458, 459	sylibr 235	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑥 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
461	460	ralrimiva 3131	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ X 𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))𝑥 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
462		dfss3 3904	. . . . . . . . . . . . . . 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)))
463	461, 462	sylibr 235	. . . . . . . . . . . . . 14 ⊢ (𝜑 → X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
464		ovexd 7392	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℝ ↑_m 𝑌) ∈ V)
465	228	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ℕ ∈ V)
466	464, 465	elmapd 8778	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐾 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ) ↔ 𝐾:ℕ⟶(ℝ ↑_m 𝑌)))
467	109, 466	mpbird 258	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ))
468	464, 465	elmapd 8778	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐽 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ) ↔ 𝐽:ℕ⟶(ℝ ↑_m 𝑌)))
469	89, 468	mpbird 258	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐽 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ))
470	82, 71	elmapd 8778	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝐵 ↾ 𝑌) ∈ (ℝ ↑_m 𝑌) ↔ (𝐵 ↾ 𝑌):𝑌⟶ℝ))
471	204, 470	mpbird 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 ↾ 𝑌) ∈ (ℝ ↑_m 𝑌))
472	82, 71	elmapd 8778	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐴 ↾ 𝑌) ∈ (ℝ ↑_m 𝑌) ↔ (𝐴 ↾ 𝑌):𝑌⟶ℝ))
473	202, 472	mpbird 258	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 ↾ 𝑌) ∈ (ℝ ↑_m 𝑌))
474		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))))))
475		fveq1 6827	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑒 = (𝐴 ↾ 𝑌) → (𝑒‘𝑘) = ((𝐴 ↾ 𝑌)‘𝑘))
476	475	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑒‘𝑘) = ((𝐴 ↾ 𝑌)‘𝑘))
477	250	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝐴 ↾ 𝑌)‘𝑘) = (𝐴‘𝑘))
478	476, 477	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑒‘𝑘) = (𝐴‘𝑘))
479	478	oveq1d 7372	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = ((𝐴‘𝑘)[,)(𝑓‘𝑘)))
480	479	ixpeq2dva 8851	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒 = (𝐴 ↾ 𝑌) → X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)))
481	480	sseq1d 3946	. . . . . . . . . . . . . . . . . . . . . . 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))))
482		oveq1 7364	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒 = (𝐴 ↾ 𝑌) → (𝑒(𝐿‘𝑌)𝑓) = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓))
483	482	breq1d 5083	. . . . . . . . . . . . . . . . . . . . . . 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))))))
484	481, 483	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)))))))
485	484	ralbidv 3162	. . . . . . . . . . . . . . . . . . . . 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)))))))
486	485	ralbidv 3162	. . . . . . . . . . . . . . . . . . . 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)))))))
487	486	ralbidv 3162	. . . . . . . . . . . . . . . . . . 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)))))))
488	487	rspcva 3558	. . . . . . . . . . . . . . . . . 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))))))
489	473, 474, 488	syl2anc 590	. . . . . . . . . . . . . . . . 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))))))
490		fveq1 6827	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 = (𝐵 ↾ 𝑌) → (𝑓‘𝑘) = ((𝐵 ↾ 𝑌)‘𝑘))
491	490	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑓‘𝑘) = ((𝐵 ↾ 𝑌)‘𝑘))
492	251	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝐵 ↾ 𝑌)‘𝑘) = (𝐵‘𝑘))
493	491, 492	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑓‘𝑘) = (𝐵‘𝑘))
494	493	oveq2d 7373	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘)[,)(𝑓‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
495	494	ixpeq2dva 8851	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = (𝐵 ↾ 𝑌) → X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) = X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
496	495	sseq1d 3946	. . . . . . . . . . . . . . . . . . . . 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))))
497		oveq2 7365	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = (𝐵 ↾ 𝑌) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)))
498	497	breq1d 5083	. . . . . . . . . . . . . . . . . . . . 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))))))
499	496, 498	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)))))))
500	499	ralbidv 3162	. . . . . . . . . . . . . . . . . . 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)))))))
501	500	ralbidv 3162	. . . . . . . . . . . . . . . . . 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)))))))
502	501	rspcva 3558	. . . . . . . . . . . . . . . . 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))))))
503	471, 489, 502	syl2anc 590	. . . . . . . . . . . . . . . 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))))))
504		fveq1 6827	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑔 = 𝐽 → (𝑔‘𝑗) = (𝐽‘𝑗))
505	504	fveq1d 6830	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑔 = 𝐽 → ((𝑔‘𝑗)‘𝑘) = ((𝐽‘𝑗)‘𝑘))
506	505	oveq1d 7372	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔 = 𝐽 → (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))
507	506	ixpeq2dv 8852	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = 𝐽 → X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))
508	507	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)))
509	508	sseq2d 3947	. . . . . . . . . . . . . . . . . . 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))))
510	504	oveq1d 7372	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔 = 𝐽 → ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)) = ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))
511	510	mpteq2dv 5167	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = 𝐽 → (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))
512	511	fveq2d 6832	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = 𝐽 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))
513	512	breq2d 5085	. . . . . . . . . . . . . . . . . . 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))))))
514	509, 513	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)))))))
515	514	ralbidv 3162	. . . . . . . . . . . . . . . . 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)))))))
516	515	rspcva 3558	. . . . . . . . . . . . . . . 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))))))
517	469, 503, 516	syl2anc 590	. . . . . . . . . . . . . . 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))))))
518		fveq1 6827	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ = 𝐾 → (ℎ‘𝑗) = (𝐾‘𝑗))
519	518	fveq1d 6830	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ = 𝐾 → ((ℎ‘𝑗)‘𝑘) = ((𝐾‘𝑗)‘𝑘))
520	519	oveq2d 7373	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = 𝐾 → (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
521	520	ixpeq2dv 8852	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = 𝐾 → X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
522	521	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)))
523	522	sseq2d 3947	. . . . . . . . . . . . . . . . 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))))
524	518	oveq2d 7373	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = 𝐾 → ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
525	524	mpteq2dv 5167	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = 𝐾 → (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))
526	525	fveq2d 6832	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = 𝐾 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
527	526	breq2d 5085	. . . . . . . . . . . . . . . . 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))))))
528	523, 527	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)))))))
529	528	rspcva 3558	. . . . . . . . . . . . . . 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))))))
530	467, 517, 529	syl2anc 590	. . . . . . . . . . . . . 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))))))
531	463, 530	mpd 15	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
532		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))))))
533	531, 532	mpd 15	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
534	533	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
535	41	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
536	535	mpteq2dva 5166	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)) = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))
537	536	fveq2d 6832	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
538	249, 537	breq12d 5086	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ↔ ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))))
539	534, 538	mpbird 258	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
540	539	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → 𝐺 ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
541	238, 240, 241, 281, 540	ltletrd 11298	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → (𝐺 / (1 + 𝐸)) < (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
542	226, 237, 541	syl2anc 590	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) < (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
543	225, 542	pm2.61dan 818	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 / (1 + 𝐸)) < (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
544	196, 197, 198, 199, 218, 543	sge0uzfsumgt 46895	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ∃𝑚 ∈ ℕ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))
545	217	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (𝐺 / (1 + 𝐸)) ∈ ℝ)
546		fzfid 13927	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...𝑚) ∈ Fin)
547		simpl 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → 𝜑)
548		elfznn 13499	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (1...𝑚) → 𝑗 ∈ ℕ)
549	548	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → 𝑗 ∈ ℕ)
550	28, 114	sselid 3913	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ ℝ)
551	547, 549, 550	syl2anc 590	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → (𝑃‘𝑗) ∈ ℝ)
552	546, 551	fsumrecl 15688	. . . . . . . . . . . 12 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) ∈ ℝ)
553	552	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) ∈ ℝ)
554		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))
555	545, 553, 554	ltled 11286	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (𝐺 / (1 + 𝐸)) ≤ Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))
556	207	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ∈ ℝ)
557	213	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (1 + 𝐸) ∈ ℝ⁺)
558	556, 553, 557	ledivmuld 13031	. . . . . . . . . 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))))
559	555, 558	mpbid 233	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
560	559	ex 413	. . . . . . . 8 ⊢ (𝜑 → ((𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
561	560	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
562	561	adantlr 721	. . . . . 6 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ 𝑚 ∈ ℕ) → ((𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
563	562	reximdva 3152	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (∃𝑚 ∈ ℕ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
564	544, 563	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
565	193, 195, 564	syl2anc 590	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 = ∅) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
566	192, 565	pm2.61dan 818	. 2 ⊢ (𝜑 → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
567	43	3ad2ant1 1139	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑋 ∈ Fin)
568	46	3ad2ant1 1139	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑌 ⊆ 𝑋)
569	47	3ad2ant1 1139	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑍 ∈ (𝑋 ∖ 𝑌))
570	200	3ad2ant1 1139	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐴:𝑊⟶ℝ)
571	203	3ad2ant1 1139	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐵:𝑊⟶ℝ)
572	62	3ad2ant1 1139	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐶:ℕ⟶(ℝ ↑_m 𝑊))
573		eqid 2739	. . . . 5 ⊢ (𝑦 ∈ 𝑌 ↦ 0) = (𝑦 ∈ 𝑌 ↦ 0)
574		eqid 2739	. . . . 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)))
575	95	3ad2ant1 1139	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐷:ℕ⟶(ℝ ↑_m 𝑊))
576		eqid 2739	. . . . 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)))
577		fveq2 6828	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝐶‘𝑖) = (𝐶‘𝑗))
578		fveq2 6828	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗))
579	577, 578	oveq12d 7375	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)) = ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))
580	579	cbvmptv 5177	. . . . . . . 8 ⊢ (𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))
581	580	fveq2i 6831	. . . . . . 7 ⊢ (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))
582		hoidmvlelem3.r	. . . . . . 7 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
583	581, 582	eqeltrid 2843	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ)
584	583	3ad2ant1 1139	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ)
585		hoidmvlelem3.h	. . . . . 6 ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))
586		eleq1w 2822	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ 𝑌 ↔ 𝑖 ∈ 𝑌))
587		fveq2 6828	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝑐‘𝑗) = (𝑐‘𝑖))
588	587	breq1d 5083	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → ((𝑐‘𝑗) ≤ 𝑥 ↔ (𝑐‘𝑖) ≤ 𝑥))
589	588, 587	ifbieq1d 4480	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥) = if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥))
590	586, 587, 589	ifbieq12d 4484	. . . . . . . . 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)))
591	590	cbvmptv 5177	. . . . . . . 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)))
592	591	mpteq2i 5169	. . . . . . 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))))
593	592	mpteq2i 5169	. . . . . 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)))))
594	585, 593	eqtri 2762	. . . . 5 ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥)))))
595	172	3ad2ant1 1139	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐸 ∈ ℝ⁺)
596		fveq2 6828	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → (𝐶‘𝑗) = (𝐶‘𝑖))
597		fveq2 6828	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑖 → (𝐷‘𝑗) = (𝐷‘𝑖))
598	597	fveq2d 6832	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘𝑧)‘(𝐷‘𝑖)))
599	596, 598	oveq12d 7375	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))
600	599	cbvmptv 5177	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))) = (𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))
601	600	fveq2i 6831	. . . . . . . . 9 ⊢ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))) = (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖)))))
602	601	oveq2i 7368	. . . . . . . 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))))))
603	602	breq2i 5081	. . . . . . 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)))))))
604	603	rabbii 3396	. . . . . 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))))))}
605	337, 604	eqtri 2762	. . . . 5 ⊢ 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))))}
606	285	3ad2ant1 1139	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑆 ∈ 𝑈)
607	344	3ad2ant1 1139	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑆 < (𝐵‘𝑍))
608		eqid 2739	. . . . 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)))
609		simp2 1143	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑚 ∈ ℕ)
610		id 22	. . . . . . . 8 ⊢ (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
611		fveq2 6828	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → (𝑃‘𝑗) = (𝑃‘𝑖))
612	611	cbvsumv 15650	. . . . . . . . . 10 ⊢ Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) = Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)
613	612	oveq2i 7368	. . . . . . . . 9 ⊢ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) = ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖))
614	613	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)))
615	610, 614	breqtrd 5099	. . . . . . 7 ⊢ (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)))
616	615	3ad2ant3 1141	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)))
617		simpl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑚)) → 𝜑)
618		elfznn 13499	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...𝑚) → 𝑖 ∈ ℕ)
619	618	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ∈ ℕ)
620		eleq1w 2822	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ ℕ ↔ 𝑖 ∈ ℕ))
621		fveq2 6828	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑖 → (𝐽‘𝑗) = (𝐽‘𝑖))
622		fveq2 6828	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑖 → (𝐾‘𝑗) = (𝐾‘𝑖))
623	621, 622	oveq12d 7375	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑖 → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)))
624	611, 623	eqeq12d 2755	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → ((𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ↔ (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖))))
625	620, 624	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)))))
626	625, 41	chvarvv 1996	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ℕ → (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)))
627	626	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)))
628	620	anbi2d 636	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → ((𝜑 ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ 𝑖 ∈ ℕ)))
629	596	fveq1d 6830	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑖 → ((𝐶‘𝑗)‘𝑍) = ((𝐶‘𝑖)‘𝑍))
630	597	fveq1d 6830	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑖 → ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑖)‘𝑍))
631	629, 630	oveq12d 7375	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑖 → (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) = (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
632	631	eleq2d 2825	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑖 → (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))))
633	596	reseq1d 5931	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑖 → ((𝐶‘𝑗) ↾ 𝑌) = ((𝐶‘𝑖) ↾ 𝑌))
634	632, 633	ifbieq1d 4480	. . . . . . . . . . . . . . . 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))
635	621, 634	eqeq12d 2755	. . . . . . . . . . . . . . 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)))
636	628, 635	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))))
637	636, 149	chvarvv 1996	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐽‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹))
638	597	reseq1d 5931	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑖 → ((𝐷‘𝑗) ↾ 𝑌) = ((𝐷‘𝑖) ↾ 𝑌))
639	632, 638	ifbieq1d 4480	. . . . . . . . . . . . . . . 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))
640	622, 639	eqeq12d 2755	. . . . . . . . . . . . . . 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)))
641	628, 640	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))))
642	641, 438	chvarvv 1996	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐾‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹))
643	637, 642	oveq12d 7375	. . . . . . . . . . . 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)))
644	627, 643	eqtrd 2774	. . . . . . . . . . 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)))
645		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
646		ovexd 7392	. . . . . . . . . . . . 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)
647	608	fvmpt2 6948	. . . . . . . . . . . . 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)))
648	645, 646, 647	syl2anc 590	. . . . . . . . . . . 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)))
649		fvex 6841	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶‘𝑖) ∈ V
650	649	resex 5982	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶‘𝑖) ↾ 𝑌) ∈ V
651	650	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐶‘𝑖) ↾ 𝑌) ∈ V)
652	80, 143	eqeltrrid 2844	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ 0) ∈ V)
653	651, 652	ifcld 4502	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V)
654	653	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V)
655	574	fvmpt2 6948	. . . . . . . . . . . . . . 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)))
656	645, 654, 655	syl2anc 590	. . . . . . . . . . . . . 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)))
657	80	eqcomi 2748	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑌 ↦ 0) = 𝐹
658		ifeq2 4460	. . . . . . . . . . . . . . . 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))
659	657, 658	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)
660	659	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))
661	656, 660	eqtrd 2774	. . . . . . . . . . . . 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))
662		fvex 6841	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷‘𝑖) ∈ V
663	662	resex 5982	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷‘𝑖) ↾ 𝑌) ∈ V
664	663	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐷‘𝑖) ↾ 𝑌) ∈ V)
665	664, 652	ifcld 4502	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V)
666	665	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V)
667	576	fvmpt2 6948	. . . . . . . . . . . . . . 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)))
668	645, 666, 667	syl2anc 590	. . . . . . . . . . . . . 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)))
669		biid 262	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
670	669, 657	ifbieq2i 4481	. . . . . . . . . . . . . . 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)
671	670	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))
672	668, 671	eqtrd 2774	. . . . . . . . . . . . 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))
673	661, 672	oveq12d 7375	. . . . . . . . . . . 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)))
674	648, 673	eqtrd 2774	. . . . . . . . . . 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)))
675	644, 674	eqtr4d 2777	. . . . . . . . . 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))
676	617, 619, 675	syl2anc 590	. . . . . . . . 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))
677	676	3ad2antl1 1192	. . . . . . . 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))
678	677	sumeq2dv 15656	. . . . . . 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))
679	678	oveq2d 7373	. . . . . 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)))
680	616, 679	breqtrd 5099	. . . . 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)))
681		fveq2 6828	. . . . . . . 8 ⊢ (𝑗 = ℎ → (𝐷‘𝑗) = (𝐷‘ℎ))
682	681	fveq1d 6830	. . . . . . 7 ⊢ (𝑗 = ℎ → ((𝐷‘𝑗)‘𝑍) = ((𝐷‘ℎ)‘𝑍))
683	682	cbvmptv 5177	. . . . . 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))
684	683	rneqi 5880	. . . . 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))
685		fveq2 6828	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑖 → (𝐶‘ℎ) = (𝐶‘𝑖))
686	685	fveq1d 6830	. . . . . . . . . . 11 ⊢ (ℎ = 𝑖 → ((𝐶‘ℎ)‘𝑍) = ((𝐶‘𝑖)‘𝑍))
687		fveq2 6828	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑖 → (𝐷‘ℎ) = (𝐷‘𝑖))
688	687	fveq1d 6830	. . . . . . . . . . 11 ⊢ (ℎ = 𝑖 → ((𝐷‘ℎ)‘𝑍) = ((𝐷‘𝑖)‘𝑍))
689	686, 688	oveq12d 7375	. . . . . . . . . 10 ⊢ (ℎ = 𝑖 → (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍)) = (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
690	689	eleq2d 2825	. . . . . . . . 9 ⊢ (ℎ = 𝑖 → (𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))))
691	690	cbvrabv 3401	. . . . . . . 8 ⊢ {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} = {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))}
692	691	mpteq1i 5164	. . . . . . 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))
693	692	rneqi 5880	. . . . . 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))
694	693	uneq2i 4096	. . . . 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)))
695		eqid 2739	. . . . 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, < )
696	18, 567, 568, 569, 44, 570, 571, 572, 573, 574, 575, 576, 584, 594, 5, 595, 605, 606, 607, 608, 609, 680, 684, 694, 695	hoidmvlelem2 47047	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
697	696	3exp 1125	. . 3 ⊢ (𝜑 → (𝑚 ∈ ℕ → (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)))
698	697	rexlimdv 3138	. 2 ⊢ (𝜑 → (∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢))
699	566, 698	mpd 15	1 ⊢ (𝜑 → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 ∃wrex 3063 {crab 3391 Vcvv 3431 ∖ cdif 3880 ∪ cun 3881 ⊆ wss 3883 ∅c0 4262 ifcif 4455 {csn 4556 ∪ ciun 4922 class class class wbr 5073 ↦ cmpt 5154 ran crn 5620 ↾ cres 5621 Fn wfn 6481 ⟶wf 6482 ‘cfv 6486 (class class class)co 7357 ∈ cmpo 7359 ↑_m cmap 8764 Xcixp 8836 Fincfn 8884 infcinf 9345 ℝcr 11029 0cc0 11030 1c1 11031 + caddc 11033 · cmul 11035 +∞cpnf 11168 ℝ^*cxr 11170 < clt 11171 ≤ cle 11172 − cmin 11369 / cdiv 11799 ℕcn 12166 ℤcz 12516 ℝ⁺crp 12934 [,)cico 13292 [,]cicc 13293 ...cfz 13453 Σcsu 15640 ∏cprod 15860 volcvol 25449 Σ^{^}csumge0 46813

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-inf2 9554 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7621 df-om 7808 df-1st 7932 df-2nd 7933 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-dju 9817 df-card 9855 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12167 df-2 12236 df-3 12237 df-n0 12430 df-z 12517 df-uz 12781 df-q 12891 df-rp 12935 df-xneg 13055 df-xadd 13056 df-xmul 13057 df-ioo 13294 df-ico 13296 df-icc 13297 df-fz 13454 df-fzo 13601 df-fl 13743 df-seq 13956 df-exp 14016 df-hash 14285 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15442 df-rlim 15443 df-sum 15641 df-prod 15861 df-rest 17377 df-topgen 17398 df-psmet 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-top 22878 df-topon 22895 df-bases 22930 df-cmp 23371 df-ovol 25450 df-vol 25451 df-sumge0 46814

This theorem is referenced by: hoidmvlelem4 47049

W3C validator