Theorem hoidmvlelem3 47606

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 12346	. . . . 5 ⊢ 1 ∈ ℕ
2	1	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 1 ∈ ℕ)
3		0le0 12444	. . . . . 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 6885	. . . . . . . . 9 ⊢ (𝑌 = ∅ → (𝐿‘𝑌) = (𝐿‘∅))
8		reseq2 5965	. . . . . . . . . 10 ⊢ (𝑌 = ∅ → (𝐴 ↾ 𝑌) = (𝐴 ↾ ∅))
9		res0 5974	. . . . . . . . . . 11 ⊢ (𝐴 ↾ ∅) = ∅
10	9	a1i 11	. . . . . . . . . 10 ⊢ (𝑌 = ∅ → (𝐴 ↾ ∅) = ∅)
11	8, 10	eqtrd 2796	. . . . . . . . 9 ⊢ (𝑌 = ∅ → (𝐴 ↾ 𝑌) = ∅)
12		reseq2 5965	. . . . . . . . . 10 ⊢ (𝑌 = ∅ → (𝐵 ↾ 𝑌) = (𝐵 ↾ ∅))
13		res0 5974	. . . . . . . . . . 11 ⊢ (𝐵 ↾ ∅) = ∅
14	13	a1i 11	. . . . . . . . . 10 ⊢ (𝑌 = ∅ → (𝐵 ↾ ∅) = ∅)
15	12, 14	eqtrd 2796	. . . . . . . . 9 ⊢ (𝑌 = ∅ → (𝐵 ↾ 𝑌) = ∅)
16	7, 11, 15	oveq123d 7441	. . . . . . . 8 ⊢ (𝑌 = ∅ → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) = (∅(𝐿‘∅)∅))
17	16	adantl 487	. . . . . . 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 6763	. . . . . . . . 9 ⊢ ∅:∅⟶ℝ
20	19	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ∅:∅⟶ℝ)
21	18, 20, 20	hoidmv0val 47592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (∅(𝐿‘∅)∅) = 0)
22	6, 17, 21	3eqtrd 2800	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 𝐺 = 0)
23		nfcvd 2924	. . . . . . . . . . 11 ⊢ (𝜑 → Ⅎ𝑗(𝑃‘1))
24		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑗𝜑
25		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 = 1) → 𝑗 = 1)
26	25	fveq2d 6889	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 = 1) → (𝑃‘𝑗) = (𝑃‘1))
27		1red 11309	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℝ)
28		rge0ssre 13587	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ⊆ ℝ
29		id 23	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝜑)
30	1	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℕ)
31	1	elexi 3473	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
32		eleq1 2849	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 1 → (𝑗 ∈ ℕ ↔ 1 ∈ ℕ))
33	32	anbi2d 642	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 1 → ((𝜑 ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ 1 ∈ ℕ)))
34		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 1 → (𝑃‘𝑗) = (𝑃‘1))
35	34	eleq1d 2846	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 1 → ((𝑃‘𝑗) ∈ (0[,)+∞) ↔ (𝑃‘1) ∈ (0[,)+∞)))
36	33, 35	imbi12d 347	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 1 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,)+∞)) ↔ ((𝜑 ∧ 1 ∈ ℕ) → (𝑃‘1) ∈ (0[,)+∞))))
37		id 23	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
38		ovexd 7455	. . . . . . . . . . . . . . . . . 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 7005	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ℕ ∧ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ V) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
41	37, 38, 40	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
42	41	adantl 487	. . . . . . . . . . . . . . . 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 3911	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑍 ∈ 𝑋)
49		snssi 4746	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑍 ∈ 𝑋 → {𝑍} ⊆ 𝑋)
50	48, 49	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {𝑍} ⊆ 𝑋)
51	46, 50	unssd 4138	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑌 ∪ {𝑍}) ⊆ 𝑋)
52	45, 51	eqsstrd 3965	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑊 ⊆ 𝑋)
53	43, 52	ssfid 9260	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑊 ∈ Fin)
54		ssun1 4124	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑌 ⊆ (𝑌 ∪ {𝑍})
55	44	eqcomi 2770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑌 ∪ {𝑍}) = 𝑊
56	54, 55	sseqtri 3979	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑌 ⊆ 𝑊
57	56	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑌 ⊆ 𝑊)
58	53, 57	ssfid 9260	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑌 ∈ Fin)
59	58	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ∈ Fin)
60		iftrue 4488	. . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . 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 7084	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑m 𝑊))
64		elmapi 8869	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐶‘𝑗) ∈ (ℝ ↑m 𝑊) → (𝐶‘𝑗):𝑊⟶ℝ)
65	63, 64	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑊⟶ℝ)
66	54, 44	sseqtrri 3980	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑌 ⊆ 𝑊
67	66	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ⊆ 𝑊)
68	65, 67	fssresd 6749	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
69		reex 11291	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ℝ ∈ V
70	69	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ℝ ∈ V)
71	53, 57	ssexd 5286	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑌 ∈ V)
72	71	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ∈ V)
73	70, 72	elmapd 8860	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐶‘𝑗) ↾ 𝑌) ∈ (ℝ ↑m 𝑌) ↔ ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ))
74	68, 73	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗) ↾ 𝑌) ∈ (ℝ ↑m 𝑌))
75	74	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐶‘𝑗) ↾ 𝑌) ∈ (ℝ ↑m 𝑌))
76	61, 75	eqeltrd 2861	. . . . . . . . . . . . . . . . . . . . 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 4491	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
78	77	adantl 487	. . . . . . . . . . . . . . . . . . . . . 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 11311	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ∈ ℝ)
80		hoidmvlelem3.f	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝐹 = (𝑦 ∈ 𝑌 ↦ 0)
81	79, 80	fmptd 7114	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐹:𝑌⟶ℝ)
82	69	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ℝ ∈ V)
83	82, 58	elmapd 8860	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝐹 ∈ (ℝ ↑m 𝑌) ↔ 𝐹:𝑌⟶ℝ))
84	81, 83	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐹 ∈ (ℝ ↑m 𝑌))
85	84	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝐹 ∈ (ℝ ↑m 𝑌))
86	78, 85	eqeltrd 2861	. . . . . . . . . . . . . . . . . . . . 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 825	. . . . . . . . . . . . . . . . . . . 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 7114	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐽:ℕ⟶(ℝ ↑m 𝑌))
90	89	ffvelcdmda 7084	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗) ∈ (ℝ ↑m 𝑌))
91		elmapi 8869	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽‘𝑗) ∈ (ℝ ↑m 𝑌) → (𝐽‘𝑗):𝑌⟶ℝ)
92	90, 91	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗):𝑌⟶ℝ)
93		iftrue 4488	. . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . 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 7084	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑m 𝑊))
97		elmapi 8869	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐷‘𝑗) ∈ (ℝ ↑m 𝑊) → (𝐷‘𝑗):𝑊⟶ℝ)
98	96, 97	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑊⟶ℝ)
99	98, 67	fssresd 6749	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
100	70, 72	elmapd 8860	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐷‘𝑗) ↾ 𝑌) ∈ (ℝ ↑m 𝑌) ↔ ((𝐷‘𝑗) ↾ 𝑌):𝑌⟶ℝ))
101	99, 100	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗) ↾ 𝑌) ∈ (ℝ ↑m 𝑌))
102	101	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐷‘𝑗) ↾ 𝑌) ∈ (ℝ ↑m 𝑌))
103	94, 102	eqeltrd 2861	. . . . . . . . . . . . . . . . . . . . 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 4491	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
105	104	adantl 487	. . . . . . . . . . . . . . . . . . . . . 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 2861	. . . . . . . . . . . . . . . . . . . . 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 825	. . . . . . . . . . . . . . . . . . . 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 7114	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾:ℕ⟶(ℝ ↑m 𝑌))
110	109	ffvelcdmda 7084	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗) ∈ (ℝ ↑m 𝑌))
111		elmapi 8869	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾‘𝑗) ∈ (ℝ ↑m 𝑌) → (𝐾‘𝑗):𝑌⟶ℝ)
112	110, 111	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗):𝑌⟶ℝ)
113	18, 59, 92, 112	hoidmvcl 47591	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ (0[,)+∞))
114	42, 113	eqeltrd 2861	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,)+∞))
115	31, 36, 114	vtocl 3521	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 ∈ ℕ) → (𝑃‘1) ∈ (0[,)+∞))
116	29, 30, 115	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃‘1) ∈ (0[,)+∞))
117	28, 116	sselid 3929	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃‘1) ∈ ℝ)
118	117	recnd 11337	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃‘1) ∈ ℂ)
119	23, 24, 26, 27, 118	sumsnd 46042	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑗 ∈ {1} (𝑃‘𝑗) = (𝑃‘1))
120	119	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = ∅) → Σ𝑗 ∈ {1} (𝑃‘𝑗) = (𝑃‘1))
121		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑗 = 1 → (𝐽‘𝑗) = (𝐽‘1))
122		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑗 = 1 → (𝐾‘𝑗) = (𝐾‘1))
123	121, 122	oveq12d 7438	. . . . . . . . . . . 12 ⊢ (𝑗 = 1 → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)))
124		ovex 7453	. . . . . . . . . . . 12 ⊢ ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) ∈ V
125	123, 39, 124	fvmpt 6993	. . . . . . . . . . 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 7437	. . . . . . . . . . 11 ⊢ (𝑌 = ∅ → ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) = ((𝐽‘1)(𝐿‘∅)(𝐾‘1)))
129	128	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) = ((𝐽‘1)(𝐿‘∅)(𝐾‘1)))
130	121	feq1d 6691	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 1 → ((𝐽‘𝑗):𝑌⟶ℝ ↔ (𝐽‘1):𝑌⟶ℝ))
131	33, 130	imbi12d 347	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 1 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗):𝑌⟶ℝ) ↔ ((𝜑 ∧ 1 ∈ ℕ) → (𝐽‘1):𝑌⟶ℝ)))
132	68	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
133	61	feq1d 6691	. . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝐹:𝑌⟶ℝ)
136	78	feq1d 6691	. . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . 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 825	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
139		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
140		fvex 6898	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐶‘𝑗) ∈ V
141	140	resex 6018	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶‘𝑗) ↾ 𝑌) ∈ V
142	61, 141	eqeltrdi 2869	. . . . . . . . . . . . . . . . . . 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 3474	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹 ∈ V)
144	143	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐹 ∈ V)
145	144	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝐹 ∈ V)
146	78, 145	eqeltrd 2861	. . . . . . . . . . . . . . . . . . 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 825	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
148	88	fvmpt2 7005	. . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
150	149	feq1d 6691	. . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗):𝑌⟶ℝ)
152	31, 131, 151	vtocl 3521	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 ∈ ℕ) → (𝐽‘1):𝑌⟶ℝ)
153	29, 30, 152	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐽‘1):𝑌⟶ℝ)
154	153	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐽‘1):𝑌⟶ℝ)
155		id 23	. . . . . . . . . . . . . . 15 ⊢ (𝑌 = ∅ → 𝑌 = ∅)
156	155	eqcomd 2767	. . . . . . . . . . . . . 14 ⊢ (𝑌 = ∅ → ∅ = 𝑌)
157	156	feq2d 6693	. . . . . . . . . . . . 13 ⊢ (𝑌 = ∅ → ((𝐽‘1):∅⟶ℝ ↔ (𝐽‘1):𝑌⟶ℝ))
158	157	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1):∅⟶ℝ ↔ (𝐽‘1):𝑌⟶ℝ))
159	154, 158	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐽‘1):∅⟶ℝ)
160	122	feq1d 6691	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 1 → ((𝐾‘𝑗):𝑌⟶ℝ ↔ (𝐾‘1):𝑌⟶ℝ))
161	33, 160	imbi12d 347	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 1 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗):𝑌⟶ℝ) ↔ ((𝜑 ∧ 1 ∈ ℕ) → (𝐾‘1):𝑌⟶ℝ)))
162	31, 161, 112	vtocl 3521	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 ∈ ℕ) → (𝐾‘1):𝑌⟶ℝ)
163	29, 30, 162	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾‘1):𝑌⟶ℝ)
164	163	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐾‘1):𝑌⟶ℝ)
165	156	feq2d 6693	. . . . . . . . . . . . 13 ⊢ (𝑌 = ∅ → ((𝐾‘1):∅⟶ℝ ↔ (𝐾‘1):𝑌⟶ℝ))
166	165	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐾‘1):∅⟶ℝ ↔ (𝐾‘1):𝑌⟶ℝ))
167	164, 166	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐾‘1):∅⟶ℝ)
168	18, 159, 167	hoidmv0val 47592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1)(𝐿‘∅)(𝐾‘1)) = 0)
169	129, 168	eqtrd 2796	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) = 0)
170	120, 127, 169	3eqtrd 2800	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = ∅) → Σ𝑗 ∈ {1} (𝑃‘𝑗) = 0)
171	170	oveq2d 7436	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)) = ((1 + 𝐸) · 0))
172		hoidmvlelem3.e	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
173	172	rpred 13164	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ)
174	27, 173	readdcld 11338	. . . . . . . . . 10 ⊢ (𝜑 → (1 + 𝐸) ∈ ℝ)
175	174	recnd 11337	. . . . . . . . 9 ⊢ (𝜑 → (1 + 𝐸) ∈ ℂ)
176	175	mul01d 11509	. . . . . . . 8 ⊢ (𝜑 → ((1 + 𝐸) · 0) = 0)
177	176	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((1 + 𝐸) · 0) = 0)
178		eqidd 2762	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 0 = 0)
179	171, 177, 178	3eqtrd 2800	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)) = 0)
180	22, 179	breq12d 5116	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)) ↔ 0 ≤ 0))
181	4, 180	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)))
182		oveq2 7428	. . . . . . . . 9 ⊢ (𝑚 = 1 → (1...𝑚) = (1...1))
183	1	nnzi 12720	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
184		fzsn 13700	. . . . . . . . . . 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 2796	. . . . . . . 8 ⊢ (𝑚 = 1 → (1...𝑚) = {1})
188	187	sumeq1d 15867	. . . . . . 7 ⊢ (𝑚 = 1 → Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) = Σ𝑗 ∈ {1} (𝑃‘𝑗))
189	188	oveq2d 7436	. . . . . 6 ⊢ (𝑚 = 1 → ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) = ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)))
190	189	breq2d 5115	. . . . 5 ⊢ (𝑚 = 1 → (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) ↔ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗))))
191	190	rspcev 3577	. . . 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 596	. . 3 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
193		simpl 488	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 = ∅) → 𝜑)
194		neqne 2964	. . . . 5 ⊢ (¬ 𝑌 = ∅ → 𝑌 ≠ ∅)
195	194	adantl 487	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 = ∅) → 𝑌 ≠ ∅)
196		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑌 ≠ ∅)
197	183	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 1 ∈ ℤ)
198		nnuz 13004	. . . . . 6 ⊢ ℕ = (ℤ≥‘1)
199	114	adantlr 728	. . . . . 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 6749	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ↾ 𝑌):𝑌⟶ℝ)
203		hoidmvlelem3.b	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵:𝑊⟶ℝ)
204	203, 201	fssresd 6749	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ↾ 𝑌):𝑌⟶ℝ)
205	18, 58, 202, 204	hoidmvcl 47591	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ∈ (0[,)+∞))
206	28, 205	sselid 3929	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ∈ ℝ)
207	5, 206	eqeltrid 2865	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ℝ)
208		0red 11311	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ)
209		1rp 13124	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ+
210	209	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ+)
211	210, 172	jca 521	. . . . . . . . . . 11 ⊢ (𝜑 → (1 ∈ ℝ+ ∧ 𝐸 ∈ ℝ+))
212		rpaddcl 13144	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ+ ∧ 𝐸 ∈ ℝ+) → (1 + 𝐸) ∈ ℝ+)
213	211, 212	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (1 + 𝐸) ∈ ℝ+)
214		rpgt0 13133	. . . . . . . . . 10 ⊢ ((1 + 𝐸) ∈ ℝ+ → 0 < (1 + 𝐸))
215	213, 214	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 0 < (1 + 𝐸))
216	208, 215	gtned 11445	. . . . . . . 8 ⊢ (𝜑 → (1 + 𝐸) ≠ 0)
217	207, 174, 216	redivcld 12145	. . . . . . 7 ⊢ (𝜑 → (𝐺 / (1 + 𝐸)) ∈ ℝ)
218	217	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 / (1 + 𝐸)) ∈ ℝ)
219	217	ltpnfd 13250	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 / (1 + 𝐸)) < +∞)
220	219	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) < +∞)
221		id 23	. . . . . . . . . . 11 ⊢ ((Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞ → (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞)
222	221	eqcomd 2767	. . . . . . . . . 10 ⊢ ((Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞ → +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
223	222	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
224	220, 223	breqtrd 5131	. . . . . . . 8 ⊢ ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) < (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
225	224	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) < (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
226		simpl 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝜑 ∧ 𝑌 ≠ ∅))
227		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞)
228		nnex 12341	. . . . . . . . . . . 12 ⊢ ℕ ∈ V
229	228	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → ℕ ∈ V)
230		icossicc 13567	. . . . . . . . . . . . . 14 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
231	230, 114	sselid 3929	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,]+∞))
232		eqid 2761	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)) = (𝑗 ∈ ℕ ↦ (𝑃‘𝑗))
233	231, 232	fmptd 7114	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)):ℕ⟶(0[,]+∞))
234	233	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)):ℕ⟶(0[,]+∞))
235	229, 234	sge0repnf 47395	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ ↔ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞))
236	227, 235	mpbird 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ)
237	236	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ)
238	218	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → (𝐺 / (1 + 𝐸)) ∈ ℝ)
239	207	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → 𝐺 ∈ ℝ)
240	239	adantlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → 𝐺 ∈ ℝ)
241		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ)
242	27, 172	ltaddrpd 13197	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 < (1 + 𝐸))
243	242	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 1 < (1 + 𝐸))
244	58	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝑌 ∈ Fin)
245		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝑌 ≠ ∅)
246	202	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐴 ↾ 𝑌):𝑌⟶ℝ)
247	204	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐵 ↾ 𝑌):𝑌⟶ℝ)
248	18, 244, 245, 246, 247	hoidmvn0val 47593	. . . . . . . . . . . . . 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 6904	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑌 → ((𝐴 ↾ 𝑌)‘𝑘) = (𝐴‘𝑘))
251		fvres 6904	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑌 → ((𝐵 ↾ 𝑌)‘𝑘) = (𝐵‘𝑘))
252	250, 251	oveq12d 7438	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ 𝑌 → (((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
253	252	fveq2d 6889	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ 𝑌 → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
254	253	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
255	200	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐴:𝑊⟶ℝ)
256		elun1 4128	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑌 → 𝑘 ∈ (𝑌 ∪ {𝑍}))
257	256, 44	eleqtrrdi 2872	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑌 → 𝑘 ∈ 𝑊)
258	257	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑊)
259	255, 258	ffvelcdmd 7085	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐴‘𝑘) ∈ ℝ)
260	203	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐵:𝑊⟶ℝ)
261	260, 258	ffvelcdmd 7085	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐵‘𝑘) ∈ ℝ)
262		volico 46992	. . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . 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 603	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐴‘𝑘) < (𝐵‘𝑘))
266	265	iftrued 4490	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
267	254, 263, 266	3eqtrd 2800	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
268	267	prodeq2dv 16090	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)))
269	268	eqcomd 2767	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))))
270	269	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))))
271	248, 249, 270	3eqtr4d 2806	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 = ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)))
272		difrp 13160	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ+))
273	259, 261, 272	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ+))
274	265, 273	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ+)
275	58, 274	fprodrpcl 16123	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ+)
276	275	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ+)
277	271, 276	eqeltrd 2861	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 ∈ ℝ+)
278	213	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (1 + 𝐸) ∈ ℝ+)
279	277, 278	ltdivgt1 46367	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (1 < (1 + 𝐸) ↔ (𝐺 / (1 + 𝐸)) < 𝐺))
280	243, 279	mpbid 235	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 / (1 + 𝐸)) < 𝐺)
281	280	adantr 486	. . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . 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 6900	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑥‘𝑘) ∈ V)
285		hoidmvlelem3.s	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑆 ∈ 𝑈)
286	285	elexd 3474	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑆 ∈ V)
287	284, 286	ifcld 4529	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
288	287	ralrimivw 3159	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ∀𝑘 ∈ 𝑊 if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
289	288	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∀𝑘 ∈ 𝑊 if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
290		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
291	290	fnmpt 6679	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑘 ∈ 𝑊 if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) Fn 𝑊)
292	289, 291	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) Fn 𝑊)
293		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
294		mptexg 7227	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑊 ∈ Fin → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) ∈ V)
295	53, 294	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) ∈ V)
296	295	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 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 7005	. . . . . . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))
300	299	fneq1d 6632	. . . . . . . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) Fn 𝑊)
302		nfv 1947	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘𝜑
303		nfcv 2923	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑘𝑥
304		nfixp1 8946	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑘X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
305	303, 304	nfel 2937	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
306	302, 305	nfan 1932	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
307	299	fveq1d 6887	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥)‘𝑘) = ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘))
308	307	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑊)
310	287	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
311	290	fvmpt2 7005	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
313	312	adantlr 728	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
315		iftrue 4488	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ 𝑌 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = (𝑥‘𝑘))
316	315	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = (𝑥‘𝑘))
317		vex 3455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 𝑥 ∈ V
318	317	elixp 8932	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↔ (𝑥 Fn 𝑌 ∧ ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))))
319	318	simprbi 503	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) → ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
320	319	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
321		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑌)
322		rspa 3252	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
323	320, 321, 322	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
324	323	ad4ant24 767	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
325	316, 324	eqeltrd 2861	. . . . . . . . . . . . . . . . . . . . . . . . . 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 4621	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑍 ∈ (𝑋 ∖ 𝑌) → 𝑍 ∈ {𝑍})
327	47, 326	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → 𝑍 ∈ {𝑍})
328		elun2 4129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑌 ∪ {𝑍}))
329	327, 328	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → 𝑍 ∈ (𝑌 ∪ {𝑍}))
330	55	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → (𝑌 ∪ {𝑍}) = 𝑊)
331	329, 330	eleqtrd 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
332	200, 331	ffvelcdmd 7085	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ)
333	332	rexrd 11359	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ*)
334	203, 331	ffvelcdmd 7085	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ)
335	334	rexrd 11359	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ*)
336		iccssxr 13561	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4028	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3977	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍))
340	339, 285	sselid 3929	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
341	336, 340	sselid 3929	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝑆 ∈ ℝ*)
342		iccgelb 13533	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐴‘𝑍) ∈ ℝ* ∧ (𝐵‘𝑍) ∈ ℝ* ∧ 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → (𝐴‘𝑍) ≤ 𝑆)
343	333, 335, 340, 342	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → (𝐴‘𝑍) ≤ 𝑆)
344		hoidmvlelem3.sb	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝑆 < (𝐵‘𝑍))
345	333, 335, 341, 343, 344	elicod 13526	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
346	345	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → 𝑆 ∈ ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
347		iffalse 4491	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ 𝑘 ∈ 𝑌 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = 𝑆)
348	347	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = 𝑆)
349	44	eleq2i 2853	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑘 ∈ 𝑊 ↔ 𝑘 ∈ (𝑌 ∪ {𝑍}))
350	349	birani 509	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 ∈ (𝑌 ∪ {𝑍}))
351		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → ¬ 𝑘 ∈ 𝑌)
352		elunnel1 4101	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑘 ∈ (𝑌 ∪ {𝑍}) ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 ∈ {𝑍})
353	350, 351, 352	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 ∈ {𝑍})
354		elsni 4601	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 ∈ {𝑍} → 𝑘 = 𝑍)
355	353, 354	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 = 𝑍)
356		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 = 𝑍 → (𝐴‘𝑘) = (𝐴‘𝑍))
357		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 = 𝑍 → (𝐵‘𝑘) = (𝐵‘𝑍))
358	356, 357	oveq12d 7438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 = 𝑍 → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
359	355, 358	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
360	359	adantll 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
361	348, 360	eleq12d 2855	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
363	362	adantllr 732	. . . . . . . . . . . . . . . . . . . . . . . . . 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 825	. . . . . . . . . . . . . . . . . . . . . . . . 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 2861	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
366	365	ex 418	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑘 ∈ 𝑊 → ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))))
367	306, 366	ralrimi 3261	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
368	301, 367	jca 521	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥) Fn 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))))
369		fvex 6898	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑂‘𝑥) ∈ V
370	369	elixp 8932	. . . . . . . . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
372	283, 371	sseldd 3932	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
373		eliun 4955	. . . . . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∃𝑗 ∈ ℕ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
375		ixpfn 8931	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) → 𝑥 Fn 𝑌)
376	375	adantl 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑥 Fn 𝑌)
377	376	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . 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 1947	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑘 𝑗 ∈ ℕ
379	306, 378	nfan 1932	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ)
380		nfcv 2923	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑘(𝑂‘𝑥)
381		nfixp1 8946	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑘X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))
382	380, 381	nfel 2937	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘(𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))
383	379, 382	nfan 1932	. . . . . . . . . . . . . . . . . . . . . . 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 1150	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
386	258, 385, 311	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
387	386	3adant2 1149	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1153	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = (𝑥‘𝑘))
389	384, 387, 388	3eqtrrd 2801	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) = ((𝑂‘𝑥)‘𝑘))
390	389	ad5ant125 1390	. . . . . . . . . . . . . . . . . . . . . . . . . 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 8932	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 509	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 501	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑊)
395		rspa 3252	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2861	. . . . . . . . . . . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6887	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥)‘𝑍) = ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑍))
402		eqidd 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))
403		eleq1 2849	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑘 = 𝑍 → (𝑘 ∈ 𝑌 ↔ 𝑍 ∈ 𝑌))
404		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑘 = 𝑍 → (𝑥‘𝑘) = (𝑥‘𝑍))
405	403, 404	ifbieq1d 4507	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑘 = 𝑍 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆))
406	405	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ 𝑘 = 𝑍) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆))
407		fvexd 6900	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → (𝑥‘𝑍) ∈ V)
408	407, 286	ifcld 4529	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆) ∈ V)
409	402, 406, 331, 408	fvmptd 7001	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑍) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆))
410	409	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3912	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
412	411	iffalsed 4493	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆) = 𝑆)
413	412	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆) = 𝑆)
414	401, 410, 413	3eqtrrd 2801	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑆 = ((𝑂‘𝑥)‘𝑍))
415	414	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 503	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑘 = 𝑍 → ((𝑂‘𝑥)‘𝑘) = ((𝑂‘𝑥)‘𝑍))
420		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑘 = 𝑍 → ((𝐶‘𝑗)‘𝑘) = ((𝐶‘𝑗)‘𝑍))
421		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑘 = 𝑍 → ((𝐷‘𝑗)‘𝑘) = ((𝐷‘𝑗)‘𝑍))
422	420, 421	oveq12d 7438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑘 = 𝑍 → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
423	419, 422	eleq12d 2855	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑘 = 𝑍 → (((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ↔ ((𝑂‘𝑥)‘𝑍) ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))))
424	423	rspcva 3575	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2861	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1150	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1153	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐽‘𝑗) = ((𝐶‘𝑗) ↾ 𝑌))
430	429	fveq1d 6887	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐽‘𝑗)‘𝑘) = (((𝐶‘𝑗) ↾ 𝑌)‘𝑘))
431	399, 400, 426, 430	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6904	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ 𝑌 → (((𝐶‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
434	433	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 3474	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
437	108	fvmpt2 7005	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
439	438	3adant3 1150	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1153	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐾‘𝑗) = ((𝐷‘𝑗) ↾ 𝑌))
442	441	fveq1d 6887	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐾‘𝑗)‘𝑘) = (((𝐷‘𝑗) ↾ 𝑌)‘𝑘))
443	399, 400, 426, 442	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6904	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ 𝑌 → (((𝐷‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
446	445	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 7438	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . . . . . . . . . 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 2863	. . . . . . . . . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . . . . . . . . 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 3261	. . . . . . . . . . . . . . . . . . . . . 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 521	. . . . . . . . . . . . . . . . . . . . 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 8932	. . . . . . . . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . . . . 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 3176	. . . . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∃𝑗 ∈ ℕ 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
459		eliun 4955	. . . . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑥 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
461	460	ralrimiva 3155	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ X 𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))𝑥 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
462		dfss3 3920	. . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . 14 ⊢ (𝜑 → X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
464		ovexd 7455	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℝ ↑m 𝑌) ∈ V)
465	228	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ℕ ∈ V)
466	464, 465	elmapd 8860	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐾 ∈ ((ℝ ↑m 𝑌) ↑m ℕ) ↔ 𝐾:ℕ⟶(ℝ ↑m 𝑌)))
467	109, 466	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ ((ℝ ↑m 𝑌) ↑m ℕ))
468	464, 465	elmapd 8860	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐽 ∈ ((ℝ ↑m 𝑌) ↑m ℕ) ↔ 𝐽:ℕ⟶(ℝ ↑m 𝑌)))
469	89, 468	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐽 ∈ ((ℝ ↑m 𝑌) ↑m ℕ))
470	82, 71	elmapd 8860	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝐵 ↾ 𝑌) ∈ (ℝ ↑m 𝑌) ↔ (𝐵 ↾ 𝑌):𝑌⟶ℝ))
471	204, 470	mpbird 260	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 ↾ 𝑌) ∈ (ℝ ↑m 𝑌))
472	82, 71	elmapd 8860	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐴 ↾ 𝑌) ∈ (ℝ ↑m 𝑌) ↔ (𝐴 ↾ 𝑌):𝑌⟶ℝ))
473	202, 472	mpbird 260	. . . . . . . . . . . . . . . . . 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 6884	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑒 = (𝐴 ↾ 𝑌) → (𝑒‘𝑘) = ((𝐴 ↾ 𝑌)‘𝑘))
476	475	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑒‘𝑘) = ((𝐴 ↾ 𝑌)‘𝑘))
477	250	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝐴 ↾ 𝑌)‘𝑘) = (𝐴‘𝑘))
478	476, 477	eqtrd 2796	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑒‘𝑘) = (𝐴‘𝑘))
479	478	oveq1d 7435	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = ((𝐴‘𝑘)[,)(𝑓‘𝑘)))
480	479	ixpeq2dva 8940	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒 = (𝐴 ↾ 𝑌) → X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)))
481	480	sseq1d 3962	. . . . . . . . . . . . . . . . . . . . . . 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 7427	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒 = (𝐴 ↾ 𝑌) → (𝑒(𝐿‘𝑌)𝑓) = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓))
483	482	breq1d 5113	. . . . . . . . . . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . . . . . . . . . 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 3186	. . . . . . . . . . . . . . . . . . . . 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 3186	. . . . . . . . . . . . . . . . . . . 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 3186	. . . . . . . . . . . . . . . . . . 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 3575	. . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . 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 6884	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 = (𝐵 ↾ 𝑌) → (𝑓‘𝑘) = ((𝐵 ↾ 𝑌)‘𝑘))
491	490	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑓‘𝑘) = ((𝐵 ↾ 𝑌)‘𝑘))
492	251	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝐵 ↾ 𝑌)‘𝑘) = (𝐵‘𝑘))
493	491, 492	eqtrd 2796	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑓‘𝑘) = (𝐵‘𝑘))
494	493	oveq2d 7436	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘)[,)(𝑓‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
495	494	ixpeq2dva 8940	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = (𝐵 ↾ 𝑌) → X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) = X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
496	495	sseq1d 3962	. . . . . . . . . . . . . . . . . . . . 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 7428	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = (𝐵 ↾ 𝑌) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)))
498	497	breq1d 5113	. . . . . . . . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . . . . . . . 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 3186	. . . . . . . . . . . . . . . . . . 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 3186	. . . . . . . . . . . . . . . . . 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 3575	. . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . 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 6884	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑔 = 𝐽 → (𝑔‘𝑗) = (𝐽‘𝑗))
505	504	fveq1d 6887	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑔 = 𝐽 → ((𝑔‘𝑗)‘𝑘) = ((𝐽‘𝑗)‘𝑘))
506	505	oveq1d 7435	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔 = 𝐽 → (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))
507	506	ixpeq2dv 8941	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = 𝐽 → X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))
508	507	iuneq2d 4981	. . . . . . . . . . . . . . . . . . . 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 3963	. . . . . . . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔 = 𝐽 → ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)) = ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))
511	510	mpteq2dv 5199	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = 𝐽 → (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))
512	511	fveq2d 6889	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = 𝐽 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))
513	512	breq2d 5115	. . . . . . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . . . . . 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 3186	. . . . . . . . . . . . . . . . 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 3575	. . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . 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 6884	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ = 𝐾 → (ℎ‘𝑗) = (𝐾‘𝑗))
519	518	fveq1d 6887	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ = 𝐾 → ((ℎ‘𝑗)‘𝑘) = ((𝐾‘𝑗)‘𝑘))
520	519	oveq2d 7436	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = 𝐾 → (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
521	520	ixpeq2dv 8941	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = 𝐾 → X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
522	521	iuneq2d 4981	. . . . . . . . . . . . . . . . . 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 3963	. . . . . . . . . . . . . . . . 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 7436	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = 𝐾 → ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
525	524	mpteq2dv 5199	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = 𝐾 → (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))
526	525	fveq2d 6889	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = 𝐾 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
527	526	breq2d 5115	. . . . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . . . 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 3575	. . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 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 16	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
532		idd 25	. . . . . . . . . . . . 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 16	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
534	533	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
535	41	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
536	535	mpteq2dva 5198	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)) = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))
537	536	fveq2d 6889	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
538	249, 537	breq12d 5116	. . . . . . . . . . 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 260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
540	539	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → 𝐺 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
541	238, 240, 241, 281, 540	ltletrd 11470	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → (𝐺 / (1 + 𝐸)) < (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
542	226, 237, 541	syl2anc 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) < (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
543	225, 542	pm2.61dan 825	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 / (1 + 𝐸)) < (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
544	196, 197, 198, 199, 218, 543	sge0uzfsumgt 47453	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ∃𝑚 ∈ ℕ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))
545	217	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (𝐺 / (1 + 𝐸)) ∈ ℝ)
546		fzfid 14116	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...𝑚) ∈ Fin)
547		simpl 488	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → 𝜑)
548		elfznn 13687	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (1...𝑚) → 𝑗 ∈ ℕ)
549	548	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → 𝑗 ∈ ℕ)
550	28, 114	sselid 3929	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ ℝ)
551	547, 549, 550	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → (𝑃‘𝑗) ∈ ℝ)
552	546, 551	fsumrecl 15900	. . . . . . . . . . . 12 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) ∈ ℝ)
553	552	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) ∈ ℝ)
554		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))
555	545, 553, 554	ltled 11458	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (𝐺 / (1 + 𝐸)) ≤ Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))
556	207	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ∈ ℝ)
557	213	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (1 + 𝐸) ∈ ℝ+)
558	556, 553, 557	ledivmuld 13217	. . . . . . . . . 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 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
560	559	ex 418	. . . . . . . 8 ⊢ (𝜑 → ((𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
561	560	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
562	561	adantlr 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ 𝑚 ∈ ℕ) → ((𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
563	562	reximdva 3176	. . . . 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 16	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
565	193, 195, 564	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 = ∅) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
566	192, 565	pm2.61dan 825	. 2 ⊢ (𝜑 → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
567	43	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑋 ∈ Fin)
568	46	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑌 ⊆ 𝑋)
569	47	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑍 ∈ (𝑋 ∖ 𝑌))
570	200	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐴:𝑊⟶ℝ)
571	203	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐵:𝑊⟶ℝ)
572	62	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐶:ℕ⟶(ℝ ↑m 𝑊))
573		eqid 2761	. . . . 5 ⊢ (𝑦 ∈ 𝑌 ↦ 0) = (𝑦 ∈ 𝑌 ↦ 0)
574		eqid 2761	. . . . 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 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐷:ℕ⟶(ℝ ↑m 𝑊))
576		eqid 2761	. . . . 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 6885	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝐶‘𝑖) = (𝐶‘𝑗))
578		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗))
579	577, 578	oveq12d 7438	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)) = ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))
580	579	cbvmptv 5209	. . . . . . . 8 ⊢ (𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))
581	580	fveq2i 6888	. . . . . . 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 2865	. . . . . 6 ⊢ (𝜑 → (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ)
584	583	3ad2ant1 1151	. . . . 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 2844	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ 𝑌 ↔ 𝑖 ∈ 𝑌))
587		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝑐‘𝑗) = (𝑐‘𝑖))
588	587	breq1d 5113	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → ((𝑐‘𝑗) ≤ 𝑥 ↔ (𝑐‘𝑖) ≤ 𝑥))
589	588, 587	ifbieq1d 4507	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥) = if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥))
590	586, 587, 589	ifbieq12d 4511	. . . . . . . . 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 5209	. . . . . . . 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 5201	. . . . . . 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 5201	. . . . . 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 2784	. . . . 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 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐸 ∈ ℝ+)
596		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → (𝐶‘𝑗) = (𝐶‘𝑖))
597		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑖 → (𝐷‘𝑗) = (𝐷‘𝑖))
598	597	fveq2d 6889	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘𝑧)‘(𝐷‘𝑖)))
599	596, 598	oveq12d 7438	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))
600	599	cbvmptv 5209	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))) = (𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))
601	600	fveq2i 6888	. . . . . . . . 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 7431	. . . . . . . 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 5111	. . . . . . 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 3418	. . . . . 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 2784	. . . . 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 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑆 ∈ 𝑈)
607	344	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑆 < (𝐵‘𝑍))
608		eqid 2761	. . . . 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 1155	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑚 ∈ ℕ)
610		id 23	. . . . . . . 8 ⊢ (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
611		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → (𝑃‘𝑗) = (𝑃‘𝑖))
612	611	cbvsumv 15863	. . . . . . . . . 10 ⊢ Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) = Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)
613	612	oveq2i 7431	. . . . . . . . 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 5131	. . . . . . 7 ⊢ (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)))
616	615	3ad2ant3 1153	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)))
617		simpl 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑚)) → 𝜑)
618		elfznn 13687	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...𝑚) → 𝑖 ∈ ℕ)
619	618	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ∈ ℕ)
620		eleq1w 2844	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ ℕ ↔ 𝑖 ∈ ℕ))
621		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑖 → (𝐽‘𝑗) = (𝐽‘𝑖))
622		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑖 → (𝐾‘𝑗) = (𝐾‘𝑖))
623	621, 622	oveq12d 7438	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑖 → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)))
624	611, 623	eqeq12d 2777	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → ((𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ↔ (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖))))
625	620, 624	imbi12d 347	. . . . . . . . . . . . . 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 2022	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ℕ → (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)))
627	626	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)))
628	620	anbi2d 642	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → ((𝜑 ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ 𝑖 ∈ ℕ)))
629	596	fveq1d 6887	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑖 → ((𝐶‘𝑗)‘𝑍) = ((𝐶‘𝑖)‘𝑍))
630	597	fveq1d 6887	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑖 → ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑖)‘𝑍))
631	629, 630	oveq12d 7438	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑖 → (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) = (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
632	631	eleq2d 2847	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑖 → (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))))
633	596	reseq1d 5969	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑖 → ((𝐶‘𝑗) ↾ 𝑌) = ((𝐶‘𝑖) ↾ 𝑌))
634	632, 633	ifbieq1d 4507	. . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . 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 2022	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐽‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹))
638	597	reseq1d 5969	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑖 → ((𝐷‘𝑗) ↾ 𝑌) = ((𝐷‘𝑖) ↾ 𝑌))
639	632, 638	ifbieq1d 4507	. . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . 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 2022	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐾‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹))
643	637, 642	oveq12d 7438	. . . . . . . . . . . 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 2796	. . . . . . . . . . 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 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
646		ovexd 7455	. . . . . . . . . . . . 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 7005	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 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 6898	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶‘𝑖) ∈ V
650	649	resex 6018	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶‘𝑖) ↾ 𝑌) ∈ V
651	650	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐶‘𝑖) ↾ 𝑌) ∈ V)
652	80, 143	eqeltrrid 2866	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ 0) ∈ V)
653	651, 652	ifcld 4529	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V)
654	653	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V)
655	574	fvmpt2 7005	. . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑌 ↦ 0) = 𝐹
658		ifeq2 4487	. . . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . 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 6898	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷‘𝑖) ∈ V
663	662	resex 6018	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷‘𝑖) ↾ 𝑌) ∈ V
664	663	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐷‘𝑖) ↾ 𝑌) ∈ V)
665	664, 652	ifcld 4529	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V)
666	665	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V)
667	576	fvmpt2 7005	. . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 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 264	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
670	669, 657	ifbieq2i 4508	. . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . 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 7438	. . . . . . . . . . . 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 2796	. . . . . . . . . . 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 2799	. . . . . . . . . 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 596	. . . . . . . . 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 1204	. . . . . . . 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 15869	. . . . . . 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 7436	. . . . . 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 5131	. . . . 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 6885	. . . . . . . 8 ⊢ (𝑗 = ℎ → (𝐷‘𝑗) = (𝐷‘ℎ))
682	681	fveq1d 6887	. . . . . . 7 ⊢ (𝑗 = ℎ → ((𝐷‘𝑗)‘𝑍) = ((𝐷‘ℎ)‘𝑍))
683	682	cbvmptv 5209	. . . . . 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 5919	. . . . 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 6885	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑖 → (𝐶‘ℎ) = (𝐶‘𝑖))
686	685	fveq1d 6887	. . . . . . . . . . 11 ⊢ (ℎ = 𝑖 → ((𝐶‘ℎ)‘𝑍) = ((𝐶‘𝑖)‘𝑍))
687		fveq2 6885	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑖 → (𝐷‘ℎ) = (𝐷‘𝑖))
688	687	fveq1d 6887	. . . . . . . . . . 11 ⊢ (ℎ = 𝑖 → ((𝐷‘ℎ)‘𝑍) = ((𝐷‘𝑖)‘𝑍))
689	686, 688	oveq12d 7438	. . . . . . . . . 10 ⊢ (ℎ = 𝑖 → (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍)) = (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
690	689	eleq2d 2847	. . . . . . . . 9 ⊢ (ℎ = 𝑖 → (𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))))
691	690	cbvrabv 3423	. . . . . . . 8 ⊢ {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} = {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))}
692	691	mpteq1i 5196	. . . . . . 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 5919	. . . . . 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 4112	. . . . 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 2761	. . . . 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 47605	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
697	696	3exp 1137	. . 3 ⊢ (𝜑 → (𝑚 ∈ ℕ → (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)))
698	697	rexlimdv 3162	. 2 ⊢ (𝜑 → (∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢))
699	566, 698	mpd 16	1 ⊢ (𝜑 → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  {crab 3413  Vcvv 3451   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  ∅c0 4279  ifcif 4482  {csn 4584  ∪ ciun 4951   class class class wbr 5103   ↦ cmpt 5186  ran crn 5652   ↾ cres 5653   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422   ↑_m cmap 8847  Xcixp 8925  Fincfn 8973  infcinf 9433  ℝcr 11199  0cc0 11200  1c1 11201   + caddc 11203   · cmul 11205  +∞cpnf 11340  ℝ^*cxr 11342   < clt 11343   ≤ cle 11344   − cmin 11541   / cdiv 11973  ℕcn 12335  ℤcz 12693  ℝ⁺crp 13120  [,)cico 13478  [,]cicc 13479  ...cfz 13639  Σcsu 15853  ∏cprod 16072  volcvol 25784  Σ^{^}csumge0 47371

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-inf2 9642  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-map 8849  df-pm 8850  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-fi 9403  df-sup 9434  df-inf 9435  df-oi 9504  df-dju 9982  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-n0 12607  df-z 12694  df-uz 12966  df-q 13076  df-rp 13121  df-xneg 13241  df-xadd 13242  df-xmul 13243  df-ioo 13480  df-ico 13482  df-icc 13483  df-fz 13640  df-fzo 13789  df-fl 13932  df-seq 14145  df-exp 14205  df-hash 14475  df-cj 15266  df-re 15267  df-im 15268  df-sqrt 15402  df-abs 15403  df-clim 15655  df-rlim 15656  df-sum 15854  df-prod 16073  df-rest 17593  df-topgen 17614  df-psmet 21670  df-xmet 21671  df-met 21672  df-bl 21673  df-mopn 21674  df-top 23212  df-topon 23229  df-bases 23264  df-cmp 23705  df-ovol 25785  df-vol 25786  df-sumge0 47372

This theorem is used by:  hoidmvlelem4  47607