hoidmvlelem1 - Mathbox for Glauco Siliprandi

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Theorem hoidmvlelem1 46610

Description: The supremum of 𝑈 belongs to 𝑈. Step (c) in the proof of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

**Hypotheses**
Ref	Expression
hoidmvlelem1.l	⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑_m 𝑥), 𝑏 ∈ (ℝ ↑_m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
hoidmvlelem1.x	⊢ (𝜑 → 𝑋 ∈ Fin)
hoidmvlelem1.y	⊢ (𝜑 → 𝑌 ⊆ 𝑋)
hoidmvlelem1.z	⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))
hoidmvlelem1.w	⊢ 𝑊 = (𝑌 ∪ {𝑍})
hoidmvlelem1.a	⊢ (𝜑 → 𝐴:𝑊⟶ℝ)
hoidmvlelem1.b	⊢ (𝜑 → 𝐵:𝑊⟶ℝ)
hoidmvlelem1.c	⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑_m 𝑊))
hoidmvlelem1.d	⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑_m 𝑊))
hoidmvlelem1.r	⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
hoidmvlelem1.h	⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))
hoidmvlelem1.g	⊢ 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))
hoidmvlelem1.e	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
hoidmvlelem1.u	⊢ 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))}
hoidmvlelem1.s	⊢ 𝑆 = sup(𝑈, ℝ, < )
hoidmvlelem1.ab	⊢ (𝜑 → (𝐴‘𝑍) < (𝐵‘𝑍))

**Assertion**
Ref	Expression
hoidmvlelem1	⊢ (𝜑 → 𝑆 ∈ 𝑈)

Distinct variable groups: 𝐴,𝑎,𝑏 𝑧,𝐻 𝜑,𝑎,𝑏 𝑧,𝑈 𝑧,𝐸 𝑧,𝑆 𝑧,𝐺 𝑗,𝑊,𝑧 𝑆,𝑐 𝑈,𝑐 𝐻,𝑐 𝐿,𝑐 𝑊,𝑐,𝑥 𝐸,𝑐 𝑧,𝐷 𝑗,𝑐,𝑘,𝜑,𝑥 𝐺,𝑐 𝐷,𝑐 𝑌,𝑐 𝐻,𝑎,𝑏,𝑘 𝑌,𝑎,𝑏,𝑗,𝑘,𝑥 𝑆,𝑎,𝑏,𝑗,𝑘,𝑥 𝑈,𝑎,𝑏,𝑗,𝑘,𝑥 𝑧,𝐿 𝐷,𝑎,𝑏,𝑘 𝐵,𝑐 𝐵,𝑎,𝑏,𝑘 𝑍,𝑎,𝑏 𝑧,𝑍 𝑍,𝑐,𝑗,𝑘,𝑥 𝐶,𝑐 𝑧,𝐶 𝑧,𝐵 𝐴,𝑐,𝑗,𝑘,𝑥 𝑧,𝐴 𝑊,𝑎,𝑏,𝑘 𝐶,𝑎,𝑏,𝑘 Allowed substitution hints: 𝜑(𝑧) 𝐵(𝑥,𝑗) 𝐶(𝑥,𝑗) 𝐷(𝑥,𝑗) 𝐸(𝑥,𝑗,𝑘,𝑎,𝑏) 𝐺(𝑥,𝑗,𝑘,𝑎,𝑏) 𝐻(𝑥,𝑗) 𝐿(𝑥,𝑗,𝑘,𝑎,𝑏) 𝑋(𝑥,𝑧,𝑗,𝑘,𝑎,𝑏,𝑐) 𝑌(𝑧)

Proof of Theorem hoidmvlelem1 Dummy variables ℎ 𝑤 𝑦 𝑢 𝑟 𝑠 𝑡 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hoidmvlelem1.s	. . . . . 6 ⊢ 𝑆 = sup(𝑈, ℝ, < )
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑆 = sup(𝑈, ℝ, < ))
3		hoidmvlelem1.a	. . . . . . 7 ⊢ (𝜑 → 𝐴:𝑊⟶ℝ)
4		hoidmvlelem1.z	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))
5		snidg 4660	. . . . . . . . . 10 ⊢ (𝑍 ∈ (𝑋 ∖ 𝑌) → 𝑍 ∈ {𝑍})
6	4, 5	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ {𝑍})
7		elun2 4183	. . . . . . . . 9 ⊢ (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑌 ∪ {𝑍}))
8	6, 7	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ (𝑌 ∪ {𝑍}))
9		hoidmvlelem1.w	. . . . . . . 8 ⊢ 𝑊 = (𝑌 ∪ {𝑍})
10	8, 9	eleqtrrdi 2852	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
11	3, 10	ffvelcdmd 7105	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ)
12		hoidmvlelem1.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑊⟶ℝ)
13	12, 10	ffvelcdmd 7105	. . . . . 6 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ)
14		hoidmvlelem1.u	. . . . . . . 8 ⊢ 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))}
15		ssrab2 4080	. . . . . . . 8 ⊢ {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))
16	14, 15	eqsstri 4030	. . . . . . 7 ⊢ 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍))
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
18	11	leidd 11829	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴‘𝑍) ≤ (𝐴‘𝑍))
19		hoidmvlelem1.ab	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴‘𝑍) < (𝐵‘𝑍))
20	11, 13, 19	ltled 11409	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴‘𝑍) ≤ (𝐵‘𝑍))
21	11, 13, 11, 18, 20	eliccd 45517	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
22	11	recnd 11289	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℂ)
23	22	subidd 11608	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴‘𝑍) − (𝐴‘𝑍)) = 0)
24	23	oveq2d 7447	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 · ((𝐴‘𝑍) − (𝐴‘𝑍))) = (𝐺 · 0))
25		rge0ssre 13496	. . . . . . . . . . . . . . 15 ⊢ (0[,)+∞) ⊆ ℝ
26		hoidmvlelem1.g	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))
27		hoidmvlelem1.l	. . . . . . . . . . . . . . . . 17 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑_m 𝑥), 𝑏 ∈ (ℝ ↑_m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
28		hoidmvlelem1.x	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋 ∈ Fin)
29		hoidmvlelem1.y	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
30	28, 29	ssfid 9301	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑌 ∈ Fin)
31		ssun1 4178	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑌 ⊆ (𝑌 ∪ {𝑍})
32	31, 9	sseqtrri 4033	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑌 ⊆ 𝑊
33	32	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑌 ⊆ 𝑊)
34	3, 33	fssresd 6775	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ↾ 𝑌):𝑌⟶ℝ)
35	12, 33	fssresd 6775	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 ↾ 𝑌):𝑌⟶ℝ)
36	27, 30, 34, 35	hoidmvcl 46597	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ∈ (0[,)+∞))
37	26, 36	eqeltrid 2845	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ∈ (0[,)+∞))
38	25, 37	sselid 3981	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ ℝ)
39	38	recnd 11289	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ ℂ)
40	39	mul01d 11460	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 · 0) = 0)
41	24, 40	eqtrd 2777	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 · ((𝐴‘𝑍) − (𝐴‘𝑍))) = 0)
42		1red 11262	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℝ)
43		hoidmvlelem1.e	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
44	43	rpred 13077	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ)
45	42, 44	readdcld 11290	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 + 𝐸) ∈ ℝ)
46		0red 11264	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℝ)
47		0lt1 11785	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
48	47	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 1)
49	42, 43	ltaddrpd 13110	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 < (1 + 𝐸))
50	46, 42, 45, 48, 49	lttrd 11422	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < (1 + 𝐸))
51	46, 45, 50	ltled 11409	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (1 + 𝐸))
52		nnex 12272	. . . . . . . . . . . . . . 15 ⊢ ℕ ∈ V
53	52	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℕ ∈ V)
54		icossicc 13476	. . . . . . . . . . . . . . . 16 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
55		snfi 9083	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑍} ∈ Fin
56	55	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → {𝑍} ∈ Fin)
57		unfi 9211	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑌 ∈ Fin ∧ {𝑍} ∈ Fin) → (𝑌 ∪ {𝑍}) ∈ Fin)
58	30, 56, 57	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑌 ∪ {𝑍}) ∈ Fin)
59	9, 58	eqeltrid 2845	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑊 ∈ Fin)
60	59	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
61		hoidmvlelem1.c	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑_m 𝑊))
62	61	ffvelcdmda 7104	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑_m 𝑊))
63		elmapi 8889	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶‘𝑗) ∈ (ℝ ↑_m 𝑊) → (𝐶‘𝑗):𝑊⟶ℝ)
64	62, 63	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑊⟶ℝ)
65		hoidmvlelem1.h	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))
66		eleq1w 2824	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = ℎ → (𝑗 ∈ 𝑌 ↔ ℎ ∈ 𝑌))
67		fveq2 6906	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = ℎ → (𝑐‘𝑗) = (𝑐‘ℎ))
68	67	breq1d 5153	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = ℎ → ((𝑐‘𝑗) ≤ 𝑥 ↔ (𝑐‘ℎ) ≤ 𝑥))
69	68, 67	ifbieq1d 4550	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = ℎ → if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥) = if((𝑐‘ℎ) ≤ 𝑥, (𝑐‘ℎ), 𝑥))
70	66, 67, 69	ifbieq12d 4554	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = ℎ → if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)) = if(ℎ ∈ 𝑌, (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑥, (𝑐‘ℎ), 𝑥)))
71	70	cbvmptv 5255	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥))) = (ℎ ∈ 𝑊 ↦ if(ℎ ∈ 𝑌, (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑥, (𝑐‘ℎ), 𝑥)))
72	71	mpteq2i 5247	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ (ℝ ↑_m 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))) = (𝑐 ∈ (ℝ ↑_m 𝑊) ↦ (ℎ ∈ 𝑊 ↦ if(ℎ ∈ 𝑌, (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑥, (𝑐‘ℎ), 𝑥))))
73	72	mpteq2i 5247	. . . . . . . . . . . . . . . . . . 19 ⊢ (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) ↦ (ℎ ∈ 𝑊 ↦ if(ℎ ∈ 𝑌, (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑥, (𝑐‘ℎ), 𝑥)))))
74	65, 73	eqtri 2765	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑊) ↦ (ℎ ∈ 𝑊 ↦ if(ℎ ∈ 𝑌, (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑥, (𝑐‘ℎ), 𝑥)))))
75	11	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐴‘𝑍) ∈ ℝ)
76		hoidmvlelem1.d	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑_m 𝑊))
77	76	ffvelcdmda 7104	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑_m 𝑊))
78		elmapi 8889	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐷‘𝑗) ∈ (ℝ ↑_m 𝑊) → (𝐷‘𝑗):𝑊⟶ℝ)
79	77, 78	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑊⟶ℝ)
80	74, 75, 60, 79	hsphoif 46591	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)):𝑊⟶ℝ)
81	27, 60, 64, 80	hoidmvcl 46597	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))) ∈ (0[,)+∞))
82	54, 81	sselid 3981	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))) ∈ (0[,]+∞))
83	82	fmpttd 7135	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)))):ℕ⟶(0[,]+∞))
84	53, 83	sge0cl 46396	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ∈ (0[,]+∞))
85	53, 83	sge0xrcl 46400	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ∈ ℝ^*)
86		pnfxr 11315	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ^*
87	86	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → +∞ ∈ ℝ^*)
88		hoidmvlelem1.r	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
89	88	rexrd 11311	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ^*)
90		nfv 1914	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗𝜑
91	27, 60, 64, 79	hoidmvcl 46597	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) ∈ (0[,)+∞))
92	54, 91	sselid 3981	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) ∈ (0[,]+∞))
93	4	eldifbd 3964	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
94	10, 93	eldifd 3962	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑍 ∈ (𝑊 ∖ 𝑌))
95	94	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑍 ∈ (𝑊 ∖ 𝑌))
96	27, 60, 95, 9, 75, 74, 64, 79	hsphoidmvle 46601	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))
97	90, 53, 82, 92, 96	sge0lempt 46425	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))
98	88	ltpnfd 13163	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) < +∞)
99	85, 89, 87, 97, 98	xrlelttrd 13202	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) < +∞)
100	85, 87, 99	xrltned 45368	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ≠ +∞)
101		ge0xrre 45544	. . . . . . . . . . . . 13 ⊢ (((Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ∈ (0[,]+∞) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ≠ +∞) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ∈ ℝ)
102	84, 100, 101	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ∈ ℝ)
103	53, 83	sge0ge0 46399	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))))
104		mulge0 11781	. . . . . . . . . . . 12 ⊢ ((((1 + 𝐸) ∈ ℝ ∧ 0 ≤ (1 + 𝐸)) ∧ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ∈ ℝ ∧ 0 ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))))) → 0 ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)))))))
105	45, 51, 102, 103, 104	syl22anc 839	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)))))))
106	41, 105	eqbrtrd 5165	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 · ((𝐴‘𝑍) − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)))))))
107	21, 106	jca 511	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴‘𝑍) ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∧ (𝐺 · ((𝐴‘𝑍) − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))))))
108		oveq1 7438	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐴‘𝑍) → (𝑧 − (𝐴‘𝑍)) = ((𝐴‘𝑍) − (𝐴‘𝑍)))
109	108	oveq2d 7447	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐴‘𝑍) → (𝐺 · (𝑧 − (𝐴‘𝑍))) = (𝐺 · ((𝐴‘𝑍) − (𝐴‘𝑍))))
110		fveq2 6906	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐴‘𝑍) → (𝐻‘𝑧) = (𝐻‘(𝐴‘𝑍)))
111	110	fveq1d 6908	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐴‘𝑍) → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)))
112	111	oveq2d 7447	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐴‘𝑍) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))
113	112	mpteq2dv 5244	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐴‘𝑍) → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)))))
114	113	fveq2d 6910	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐴‘𝑍) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))))
115	114	oveq2d 7447	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐴‘𝑍) → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) = ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)))))))
116	109, 115	breq12d 5156	. . . . . . . . . 10 ⊢ (𝑧 = (𝐴‘𝑍) → ((𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) ↔ (𝐺 · ((𝐴‘𝑍) − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))))))
117	116	elrab 3692	. . . . . . . . 9 ⊢ ((𝐴‘𝑍) ∈ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} ↔ ((𝐴‘𝑍) ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∧ (𝐺 · ((𝐴‘𝑍) − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))))))
118	107, 117	sylibr 234	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘𝑍) ∈ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))})
119	118, 14	eleqtrrdi 2852	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝑍) ∈ 𝑈)
120		ne0i 4341	. . . . . . 7 ⊢ ((𝐴‘𝑍) ∈ 𝑈 → 𝑈 ≠ ∅)
121	119, 120	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈 ≠ ∅)
122	11, 13, 17, 121	supicc 13541	. . . . 5 ⊢ (𝜑 → sup(𝑈, ℝ, < ) ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
123	2, 122	eqeltrd 2841	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
124	2	oveq1d 7446	. . . . . . 7 ⊢ (𝜑 → (𝑆 − (𝐴‘𝑍)) = (sup(𝑈, ℝ, < ) − (𝐴‘𝑍)))
125	124	oveq2d 7447	. . . . . 6 ⊢ (𝜑 → (𝐺 · (𝑆 − (𝐴‘𝑍))) = (𝐺 · (sup(𝑈, ℝ, < ) − (𝐴‘𝑍))))
126	11, 13	iccssred 13474	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴‘𝑍)[,](𝐵‘𝑍)) ⊆ ℝ)
127	17, 126	sstrd 3994	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ⊆ ℝ)
128	11, 13	jca 511	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴‘𝑍) ∈ ℝ ∧ (𝐵‘𝑍) ∈ ℝ))
129		iccsupr 13482	. . . . . . . . . 10 ⊢ ((((𝐴‘𝑍) ∈ ℝ ∧ (𝐵‘𝑍) ∈ ℝ) ∧ 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∧ (𝐴‘𝑍) ∈ 𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑈 𝑧 ≤ 𝑦))
130	128, 17, 119, 129	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑈 𝑧 ≤ 𝑦))
131	130	simp3d 1145	. . . . . . . 8 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑈 𝑧 ≤ 𝑦)
132		eqid 2737	. . . . . . . 8 ⊢ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} = {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}
133	127, 121, 131, 11, 132	supsubc 45364	. . . . . . 7 ⊢ (𝜑 → (sup(𝑈, ℝ, < ) − (𝐴‘𝑍)) = sup({𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}, ℝ, < ))
134	133	oveq2d 7447	. . . . . 6 ⊢ (𝜑 → (𝐺 · (sup(𝑈, ℝ, < ) − (𝐴‘𝑍))) = (𝐺 · sup({𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}, ℝ, < )))
135	46	rexrd 11311	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ^*)
136		icogelb 13438	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐺 ∈ (0[,)+∞)) → 0 ≤ 𝐺)
137	135, 87, 37, 136	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝐺)
138		vex 3484	. . . . . . . . . . . 12 ⊢ 𝑟 ∈ V
139		eqeq1 2741	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑟 → (𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ 𝑟 = (𝑢 − (𝐴‘𝑍))))
140	139	rexbidv 3179	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑟 → (∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ ∃𝑢 ∈ 𝑈 𝑟 = (𝑢 − (𝐴‘𝑍))))
141	138, 140	elab 3679	. . . . . . . . . . 11 ⊢ (𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ↔ ∃𝑢 ∈ 𝑈 𝑟 = (𝑢 − (𝐴‘𝑍)))
142	141	biimpi 216	. . . . . . . . . 10 ⊢ (𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} → ∃𝑢 ∈ 𝑈 𝑟 = (𝑢 − (𝐴‘𝑍)))
143	142	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → ∃𝑢 ∈ 𝑈 𝑟 = (𝑢 − (𝐴‘𝑍)))
144		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑢𝜑
145		nfcv 2905	. . . . . . . . . . . 12 ⊢ Ⅎ𝑢𝑟
146		nfre1 3285	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑢∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))
147	146	nfab 2911	. . . . . . . . . . . 12 ⊢ Ⅎ𝑢{𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}
148	145, 147	nfel 2920	. . . . . . . . . . 11 ⊢ Ⅎ𝑢 𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}
149	144, 148	nfan 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑢(𝜑 ∧ 𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))})
150		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑢0 ≤ 𝑟
151	11	rexrd 11311	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ^*)
152	151	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴‘𝑍) ∈ ℝ^*)
153	13	rexrd 11311	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ^*)
154	153	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐵‘𝑍) ∈ ℝ^*)
155	17	sselda 3983	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
156		iccgelb 13443	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴‘𝑍) ∈ ℝ^* ∧ (𝐵‘𝑍) ∈ ℝ^* ∧ 𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → (𝐴‘𝑍) ≤ 𝑢)
157	152, 154, 155, 156	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴‘𝑍) ≤ 𝑢)
158	127	sselda 3983	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ ℝ)
159	11	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴‘𝑍) ∈ ℝ)
160	158, 159	subge0d 11853	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (0 ≤ (𝑢 − (𝐴‘𝑍)) ↔ (𝐴‘𝑍) ≤ 𝑢))
161	157, 160	mpbird 257	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 0 ≤ (𝑢 − (𝐴‘𝑍)))
162	161	3adant3 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑟 = (𝑢 − (𝐴‘𝑍))) → 0 ≤ (𝑢 − (𝐴‘𝑍)))
163		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = (𝑢 − (𝐴‘𝑍)) → 𝑟 = (𝑢 − (𝐴‘𝑍)))
164	163	eqcomd 2743	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑢 − (𝐴‘𝑍)) → (𝑢 − (𝐴‘𝑍)) = 𝑟)
165	164	3ad2ant3 1136	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑟 = (𝑢 − (𝐴‘𝑍))) → (𝑢 − (𝐴‘𝑍)) = 𝑟)
166	162, 165	breqtrd 5169	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑟 = (𝑢 − (𝐴‘𝑍))) → 0 ≤ 𝑟)
167	166	3exp 1120	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑟 = (𝑢 − (𝐴‘𝑍)) → 0 ≤ 𝑟)))
168	167	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → (𝑢 ∈ 𝑈 → (𝑟 = (𝑢 − (𝐴‘𝑍)) → 0 ≤ 𝑟)))
169	149, 150, 168	rexlimd 3266	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → (∃𝑢 ∈ 𝑈 𝑟 = (𝑢 − (𝐴‘𝑍)) → 0 ≤ 𝑟))
170	143, 169	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → 0 ≤ 𝑟)
171	170	ralrimiva 3146	. . . . . . 7 ⊢ (𝜑 → ∀𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}0 ≤ 𝑟)
172		simp3 1139	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑤 = (𝑢 − (𝐴‘𝑍))) → 𝑤 = (𝑢 − (𝐴‘𝑍)))
173	158, 159	resubcld 11691	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 − (𝐴‘𝑍)) ∈ ℝ)
174	173	3adant3 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑤 = (𝑢 − (𝐴‘𝑍))) → (𝑢 − (𝐴‘𝑍)) ∈ ℝ)
175	172, 174	eqeltrd 2841	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑤 = (𝑢 − (𝐴‘𝑍))) → 𝑤 ∈ ℝ)
176	175	3exp 1120	. . . . . . . . . 10 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑤 = (𝑢 − (𝐴‘𝑍)) → 𝑤 ∈ ℝ)))
177	176	rexlimdv 3153	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) → 𝑤 ∈ ℝ))
178	177	alrimiv 1927	. . . . . . . 8 ⊢ (𝜑 → ∀𝑤(∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) → 𝑤 ∈ ℝ))
179		abss 4063	. . . . . . . 8 ⊢ ({𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ⊆ ℝ ↔ ∀𝑤(∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) → 𝑤 ∈ ℝ))
180	178, 179	sylibr 234	. . . . . . 7 ⊢ (𝜑 → {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ⊆ ℝ)
181	23	eqcomd 2743	. . . . . . . . . 10 ⊢ (𝜑 → 0 = ((𝐴‘𝑍) − (𝐴‘𝑍)))
182		oveq1 7438	. . . . . . . . . . 11 ⊢ (𝑢 = (𝐴‘𝑍) → (𝑢 − (𝐴‘𝑍)) = ((𝐴‘𝑍) − (𝐴‘𝑍)))
183	182	rspceeqv 3645	. . . . . . . . . 10 ⊢ (((𝐴‘𝑍) ∈ 𝑈 ∧ 0 = ((𝐴‘𝑍) − (𝐴‘𝑍))) → ∃𝑢 ∈ 𝑈 0 = (𝑢 − (𝐴‘𝑍)))
184	119, 181, 183	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑢 ∈ 𝑈 0 = (𝑢 − (𝐴‘𝑍)))
185		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑤 = 0 → (𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ 0 = (𝑢 − (𝐴‘𝑍))))
186	185	rexbidv 3179	. . . . . . . . 9 ⊢ (𝑤 = 0 → (∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ ∃𝑢 ∈ 𝑈 0 = (𝑢 − (𝐴‘𝑍))))
187	46, 184, 186	elabd 3681	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))})
188		ne0i 4341	. . . . . . . 8 ⊢ (0 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} → {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ≠ ∅)
189	187, 188	syl 17	. . . . . . 7 ⊢ (𝜑 → {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ≠ ∅)
190	13, 11	resubcld 11691	. . . . . . . 8 ⊢ (𝜑 → ((𝐵‘𝑍) − (𝐴‘𝑍)) ∈ ℝ)
191		vex 3484	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ V
192		eqeq1 2741	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑠 → (𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ 𝑠 = (𝑢 − (𝐴‘𝑍))))
193	192	rexbidv 3179	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑠 → (∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ ∃𝑢 ∈ 𝑈 𝑠 = (𝑢 − (𝐴‘𝑍))))
194	191, 193	elab 3679	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ↔ ∃𝑢 ∈ 𝑈 𝑠 = (𝑢 − (𝐴‘𝑍)))
195	194	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} → ∃𝑢 ∈ 𝑈 𝑠 = (𝑢 − (𝐴‘𝑍)))
196	195	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → ∃𝑢 ∈ 𝑈 𝑠 = (𝑢 − (𝐴‘𝑍)))
197		nfcv 2905	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑢𝑠
198	197, 147	nfel 2920	. . . . . . . . . . . 12 ⊢ Ⅎ𝑢 𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}
199	144, 198	nfan 1899	. . . . . . . . . . 11 ⊢ Ⅎ𝑢(𝜑 ∧ 𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))})
200		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑢 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍))
201		simp3 1139	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → 𝑠 = (𝑢 − (𝐴‘𝑍)))
202	159	3adant3 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → (𝐴‘𝑍) ∈ ℝ)
203	13	3ad2ant1 1134	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → (𝐵‘𝑍) ∈ ℝ)
204	155	3adant3 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → 𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
205	21	3ad2ant1 1134	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → (𝐴‘𝑍) ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
206	202, 203, 204, 205	iccsuble 45532	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → (𝑢 − (𝐴‘𝑍)) ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))
207	201, 206	eqbrtrd 5165	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))
208	207	3exp 1120	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑠 = (𝑢 − (𝐴‘𝑍)) → 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))))
209	208	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → (𝑢 ∈ 𝑈 → (𝑠 = (𝑢 − (𝐴‘𝑍)) → 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))))
210	199, 200, 209	rexlimd 3266	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → (∃𝑢 ∈ 𝑈 𝑠 = (𝑢 − (𝐴‘𝑍)) → 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍))))
211	196, 210	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))
212	211	ralrimiva 3146	. . . . . . . 8 ⊢ (𝜑 → ∀𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))
213		brralrspcev 5203	. . . . . . . 8 ⊢ ((((𝐵‘𝑍) − (𝐴‘𝑍)) ∈ ℝ ∧ ∀𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍))) → ∃𝑟 ∈ ℝ ∀𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑠 ≤ 𝑟)
214	190, 212, 213	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑠 ≤ 𝑟)
215		eqid 2737	. . . . . . . 8 ⊢ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)} = {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}
216		biid 261	. . . . . . . 8 ⊢ (((𝐺 ∈ ℝ ∧ 0 ≤ 𝐺 ∧ ∀𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}0 ≤ 𝑟) ∧ ({𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ⊆ ℝ ∧ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ≠ ∅ ∧ ∃𝑟 ∈ ℝ ∀𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑠 ≤ 𝑟)) ↔ ((𝐺 ∈ ℝ ∧ 0 ≤ 𝐺 ∧ ∀𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}0 ≤ 𝑟) ∧ ({𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ⊆ ℝ ∧ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ≠ ∅ ∧ ∃𝑟 ∈ ℝ ∀𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑠 ≤ 𝑟)))
217	215, 216	supmul1 12237	. . . . . . 7 ⊢ (((𝐺 ∈ ℝ ∧ 0 ≤ 𝐺 ∧ ∀𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}0 ≤ 𝑟) ∧ ({𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ⊆ ℝ ∧ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ≠ ∅ ∧ ∃𝑟 ∈ ℝ ∀𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑠 ≤ 𝑟)) → (𝐺 · sup({𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}, ℝ, < )) = sup({𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}, ℝ, < ))
218	38, 137, 171, 180, 189, 214, 217	syl33anc 1387	. . . . . 6 ⊢ (𝜑 → (𝐺 · sup({𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}, ℝ, < )) = sup({𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}, ℝ, < ))
219	125, 134, 218	3eqtrd 2781	. . . . 5 ⊢ (𝜑 → (𝐺 · (𝑆 − (𝐴‘𝑍))) = sup({𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}, ℝ, < ))
220		vex 3484	. . . . . . . . . . . 12 ⊢ 𝑐 ∈ V
221		eqeq1 2741	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑐 → (𝑣 = (𝐺 · 𝑡) ↔ 𝑐 = (𝐺 · 𝑡)))
222	221	rexbidv 3179	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑐 → (∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡) ↔ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑐 = (𝐺 · 𝑡)))
223	220, 222	elab 3679	. . . . . . . . . . 11 ⊢ (𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)} ↔ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑐 = (𝐺 · 𝑡))
224	223	biimpi 216	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)} → ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑐 = (𝐺 · 𝑡))
225		nfv 1914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))
226		vex 3484	. . . . . . . . . . . . . . . . 17 ⊢ 𝑡 ∈ V
227		eqeq1 2741	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑡 → (𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ 𝑡 = (𝑢 − (𝐴‘𝑍))))
228	227	rexbidv 3179	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑡 → (∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ ∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍))))
229	226, 228	elab 3679	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ↔ ∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)))
230	229	biimpi 216	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} → ∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)))
231	230	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ∧ 𝑐 = (𝐺 · 𝑡)) → ∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)))
232		simpl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 = (𝐺 · 𝑡) ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → 𝑐 = (𝐺 · 𝑡))
233		oveq2 7439	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = (𝑢 − (𝐴‘𝑍)) → (𝐺 · 𝑡) = (𝐺 · (𝑢 − (𝐴‘𝑍))))
234	233	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 = (𝐺 · 𝑡) ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → (𝐺 · 𝑡) = (𝐺 · (𝑢 − (𝐴‘𝑍))))
235	232, 234	eqtrd 2777	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 = (𝐺 · 𝑡) ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))))
236	235	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝐺 · 𝑡) → (𝑡 = (𝑢 − (𝐴‘𝑍)) → 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))))
237	236	reximdv 3170	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = (𝐺 · 𝑡) → (∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)) → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))))
238	237	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ∧ 𝑐 = (𝐺 · 𝑡)) → (∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)) → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))))
239	231, 238	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ∧ 𝑐 = (𝐺 · 𝑡)) → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))))
240	239	ex 412	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} → (𝑐 = (𝐺 · 𝑡) → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))))
241	225, 240	rexlimi 3259	. . . . . . . . . . 11 ⊢ (∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑐 = (𝐺 · 𝑡) → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))))
242	241	a1i 11	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)} → (∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑐 = (𝐺 · 𝑡) → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))))
243	224, 242	mpd 15	. . . . . . . . 9 ⊢ (𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)} → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))))
244	243	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}) → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))))
245		simp3 1139	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))) → 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))))
246	38	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝐺 ∈ ℝ)
247	246, 173	remulcld 11291	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐺 · (𝑢 − (𝐴‘𝑍))) ∈ ℝ)
248	45	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (1 + 𝐸) ∈ ℝ)
249	52	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ℕ ∈ V)
250	60	adantlr 715	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
251	64	adantlr 715	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑊⟶ℝ)
252	158	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → 𝑢 ∈ ℝ)
253	79	adantlr 715	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑊⟶ℝ)
254	74, 252, 250, 253	hsphoif 46591	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐻‘𝑢)‘(𝐷‘𝑗)):𝑊⟶ℝ)
255	27, 250, 251, 254	hoidmvcl 46597	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
256	54, 255	sselid 3981	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
257	256	fmpttd 7135	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗)))):ℕ⟶(0[,]+∞))
258	249, 257	sge0cl 46396	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ∈ (0[,]+∞))
259	249, 257	sge0xrcl 46400	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ∈ ℝ^*)
260	86	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → +∞ ∈ ℝ^*)
261	89	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ^*)
262		nfv 1914	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑢 ∈ 𝑈)
263	92	adantlr 715	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) ∈ (0[,]+∞))
264	95	adantlr 715	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → 𝑍 ∈ (𝑊 ∖ 𝑌))
265	27, 250, 264, 9, 252, 74, 251, 253	hsphoidmvle 46601	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))
266	262, 249, 256, 263, 265	sge0lempt 46425	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))
267	98	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) < +∞)
268	259, 261, 260, 266, 267	xrlelttrd 13202	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) < +∞)
269	259, 260, 268	xrltned 45368	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ≠ +∞)
270		ge0xrre 45544	. . . . . . . . . . . . . . . 16 ⊢ (((Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ∈ (0[,]+∞) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ≠ +∞) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ∈ ℝ)
271	258, 269, 270	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ∈ ℝ)
272	248, 271	remulcld 11291	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗)))))) ∈ ℝ)
273	126, 123	sseldd 3984	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 ∈ ℝ)
274	27, 30, 94, 9, 61, 76, 88, 65, 273	sge0hsphoire 46604	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℝ)
275	45, 274	remulcld 11291	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) ∈ ℝ)
276	275	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) ∈ ℝ)
277	14	eleq2i 2833	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ 𝑈 ↔ 𝑢 ∈ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))})
278	277	biimpi 216	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ 𝑈 → 𝑢 ∈ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))})
279		oveq1 7438	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑢 → (𝑧 − (𝐴‘𝑍)) = (𝑢 − (𝐴‘𝑍)))
280	279	oveq2d 7447	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑢 → (𝐺 · (𝑧 − (𝐴‘𝑍))) = (𝐺 · (𝑢 − (𝐴‘𝑍))))
281		fveq2 6906	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 = 𝑢 → (𝐻‘𝑧) = (𝐻‘𝑢))
282	281	fveq1d 6908	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = 𝑢 → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘𝑢)‘(𝐷‘𝑗)))
283	282	oveq2d 7447	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑢 → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))
284	283	mpteq2dv 5244	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑢 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗)))))
285	284	fveq2d 6910	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑢 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))))
286	285	oveq2d 7447	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑢 → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) = ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗)))))))
287	280, 286	breq12d 5156	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑢 → ((𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) ↔ (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))))))
288	287	elrab 3692	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} ↔ (𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∧ (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))))))
289	278, 288	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ 𝑈 → (𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∧ (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))))))
290	289	simprd 495	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑈 → (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗)))))))
291	290	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗)))))))
292	274	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℝ)
293	51	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 0 ≤ (1 + 𝐸))
294	273	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑆 ∈ ℝ)
295	74, 294, 60, 79	hsphoif 46591	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐻‘𝑆)‘(𝐷‘𝑗)):𝑊⟶ℝ)
296	27, 60, 64, 295	hoidmvcl 46597	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
297	54, 296	sselid 3981	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
298	297	adantlr 715	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
299	294	adantlr 715	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → 𝑆 ∈ ℝ)
300	127	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ⊆ ℝ)
301	121	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ≠ ∅)
302	131	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑈 𝑧 ≤ 𝑦)
303		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ 𝑈)
304		suprub 12229	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑈 𝑧 ≤ 𝑦) ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
305	300, 301, 302, 303, 304	syl31anc 1375	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
306	305, 1	breqtrrdi 5185	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ 𝑆)
307	306	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → 𝑢 ≤ 𝑆)
308	27, 250, 264, 9, 252, 299, 307, 74, 251, 253	hsphoidmvle2 46600	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))
309	262, 249, 256, 298, 308	sge0lempt 46425	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))
310	271, 292, 248, 293, 309	lemul2ad 12208	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗)))))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
311	247, 272, 276, 291, 310	letrd 11418	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
312	311	3adant3 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))) → (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
313	245, 312	eqbrtrd 5165	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))) → 𝑐 ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
314	313	3exp 1120	. . . . . . . . . 10 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))) → 𝑐 ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))))
315	314	rexlimdv 3153	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))) → 𝑐 ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))))
316	315	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}) → (∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))) → 𝑐 ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))))
317	244, 316	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}) → 𝑐 ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
318	317	ralrimiva 3146	. . . . . 6 ⊢ (𝜑 → ∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
319	224	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}) → ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑐 = (𝐺 · 𝑡))
320		nfv 1914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡𝜑
321		nfcv 2905	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡𝑐
322		nfre1 3285	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)
323	322	nfab 2911	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡{𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}
324	321, 323	nfel 2920	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}
325	320, 324	nfan 1899	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)})
326		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑡 𝑐 ∈ ℝ
327	230	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → ∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)))
328		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) ∧ 𝑐 = (𝐺 · 𝑡)) → 𝑐 = (𝐺 · 𝑡))
329	246	3adant3 1133	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → 𝐺 ∈ ℝ)
330		simp3 1139	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → 𝑡 = (𝑢 − (𝐴‘𝑍)))
331	173	3adant3 1133	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → (𝑢 − (𝐴‘𝑍)) ∈ ℝ)
332	330, 331	eqeltrd 2841	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → 𝑡 ∈ ℝ)
333	329, 332	remulcld 11291	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → (𝐺 · 𝑡) ∈ ℝ)
334	333	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) ∧ 𝑐 = (𝐺 · 𝑡)) → (𝐺 · 𝑡) ∈ ℝ)
335	328, 334	eqeltrd 2841	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) ∧ 𝑐 = (𝐺 · 𝑡)) → 𝑐 ∈ ℝ)
336	335	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → (𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ))
337	336	3exp 1120	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑡 = (𝑢 − (𝐴‘𝑍)) → (𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ))))
338	337	rexlimdv 3153	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)) → (𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ)))
339	338	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → (∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)) → (𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ)))
340	327, 339	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → (𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ))
341	340	ex 412	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} → (𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ)))
342	341	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}) → (𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} → (𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ)))
343	325, 326, 342	rexlimd 3266	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}) → (∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ))
344	319, 343	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}) → 𝑐 ∈ ℝ)
345	344	ralrimiva 3146	. . . . . . . 8 ⊢ (𝜑 → ∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ∈ ℝ)
346		dfss3 3972	. . . . . . . 8 ⊢ ({𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)} ⊆ ℝ ↔ ∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ∈ ℝ)
347	345, 346	sylibr 234	. . . . . . 7 ⊢ (𝜑 → {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)} ⊆ ℝ)
348	40	eqcomd 2743	. . . . . . . . . 10 ⊢ (𝜑 → 0 = (𝐺 · 0))
349		oveq2 7439	. . . . . . . . . . 11 ⊢ (𝑡 = 0 → (𝐺 · 𝑡) = (𝐺 · 0))
350	349	rspceeqv 3645	. . . . . . . . . 10 ⊢ ((0 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ∧ 0 = (𝐺 · 0)) → ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}0 = (𝐺 · 𝑡))
351	187, 348, 350	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}0 = (𝐺 · 𝑡))
352		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑣 = 0 → (𝑣 = (𝐺 · 𝑡) ↔ 0 = (𝐺 · 𝑡)))
353	352	rexbidv 3179	. . . . . . . . 9 ⊢ (𝑣 = 0 → (∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡) ↔ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}0 = (𝐺 · 𝑡)))
354	46, 351, 353	elabd 3681	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)})
355		ne0i 4341	. . . . . . . 8 ⊢ (0 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)} → {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)} ≠ ∅)
356	354, 355	syl 17	. . . . . . 7 ⊢ (𝜑 → {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)} ≠ ∅)
357	38, 190	remulcld 11291	. . . . . . . 8 ⊢ (𝜑 → (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))) ∈ ℝ)
358	190	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ((𝐵‘𝑍) − (𝐴‘𝑍)) ∈ ℝ)
359	137	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 0 ≤ 𝐺)
360	13	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐵‘𝑍) ∈ ℝ)
361		iccleub 13442	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴‘𝑍) ∈ ℝ^* ∧ (𝐵‘𝑍) ∈ ℝ^* ∧ 𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → 𝑢 ≤ (𝐵‘𝑍))
362	152, 154, 155, 361	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ (𝐵‘𝑍))
363	158, 360, 159, 362	lesub1dd 11879	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 − (𝐴‘𝑍)) ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))
364	173, 358, 246, 359, 363	lemul2ad 12208	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
365	364	3adant3 1133	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))) → (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
366	245, 365	eqbrtrd 5165	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))) → 𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
367	366	3exp 1120	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))) → 𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))))
368	367	rexlimdv 3153	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))) → 𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍)))))
369	368	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}) → (∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))) → 𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍)))))
370	244, 369	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}) → 𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
371	370	ralrimiva 3146	. . . . . . . 8 ⊢ (𝜑 → ∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
372		brralrspcev 5203	. . . . . . . 8 ⊢ (((𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))) ∈ ℝ ∧ ∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍)))) → ∃𝑦 ∈ ℝ ∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ≤ 𝑦)
373	357, 371, 372	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ≤ 𝑦)
374		suprleub 12234	. . . . . . 7 ⊢ ((({𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)} ⊆ ℝ ∧ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)} ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ≤ 𝑦) ∧ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) ∈ ℝ) → (sup({𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}, ℝ, < ) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) ↔ ∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))))
375	347, 356, 373, 275, 374	syl31anc 1375	. . . . . 6 ⊢ (𝜑 → (sup({𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}, ℝ, < ) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) ↔ ∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))))
376	318, 375	mpbird 257	. . . . 5 ⊢ (𝜑 → sup({𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}, ℝ, < ) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
377	219, 376	eqbrtrd 5165	. . . 4 ⊢ (𝜑 → (𝐺 · (𝑆 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
378	123, 377	jca 511	. . 3 ⊢ (𝜑 → (𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∧ (𝐺 · (𝑆 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))))
379		oveq1 7438	. . . . . 6 ⊢ (𝑧 = 𝑆 → (𝑧 − (𝐴‘𝑍)) = (𝑆 − (𝐴‘𝑍)))
380	379	oveq2d 7447	. . . . 5 ⊢ (𝑧 = 𝑆 → (𝐺 · (𝑧 − (𝐴‘𝑍))) = (𝐺 · (𝑆 − (𝐴‘𝑍))))
381		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑧 = 𝑆 → (𝐻‘𝑧) = (𝐻‘𝑆))
382	381	fveq1d 6908	. . . . . . . . 9 ⊢ (𝑧 = 𝑆 → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘𝑆)‘(𝐷‘𝑗)))
383	382	oveq2d 7447	. . . . . . . 8 ⊢ (𝑧 = 𝑆 → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))
384	383	mpteq2dv 5244	. . . . . . 7 ⊢ (𝑧 = 𝑆 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))
385	384	fveq2d 6910	. . . . . 6 ⊢ (𝑧 = 𝑆 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))
386	385	oveq2d 7447	. . . . 5 ⊢ (𝑧 = 𝑆 → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) = ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
387	380, 386	breq12d 5156	. . . 4 ⊢ (𝑧 = 𝑆 → ((𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) ↔ (𝐺 · (𝑆 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))))
388	387	elrab 3692	. . 3 ⊢ (𝑆 ∈ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} ↔ (𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∧ (𝐺 · (𝑆 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))))
389	378, 388	sylibr 234	. 2 ⊢ (𝜑 → 𝑆 ∈ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))})
390	389, 14	eleqtrrdi 2852	1 ⊢ (𝜑 → 𝑆 ∈ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 ∀wal 1538 = wceq 1540 ∈ wcel 2108 {cab 2714 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3436 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 ∅c0 4333 ifcif 4525 {csn 4626 class class class wbr 5143 ↦ cmpt 5225 ↾ cres 5687 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ↑_m cmap 8866 Fincfn 8985 supcsup 9480 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 +∞cpnf 11292 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 − cmin 11492 ℕcn 12266 ℝ⁺crp 13034 [,)cico 13389 [,]cicc 13390 ∏cprod 15939 volcvol 25498 Σ^{^}csumge0 46377

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-prod 15940 df-rest 17467 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953 df-cmp 23395 df-ovol 25499 df-vol 25500 df-sumge0 46378

This theorem is referenced by: hoidmvlelem4 46613

W3C validator