hoidmvlelem1 - Mathbox for Glauco Siliprandi

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Theorem hoidmvlelem1 44023

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,𝑗,𝑘,𝑥 𝐴,𝑐,𝑗,𝑘,𝑥 𝑧,𝐴,𝑗 𝐵,𝑎,𝑏,𝑘 𝐵,𝑐 𝑧,𝐵 𝐶,𝑎,𝑏,𝑘 𝐶,𝑐 𝑧,𝐶 𝐷,𝑎,𝑏,𝑘 𝐷,𝑐 𝑧,𝐷 𝐸,𝑐 𝑧,𝐸 𝐺,𝑐 𝑧,𝐺 𝐻,𝑎,𝑏,𝑘 𝐻,𝑐 𝑧,𝐻 𝐿,𝑐 𝑧,𝐿 𝑆,𝑎,𝑏,𝑗,𝑘,𝑥 𝑆,𝑐 𝑧,𝑆 𝑈,𝑎,𝑏,𝑗,𝑘,𝑥 𝑈,𝑐 𝑧,𝑈 𝑊,𝑎,𝑏,𝑗,𝑘,𝑥 𝑊,𝑐 𝑧,𝑊 𝑌,𝑎,𝑏,𝑗,𝑘,𝑥 𝑌,𝑐 𝑍,𝑎,𝑏,𝑗,𝑘,𝑥 𝑍,𝑐 𝑧,𝑍 𝜑,𝑎,𝑏,𝑗,𝑘,𝑥 𝜑,𝑐 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 𝑢 ℎ 𝑟 𝑠 𝑡 𝑣 𝑤 𝑦 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 4592	. . . . . . . . . 10 ⊢ (𝑍 ∈ (𝑋 ∖ 𝑌) → 𝑍 ∈ {𝑍})
6	4, 5	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ {𝑍})
7		elun2 4107	. . . . . . . . 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 2850	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
11	3, 10	ffvelrnd 6944	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ)
12		hoidmvlelem1.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑊⟶ℝ)
13	12, 10	ffvelrnd 6944	. . . . . 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 4009	. . . . . . . 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 3951	. . . . . . 7 ⊢ 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍))
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
18	11	leidd 11471	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴‘𝑍) ≤ (𝐴‘𝑍))
19		hoidmvlelem1.ab	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴‘𝑍) < (𝐵‘𝑍))
20	11, 13, 19	ltled 11053	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴‘𝑍) ≤ (𝐵‘𝑍))
21	11, 13, 11, 18, 20	eliccd 42932	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
22	11	recnd 10934	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℂ)
23	22	subidd 11250	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴‘𝑍) − (𝐴‘𝑍)) = 0)
24	23	oveq2d 7271	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 · ((𝐴‘𝑍) − (𝐴‘𝑍))) = (𝐺 · 0))
25		rge0ssre 13117	. . . . . . . . . . . . . . 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 8971	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑌 ∈ Fin)
31		ssun1 4102	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑌 ⊆ (𝑌 ∪ {𝑍})
32	31, 9	sseqtrri 3954	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑌 ⊆ 𝑊
33	32	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑌 ⊆ 𝑊)
34	3, 33	fssresd 6625	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ↾ 𝑌):𝑌⟶ℝ)
35	12, 33	fssresd 6625	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 ↾ 𝑌):𝑌⟶ℝ)
36	27, 30, 34, 35	hoidmvcl 44010	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ∈ (0[,)+∞))
37	26, 36	eqeltrid 2843	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ∈ (0[,)+∞))
38	25, 37	sselid 3915	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ ℝ)
39	38	recnd 10934	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ ℂ)
40	39	mul01d 11104	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 · 0) = 0)
41	24, 40	eqtrd 2778	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 · ((𝐴‘𝑍) − (𝐴‘𝑍))) = 0)
42		1red 10907	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℝ)
43		hoidmvlelem1.e	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
44	43	rpred 12701	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ)
45	42, 44	readdcld 10935	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 + 𝐸) ∈ ℝ)
46		0red 10909	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℝ)
47		0lt1 11427	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
48	47	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 1)
49	42, 43	ltaddrpd 12734	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 < (1 + 𝐸))
50	46, 42, 45, 48, 49	lttrd 11066	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < (1 + 𝐸))
51	46, 45, 50	ltled 11053	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (1 + 𝐸))
52		nnex 11909	. . . . . . . . . . . . . . 15 ⊢ ℕ ∈ V
53	52	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℕ ∈ V)
54		icossicc 13097	. . . . . . . . . . . . . . . 16 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
55		snfi 8788	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑍} ∈ Fin
56	55	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → {𝑍} ∈ Fin)
57		unfi 8917	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑌 ∈ Fin ∧ {𝑍} ∈ Fin) → (𝑌 ∪ {𝑍}) ∈ Fin)
58	30, 56, 57	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑌 ∪ {𝑍}) ∈ Fin)
59	9, 58	eqeltrid 2843	. . . . . . . . . . . . . . . . . 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	ffvelrnda 6943	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑_m 𝑊))
63		elmapi 8595	. . . . . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = ℎ → (𝑗 ∈ 𝑌 ↔ ℎ ∈ 𝑌))
67		fveq2 6756	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = ℎ → (𝑐‘𝑗) = (𝑐‘ℎ))
68	67	breq1d 5080	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = ℎ → ((𝑐‘𝑗) ≤ 𝑥 ↔ (𝑐‘ℎ) ≤ 𝑥))
69	68, 67	ifbieq1d 4480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = ℎ → if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥) = if((𝑐‘ℎ) ≤ 𝑥, (𝑐‘ℎ), 𝑥))
70	66, 67, 69	ifbieq12d 4484	. . . . . . . . . . . . . . . . . . . . . 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 5183	. . . . . . . . . . . . . . . . . . . . 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 5175	. . . . . . . . . . . . . . . . . . . 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 5175	. . . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . 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	ffvelrnda 6943	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑_m 𝑊))
78		elmapi 8595	. . . . . . . . . . . . . . . . . . 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 44004	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)):𝑊⟶ℝ)
81	27, 60, 64, 80	hoidmvcl 44010	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))) ∈ (0[,)+∞))
82	54, 81	sselid 3915	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))) ∈ (0[,]+∞))
83	82	fmpttd 6971	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)))):ℕ⟶(0[,]+∞))
84	53, 83	sge0cl 43809	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ∈ (0[,]+∞))
85	53, 83	sge0xrcl 43813	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ∈ ℝ^*)
86		pnfxr 10960	. . . . . . . . . . . . . . 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 10956	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ^*)
90		nfv 1918	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗𝜑
91	27, 60, 64, 79	hoidmvcl 44010	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) ∈ (0[,)+∞))
92	54, 91	sselid 3915	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) ∈ (0[,]+∞))
93	4	eldifbd 3896	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
94	10, 93	eldifd 3894	. . . . . . . . . . . . . . . . . 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 44014	. . . . . . . . . . . . . . . 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 43838	. . . . . . . . . . . . . . 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 12786	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) < +∞)
99	85, 89, 87, 97, 98	xrlelttrd 12823	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) < +∞)
100	85, 87, 99	xrltned 42786	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ≠ +∞)
101		ge0xrre 42959	. . . . . . . . . . . . 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 583	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ∈ ℝ)
103	53, 83	sge0ge0 43812	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))))
104		mulge0 11423	. . . . . . . . . . . 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 835	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)))))))
106	41, 105	eqbrtrd 5092	. . . . . . . . . 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 7262	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐴‘𝑍) → (𝑧 − (𝐴‘𝑍)) = ((𝐴‘𝑍) − (𝐴‘𝑍)))
109	108	oveq2d 7271	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐴‘𝑍) → (𝐺 · (𝑧 − (𝐴‘𝑍))) = (𝐺 · ((𝐴‘𝑍) − (𝐴‘𝑍))))
110		fveq2 6756	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐴‘𝑍) → (𝐻‘𝑧) = (𝐻‘(𝐴‘𝑍)))
111	110	fveq1d 6758	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐴‘𝑍) → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)))
112	111	oveq2d 7271	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐴‘𝑍) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))
113	112	mpteq2dv 5172	. . . . . . . . . . . . 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 6760	. . . . . . . . . . . 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 7271	. . . . . . . . . . 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 5083	. . . . . . . . . 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 3617	. . . . . . . . 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 233	. . . . . . . 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 2850	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝑍) ∈ 𝑈)
120		ne0i 4265	. . . . . . 7 ⊢ ((𝐴‘𝑍) ∈ 𝑈 → 𝑈 ≠ ∅)
121	119, 120	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈 ≠ ∅)
122	11, 13, 17, 121	supicc 13162	. . . . 5 ⊢ (𝜑 → sup(𝑈, ℝ, < ) ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
123	2, 122	eqeltrd 2839	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
124	2	oveq1d 7270	. . . . . . 7 ⊢ (𝜑 → (𝑆 − (𝐴‘𝑍)) = (sup(𝑈, ℝ, < ) − (𝐴‘𝑍)))
125	124	oveq2d 7271	. . . . . 6 ⊢ (𝜑 → (𝐺 · (𝑆 − (𝐴‘𝑍))) = (𝐺 · (sup(𝑈, ℝ, < ) − (𝐴‘𝑍))))
126	11, 13	iccssred 13095	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴‘𝑍)[,](𝐵‘𝑍)) ⊆ ℝ)
127	17, 126	sstrd 3927	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ⊆ ℝ)
128	11, 13	jca 511	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴‘𝑍) ∈ ℝ ∧ (𝐵‘𝑍) ∈ ℝ))
129		iccsupr 13103	. . . . . . . . . 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 1369	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑈 𝑧 ≤ 𝑦))
131	130	simp3d 1142	. . . . . . . 8 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑈 𝑧 ≤ 𝑦)
132		eqid 2738	. . . . . . . 8 ⊢ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} = {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}
133	127, 121, 131, 11, 132	supsubc 42782	. . . . . . 7 ⊢ (𝜑 → (sup(𝑈, ℝ, < ) − (𝐴‘𝑍)) = sup({𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}, ℝ, < ))
134	133	oveq2d 7271	. . . . . 6 ⊢ (𝜑 → (𝐺 · (sup(𝑈, ℝ, < ) − (𝐴‘𝑍))) = (𝐺 · sup({𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}, ℝ, < )))
135	46	rexrd 10956	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ^*)
136		icogelb 13059	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐺 ∈ (0[,)+∞)) → 0 ≤ 𝐺)
137	135, 87, 37, 136	syl3anc 1369	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝐺)
138		vex 3426	. . . . . . . . . . . 12 ⊢ 𝑟 ∈ V
139		eqeq1 2742	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑟 → (𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ 𝑟 = (𝑢 − (𝐴‘𝑍))))
140	139	rexbidv 3225	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑟 → (∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ ∃𝑢 ∈ 𝑈 𝑟 = (𝑢 − (𝐴‘𝑍))))
141	138, 140	elab 3602	. . . . . . . . . . 11 ⊢ (𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ↔ ∃𝑢 ∈ 𝑈 𝑟 = (𝑢 − (𝐴‘𝑍)))
142	141	biimpi 215	. . . . . . . . . 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 1918	. . . . . . . . . . 11 ⊢ Ⅎ𝑢𝜑
145		nfcv 2906	. . . . . . . . . . . 12 ⊢ Ⅎ𝑢𝑟
146		nfre1 3234	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑢∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))
147	146	nfab 2912	. . . . . . . . . . . 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 1903	. . . . . . . . . 10 ⊢ Ⅎ𝑢(𝜑 ∧ 𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))})
150		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑢0 ≤ 𝑟
151	11	rexrd 10956	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ^*)
152	151	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴‘𝑍) ∈ ℝ^*)
153	13	rexrd 10956	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ^*)
154	153	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐵‘𝑍) ∈ ℝ^*)
155	17	sselda 3917	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
156		iccgelb 13064	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴‘𝑍) ∈ ℝ^* ∧ (𝐵‘𝑍) ∈ ℝ^* ∧ 𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → (𝐴‘𝑍) ≤ 𝑢)
157	152, 154, 155, 156	syl3anc 1369	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴‘𝑍) ≤ 𝑢)
158	127	sselda 3917	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ ℝ)
159	11	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴‘𝑍) ∈ ℝ)
160	158, 159	subge0d 11495	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (0 ≤ (𝑢 − (𝐴‘𝑍)) ↔ (𝐴‘𝑍) ≤ 𝑢))
161	157, 160	mpbird 256	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 0 ≤ (𝑢 − (𝐴‘𝑍)))
162	161	3adant3 1130	. . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑢 − (𝐴‘𝑍)) → (𝑢 − (𝐴‘𝑍)) = 𝑟)
165	164	3ad2ant3 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑟 = (𝑢 − (𝐴‘𝑍))) → (𝑢 − (𝐴‘𝑍)) = 𝑟)
166	162, 165	breqtrd 5096	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑟 = (𝑢 − (𝐴‘𝑍))) → 0 ≤ 𝑟)
167	166	3exp 1117	. . . . . . . . . . 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 3245	. . . . . . . . 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 3107	. . . . . . 7 ⊢ (𝜑 → ∀𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}0 ≤ 𝑟)
172		simp3 1136	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑤 = (𝑢 − (𝐴‘𝑍))) → 𝑤 = (𝑢 − (𝐴‘𝑍)))
173	158, 159	resubcld 11333	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 − (𝐴‘𝑍)) ∈ ℝ)
174	173	3adant3 1130	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑤 = (𝑢 − (𝐴‘𝑍))) → (𝑢 − (𝐴‘𝑍)) ∈ ℝ)
175	172, 174	eqeltrd 2839	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑤 = (𝑢 − (𝐴‘𝑍))) → 𝑤 ∈ ℝ)
176	175	3exp 1117	. . . . . . . . . 10 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑤 = (𝑢 − (𝐴‘𝑍)) → 𝑤 ∈ ℝ)))
177	176	rexlimdv 3211	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) → 𝑤 ∈ ℝ))
178	177	alrimiv 1931	. . . . . . . 8 ⊢ (𝜑 → ∀𝑤(∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) → 𝑤 ∈ ℝ))
179		abss 3990	. . . . . . . 8 ⊢ ({𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ⊆ ℝ ↔ ∀𝑤(∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) → 𝑤 ∈ ℝ))
180	178, 179	sylibr 233	. . . . . . 7 ⊢ (𝜑 → {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ⊆ ℝ)
181	23	eqcomd 2744	. . . . . . . . . 10 ⊢ (𝜑 → 0 = ((𝐴‘𝑍) − (𝐴‘𝑍)))
182		oveq1 7262	. . . . . . . . . . 11 ⊢ (𝑢 = (𝐴‘𝑍) → (𝑢 − (𝐴‘𝑍)) = ((𝐴‘𝑍) − (𝐴‘𝑍)))
183	182	rspceeqv 3567	. . . . . . . . . 10 ⊢ (((𝐴‘𝑍) ∈ 𝑈 ∧ 0 = ((𝐴‘𝑍) − (𝐴‘𝑍))) → ∃𝑢 ∈ 𝑈 0 = (𝑢 − (𝐴‘𝑍)))
184	119, 181, 183	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑢 ∈ 𝑈 0 = (𝑢 − (𝐴‘𝑍)))
185		eqeq1 2742	. . . . . . . . . 10 ⊢ (𝑤 = 0 → (𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ 0 = (𝑢 − (𝐴‘𝑍))))
186	185	rexbidv 3225	. . . . . . . . 9 ⊢ (𝑤 = 0 → (∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ ∃𝑢 ∈ 𝑈 0 = (𝑢 − (𝐴‘𝑍))))
187	46, 184, 186	elabd 3605	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))})
188		ne0i 4265	. . . . . . . 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 11333	. . . . . . . 8 ⊢ (𝜑 → ((𝐵‘𝑍) − (𝐴‘𝑍)) ∈ ℝ)
191		vex 3426	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ V
192		eqeq1 2742	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑠 → (𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ 𝑠 = (𝑢 − (𝐴‘𝑍))))
193	192	rexbidv 3225	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑠 → (∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ ∃𝑢 ∈ 𝑈 𝑠 = (𝑢 − (𝐴‘𝑍))))
194	191, 193	elab 3602	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ↔ ∃𝑢 ∈ 𝑈 𝑠 = (𝑢 − (𝐴‘𝑍)))
195	194	biimpi 215	. . . . . . . . . . 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 2906	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑢𝑠
198	197, 147	nfel 2920	. . . . . . . . . . . 12 ⊢ Ⅎ𝑢 𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}
199	144, 198	nfan 1903	. . . . . . . . . . 11 ⊢ Ⅎ𝑢(𝜑 ∧ 𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))})
200		nfv 1918	. . . . . . . . . . 11 ⊢ Ⅎ𝑢 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍))
201		simp3 1136	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → 𝑠 = (𝑢 − (𝐴‘𝑍)))
202	159	3adant3 1130	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → (𝐴‘𝑍) ∈ ℝ)
203	13	3ad2ant1 1131	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → (𝐵‘𝑍) ∈ ℝ)
204	155	3adant3 1130	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → 𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
205	21	3ad2ant1 1131	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → (𝐴‘𝑍) ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
206	202, 203, 204, 205	iccsuble 42947	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → (𝑢 − (𝐴‘𝑍)) ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))
207	201, 206	eqbrtrd 5092	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))
208	207	3exp 1117	. . . . . . . . . . . 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 3245	. . . . . . . . . 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 3107	. . . . . . . 8 ⊢ (𝜑 → ∀𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))
213		brralrspcev 5130	. . . . . . . 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 583	. . . . . . 7 ⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑠 ≤ 𝑟)
215		eqid 2738	. . . . . . . 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 260	. . . . . . . 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 11874	. . . . . . 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 1383	. . . . . 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 2782	. . . . 5 ⊢ (𝜑 → (𝐺 · (𝑆 − (𝐴‘𝑍))) = sup({𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}, ℝ, < ))
220		vex 3426	. . . . . . . . . . . 12 ⊢ 𝑐 ∈ V
221		eqeq1 2742	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑐 → (𝑣 = (𝐺 · 𝑡) ↔ 𝑐 = (𝐺 · 𝑡)))
222	221	rexbidv 3225	. . . . . . . . . . . 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 3602	. . . . . . . . . . 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 215	. . . . . . . . . 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 1918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))
226		vex 3426	. . . . . . . . . . . . . . . . 17 ⊢ 𝑡 ∈ V
227		eqeq1 2742	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑡 → (𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ 𝑡 = (𝑢 − (𝐴‘𝑍))))
228	227	rexbidv 3225	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑡 → (∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ ∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍))))
229	226, 228	elab 3602	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ↔ ∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)))
230	229	biimpi 215	. . . . . . . . . . . . . . 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 7263	. . . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . 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 3201	. . . . . . . . . . . . . . 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 3243	. . . . . . . . . . 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 1136	. . . . . . . . . . . 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 10936	. . . . . . . . . . . . . 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 711	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
251	64	adantlr 711	. . . . . . . . . . . . . . . . . . . 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 711	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑊⟶ℝ)
254	74, 252, 250, 253	hsphoif 44004	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐻‘𝑢)‘(𝐷‘𝑗)):𝑊⟶ℝ)
255	27, 250, 251, 254	hoidmvcl 44010	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
256	54, 255	sselid 3915	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
257	256	fmpttd 6971	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗)))):ℕ⟶(0[,]+∞))
258	249, 257	sge0cl 43809	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ∈ (0[,]+∞))
259	249, 257	sge0xrcl 43813	. . . . . . . . . . . . . . . . 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 1918	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑢 ∈ 𝑈)
263	92	adantlr 711	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) ∈ (0[,]+∞))
264	95	adantlr 711	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → 𝑍 ∈ (𝑊 ∖ 𝑌))
265	27, 250, 264, 9, 252, 74, 251, 253	hsphoidmvle 44014	. . . . . . . . . . . . . . . . . . 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 43838	. . . . . . . . . . . . . . . . . 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 12823	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) < +∞)
269	259, 260, 268	xrltned 42786	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ≠ +∞)
270		ge0xrre 42959	. . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ∈ ℝ)
272	248, 271	remulcld 10936	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗)))))) ∈ ℝ)
273	126, 123	sseldd 3918	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 ∈ ℝ)
274	27, 30, 94, 9, 61, 76, 88, 65, 273	sge0hsphoire 44017	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℝ)
275	45, 274	remulcld 10936	. . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . . . . . 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 215	. . . . . . . . . . . . . . . . 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 7262	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑢 → (𝑧 − (𝐴‘𝑍)) = (𝑢 − (𝐴‘𝑍)))
280	279	oveq2d 7271	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑢 → (𝐺 · (𝑧 − (𝐴‘𝑍))) = (𝐺 · (𝑢 − (𝐴‘𝑍))))
281		fveq2 6756	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 = 𝑢 → (𝐻‘𝑧) = (𝐻‘𝑢))
282	281	fveq1d 6758	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = 𝑢 → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘𝑢)‘(𝐷‘𝑗)))
283	282	oveq2d 7271	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑢 → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))
284	283	mpteq2dv 5172	. . . . . . . . . . . . . . . . . . . . 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 6760	. . . . . . . . . . . . . . . . . . . 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 7271	. . . . . . . . . . . . . . . . . . 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 5083	. . . . . . . . . . . . . . . . . 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 3617	. . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . 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 44004	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐻‘𝑆)‘(𝐷‘𝑗)):𝑊⟶ℝ)
296	27, 60, 64, 295	hoidmvcl 44010	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
297	54, 296	sselid 3915	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
298	297	adantlr 711	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
299	294	adantlr 711	. . . . . . . . . . . . . . . . 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 11866	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑈 𝑧 ≤ 𝑦) ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
305	300, 301, 302, 303, 304	syl31anc 1371	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
306	305, 1	breqtrrdi 5112	. . . . . . . . . . . . . . . . . 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 44013	. . . . . . . . . . . . . . . 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 43838	. . . . . . . . . . . . . . 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 11845	. . . . . . . . . . . . . 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 11062	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
312	311	3adant3 1130	. . . . . . . . . . . 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 5092	. . . . . . . . . . 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 1117	. . . . . . . . . 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 3211	. . . . . . . . 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 3107	. . . . . 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 1918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡𝜑
321		nfcv 2906	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡𝑐
322		nfre1 3234	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)
323	322	nfab 2912	. . . . . . . . . . . . 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 1903	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)})
326		nfv 1918	. . . . . . . . . . 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 1130	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → 𝐺 ∈ ℝ)
330		simp3 1136	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → 𝑡 = (𝑢 − (𝐴‘𝑍)))
331	173	3adant3 1130	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → (𝑢 − (𝐴‘𝑍)) ∈ ℝ)
332	330, 331	eqeltrd 2839	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → 𝑡 ∈ ℝ)
333	329, 332	remulcld 10936	. . . . . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . . . 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 1117	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑡 = (𝑢 − (𝐴‘𝑍)) → (𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ))))
338	337	rexlimdv 3211	. . . . . . . . . . . . . . 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 3245	. . . . . . . . . 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 3107	. . . . . . . 8 ⊢ (𝜑 → ∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ∈ ℝ)
346		dfss3 3905	. . . . . . . 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 233	. . . . . . 7 ⊢ (𝜑 → {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)} ⊆ ℝ)
348	40	eqcomd 2744	. . . . . . . . . 10 ⊢ (𝜑 → 0 = (𝐺 · 0))
349		oveq2 7263	. . . . . . . . . . 11 ⊢ (𝑡 = 0 → (𝐺 · 𝑡) = (𝐺 · 0))
350	349	rspceeqv 3567	. . . . . . . . . 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 583	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}0 = (𝐺 · 𝑡))
352		eqeq1 2742	. . . . . . . . . 10 ⊢ (𝑣 = 0 → (𝑣 = (𝐺 · 𝑡) ↔ 0 = (𝐺 · 𝑡)))
353	352	rexbidv 3225	. . . . . . . . 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 3605	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)})
355		ne0i 4265	. . . . . . . 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 10936	. . . . . . . 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 13063	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴‘𝑍) ∈ ℝ^* ∧ (𝐵‘𝑍) ∈ ℝ^* ∧ 𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → 𝑢 ≤ (𝐵‘𝑍))
362	152, 154, 155, 361	syl3anc 1369	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ (𝐵‘𝑍))
363	158, 360, 159, 362	lesub1dd 11521	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 − (𝐴‘𝑍)) ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))
364	173, 358, 246, 359, 363	lemul2ad 11845	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
365	364	3adant3 1130	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))) → (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
366	245, 365	eqbrtrd 5092	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))) → 𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
367	366	3exp 1117	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))) → 𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))))
368	367	rexlimdv 3211	. . . . . . . . . . 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 3107	. . . . . . . 8 ⊢ (𝜑 → ∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
372		brralrspcev 5130	. . . . . . . 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 583	. . . . . . 7 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ≤ 𝑦)
374		suprleub 11871	. . . . . . 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 1371	. . . . . 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 256	. . . . 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 5092	. . . 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 7262	. . . . . 6 ⊢ (𝑧 = 𝑆 → (𝑧 − (𝐴‘𝑍)) = (𝑆 − (𝐴‘𝑍)))
380	379	oveq2d 7271	. . . . 5 ⊢ (𝑧 = 𝑆 → (𝐺 · (𝑧 − (𝐴‘𝑍))) = (𝐺 · (𝑆 − (𝐴‘𝑍))))
381		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑧 = 𝑆 → (𝐻‘𝑧) = (𝐻‘𝑆))
382	381	fveq1d 6758	. . . . . . . . 9 ⊢ (𝑧 = 𝑆 → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘𝑆)‘(𝐷‘𝑗)))
383	382	oveq2d 7271	. . . . . . . 8 ⊢ (𝑧 = 𝑆 → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))
384	383	mpteq2dv 5172	. . . . . . 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 6760	. . . . . 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 7271	. . . . 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 5083	. . . 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 3617	. . 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 233	. 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 2850	1 ⊢ (𝜑 → 𝑆 ∈ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 ∀wal 1537 = wceq 1539 ∈ wcel 2108 {cab 2715 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 {crab 3067 Vcvv 3422 ∖ cdif 3880 ∪ cun 3881 ⊆ wss 3883 ∅c0 4253 ifcif 4456 {csn 4558 class class class wbr 5070 ↦ cmpt 5153 ↾ cres 5582 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ∈ cmpo 7257 ↑_m cmap 8573 Fincfn 8691 supcsup 9129 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 +∞cpnf 10937 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 − cmin 11135 ℕcn 11903 ℝ⁺crp 12659 [,)cico 13010 [,]cicc 13011 ∏cprod 15543 volcvol 24532 Σ^{^}csumge0 43790

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-rlim 15126 df-sum 15326 df-prod 15544 df-rest 17050 df-topgen 17071 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-top 21951 df-topon 21968 df-bases 22004 df-cmp 22446 df-ovol 24533 df-vol 24534 df-sumge0 43791

This theorem is referenced by: hoidmvlelem4 44026

W3C validator