hoidmvlelem1 - Mathbox for Glauco Siliprandi

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Theorem hoidmvlelem1 47521

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 4621	. . . . . . . . . 10 ⊢ (𝑍 ∈ (𝑋 ∖ 𝑌) → 𝑍 ∈ {𝑍})
6	4, 5	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ {𝑍})
7		elun2 4129	. . . . . . . . 9 ⊢ (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑌 ∪ {𝑍}))
8	6, 7	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ (𝑌 ∪ {𝑍}))
9		hoidmvlelem1.w	. . . . . . . 8 ⊢ 𝑊 = (𝑌 ∪ {𝑍})
10	8, 9	eleqtrrdi 2871	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
11	3, 10	ffvelcdmd 7074	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ)
12		hoidmvlelem1.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑊⟶ℝ)
13	12, 10	ffvelcdmd 7074	. . . . . 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 4028	. . . . . . . 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 3977	. . . . . . 7 ⊢ 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍))
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
18	11	leidd 11837	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴‘𝑍) ≤ (𝐴‘𝑍))
19		hoidmvlelem1.ab	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴‘𝑍) < (𝐵‘𝑍))
20	11, 13, 19	ltled 11415	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴‘𝑍) ≤ (𝐵‘𝑍))
21	11, 13, 11, 18, 20	eliccd 46432	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
22	11	recnd 11294	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℂ)
23	22	subidd 11614	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴‘𝑍) − (𝐴‘𝑍)) = 0)
24	23	oveq2d 7425	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 · ((𝐴‘𝑍) − (𝐴‘𝑍))) = (𝐺 · 0))
25		rge0ssre 13542	. . . . . . . . . . . . . . 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 9239	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑌 ∈ Fin)
31		ssun1 4124	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑌 ⊆ (𝑌 ∪ {𝑍})
32	31, 9	sseqtrri 3980	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑌 ⊆ 𝑊
33	32	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑌 ⊆ 𝑊)
34	3, 33	fssresd 6738	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ↾ 𝑌):𝑌⟶ℝ)
35	12, 33	fssresd 6738	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 ↾ 𝑌):𝑌⟶ℝ)
36	27, 30, 34, 35	hoidmvcl 47508	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ∈ (0[,)+∞))
37	26, 36	eqeltrid 2864	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ∈ (0[,)+∞))
38	25, 37	sselid 3929	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ ℝ)
39	38	recnd 11294	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ ℂ)
40	39	mul01d 11466	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 · 0) = 0)
41	24, 40	eqtrd 2795	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 · ((𝐴‘𝑍) − (𝐴‘𝑍))) = 0)
42		1red 11266	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℝ)
43		hoidmvlelem1.e	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
44	43	rpred 13119	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ)
45	42, 44	readdcld 11295	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 + 𝐸) ∈ ℝ)
46		0red 11268	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℝ)
47		0lt1 11793	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
48	47	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 1)
49	42, 43	ltaddrpd 13152	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 < (1 + 𝐸))
50	46, 42, 45, 48, 49	lttrd 11428	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < (1 + 𝐸))
51	46, 45, 50	ltled 11415	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (1 + 𝐸))
52		nnex 12296	. . . . . . . . . . . . . . 15 ⊢ ℕ ∈ V
53	52	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℕ ∈ V)
54		icossicc 13522	. . . . . . . . . . . . . . . 16 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
55		snfi 9050	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑍} ∈ Fin
56	55	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → {𝑍} ∈ Fin)
57		unfi 9165	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑌 ∈ Fin ∧ {𝑍} ∈ Fin) → (𝑌 ∪ {𝑍}) ∈ Fin)
58	30, 56, 57	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑌 ∪ {𝑍}) ∈ Fin)
59	9, 58	eqeltrid 2864	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑊 ∈ Fin)
60	59	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
61		hoidmvlelem1.c	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑_m 𝑊))
62	61	ffvelcdmda 7073	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑_m 𝑊))
63		elmapi 8848	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶‘𝑗) ∈ (ℝ ↑_m 𝑊) → (𝐶‘𝑗):𝑊⟶ℝ)
64	62, 63	syl 18	. . . . . . . . . . . . . . . . 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 2843	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = ℎ → (𝑗 ∈ 𝑌 ↔ ℎ ∈ 𝑌))
67		fveq2 6874	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = ℎ → (𝑐‘𝑗) = (𝑐‘ℎ))
68	67	breq1d 5113	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = ℎ → ((𝑐‘𝑗) ≤ 𝑥 ↔ (𝑐‘ℎ) ≤ 𝑥))
69	68, 67	ifbieq1d 4507	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = ℎ → if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥) = if((𝑐‘ℎ) ≤ 𝑥, (𝑐‘ℎ), 𝑥))
70	66, 67, 69	ifbieq12d 4511	. . . . . . . . . . . . . . . . . . . . . 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 5209	. . . . . . . . . . . . . . . . . . . . 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 5201	. . . . . . . . . . . . . . . . . . . 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 5201	. . . . . . . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐴‘𝑍) ∈ ℝ)
76		hoidmvlelem1.d	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑_m 𝑊))
77	76	ffvelcdmda 7073	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑_m 𝑊))
78		elmapi 8848	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐷‘𝑗) ∈ (ℝ ↑_m 𝑊) → (𝐷‘𝑗):𝑊⟶ℝ)
79	77, 78	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑊⟶ℝ)
80	74, 75, 60, 79	hsphoif 47502	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)):𝑊⟶ℝ)
81	27, 60, 64, 80	hoidmvcl 47508	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))) ∈ (0[,)+∞))
82	54, 81	sselid 3929	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))) ∈ (0[,]+∞))
83	82	fmpttd 7104	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)))):ℕ⟶(0[,]+∞))
84	53, 83	sge0cl 47307	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ∈ (0[,]+∞))
85	53, 83	sge0xrcl 47311	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ∈ ℝ^*)
86		pnfxr 11320	. . . . . . . . . . . . . . 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 11316	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ^*)
90		nfv 1947	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗𝜑
91	27, 60, 64, 79	hoidmvcl 47508	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) ∈ (0[,)+∞))
92	54, 91	sselid 3929	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) ∈ (0[,]+∞))
93	4	eldifbd 3912	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
94	10, 93	eldifd 3910	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑍 ∈ (𝑊 ∖ 𝑌))
95	94	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑍 ∈ (𝑊 ∖ 𝑌))
96	27, 60, 95, 9, 75, 74, 64, 79	hsphoidmvle 47512	. . . . . . . . . . . . . . . 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 47336	. . . . . . . . . . . . . . 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 13205	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) < +∞)
99	85, 89, 87, 97, 98	xrlelttrd 13244	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) < +∞)
100	85, 87, 99	xrltned 46285	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ≠ +∞)
101		ge0xrre 46459	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ∈ ℝ)
103	53, 83	sge0ge0 47310	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))))
104		mulge0 11789	. . . . . . . . . . . 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 852	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)))))))
106	41, 105	eqbrtrd 5127	. . . . . . . . . 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 521	. . . . . . . . 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 7416	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐴‘𝑍) → (𝑧 − (𝐴‘𝑍)) = ((𝐴‘𝑍) − (𝐴‘𝑍)))
109	108	oveq2d 7425	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐴‘𝑍) → (𝐺 · (𝑧 − (𝐴‘𝑍))) = (𝐺 · ((𝐴‘𝑍) − (𝐴‘𝑍))))
110		fveq2 6874	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐴‘𝑍) → (𝐻‘𝑧) = (𝐻‘(𝐴‘𝑍)))
111	110	fveq1d 6876	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐴‘𝑍) → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)))
112	111	oveq2d 7425	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐴‘𝑍) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))
113	112	mpteq2dv 5199	. . . . . . . . . . . . 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 6878	. . . . . . . . . . . 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 7425	. . . . . . . . . . 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 5116	. . . . . . . . . 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 3645	. . . . . . . . 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 237	. . . . . . . 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 2871	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝑍) ∈ 𝑈)
120		ne0i 4287	. . . . . . 7 ⊢ ((𝐴‘𝑍) ∈ 𝑈 → 𝑈 ≠ ∅)
121	119, 120	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑈 ≠ ∅)
122	11, 13, 17, 121	supicc 13587	. . . . 5 ⊢ (𝜑 → sup(𝑈, ℝ, < ) ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
123	2, 122	eqeltrd 2860	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
124	2	oveq1d 7424	. . . . . . 7 ⊢ (𝜑 → (𝑆 − (𝐴‘𝑍)) = (sup(𝑈, ℝ, < ) − (𝐴‘𝑍)))
125	124	oveq2d 7425	. . . . . 6 ⊢ (𝜑 → (𝐺 · (𝑆 − (𝐴‘𝑍))) = (𝐺 · (sup(𝑈, ℝ, < ) − (𝐴‘𝑍))))
126	11, 13	iccssred 13520	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴‘𝑍)[,](𝐵‘𝑍)) ⊆ ℝ)
127	17, 126	sstrd 3941	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ⊆ ℝ)
128	11, 13	jca 521	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴‘𝑍) ∈ ℝ ∧ (𝐵‘𝑍) ∈ ℝ))
129		iccsupr 13528	. . . . . . . . . 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 1398	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑈 𝑧 ≤ 𝑦))
131	130	simp3d 1162	. . . . . . . 8 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑈 𝑧 ≤ 𝑦)
132		eqid 2760	. . . . . . . 8 ⊢ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} = {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}
133	127, 121, 131, 11, 132	supsubc 46281	. . . . . . 7 ⊢ (𝜑 → (sup(𝑈, ℝ, < ) − (𝐴‘𝑍)) = sup({𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}, ℝ, < ))
134	133	oveq2d 7425	. . . . . 6 ⊢ (𝜑 → (𝐺 · (sup(𝑈, ℝ, < ) − (𝐴‘𝑍))) = (𝐺 · sup({𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}, ℝ, < )))
135	46	rexrd 11316	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ^*)
136		icogelb 13482	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐺 ∈ (0[,)+∞)) → 0 ≤ 𝐺)
137	135, 87, 37, 136	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝐺)
138		vex 3454	. . . . . . . . . . 11 ⊢ 𝑟 ∈ V
139		eqeq1 2764	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑟 → (𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ 𝑟 = (𝑢 − (𝐴‘𝑍))))
140	139	rexbidv 3186	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑟 → (∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ ∃𝑢 ∈ 𝑈 𝑟 = (𝑢 − (𝐴‘𝑍))))
141	138, 140	elab 3633	. . . . . . . . . 10 ⊢ (𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ↔ ∃𝑢 ∈ 𝑈 𝑟 = (𝑢 − (𝐴‘𝑍)))
142	141	bilani 510	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → ∃𝑢 ∈ 𝑈 𝑟 = (𝑢 − (𝐴‘𝑍)))
143		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑢𝜑
144		nfcv 2922	. . . . . . . . . . . 12 ⊢ Ⅎ𝑢𝑟
145		nfre1 3287	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑢∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))
146	145	nfab 2928	. . . . . . . . . . . 12 ⊢ Ⅎ𝑢{𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}
147	144, 146	nfel 2936	. . . . . . . . . . 11 ⊢ Ⅎ𝑢 𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}
148	143, 147	nfan 1932	. . . . . . . . . 10 ⊢ Ⅎ𝑢(𝜑 ∧ 𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))})
149		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑢0 ≤ 𝑟
150	11	rexrd 11316	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ^*)
151	150	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴‘𝑍) ∈ ℝ^*)
152	13	rexrd 11316	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ^*)
153	152	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐵‘𝑍) ∈ ℝ^*)
154	17	sselda 3931	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
155		iccgelb 13488	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴‘𝑍) ∈ ℝ^* ∧ (𝐵‘𝑍) ∈ ℝ^* ∧ 𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → (𝐴‘𝑍) ≤ 𝑢)
156	151, 153, 154, 155	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴‘𝑍) ≤ 𝑢)
157	127	sselda 3931	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ ℝ)
158	11	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴‘𝑍) ∈ ℝ)
159	157, 158	subge0d 11861	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (0 ≤ (𝑢 − (𝐴‘𝑍)) ↔ (𝐴‘𝑍) ≤ 𝑢))
160	156, 159	mpbird 260	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 0 ≤ (𝑢 − (𝐴‘𝑍)))
161	160	3adant3 1150	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑟 = (𝑢 − (𝐴‘𝑍))) → 0 ≤ (𝑢 − (𝐴‘𝑍)))
162		id 23	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = (𝑢 − (𝐴‘𝑍)) → 𝑟 = (𝑢 − (𝐴‘𝑍)))
163	162	eqcomd 2766	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑢 − (𝐴‘𝑍)) → (𝑢 − (𝐴‘𝑍)) = 𝑟)
164	163	3ad2ant3 1153	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑟 = (𝑢 − (𝐴‘𝑍))) → (𝑢 − (𝐴‘𝑍)) = 𝑟)
165	161, 164	breqtrd 5131	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑟 = (𝑢 − (𝐴‘𝑍))) → 0 ≤ 𝑟)
166	165	3exp 1137	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑟 = (𝑢 − (𝐴‘𝑍)) → 0 ≤ 𝑟)))
167	166	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → (𝑢 ∈ 𝑈 → (𝑟 = (𝑢 − (𝐴‘𝑍)) → 0 ≤ 𝑟)))
168	148, 149, 167	rexlimd 3269	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → (∃𝑢 ∈ 𝑈 𝑟 = (𝑢 − (𝐴‘𝑍)) → 0 ≤ 𝑟))
169	142, 168	mpd 16	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → 0 ≤ 𝑟)
170	169	ralrimiva 3154	. . . . . . 7 ⊢ (𝜑 → ∀𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}0 ≤ 𝑟)
171		simp3 1156	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑤 = (𝑢 − (𝐴‘𝑍))) → 𝑤 = (𝑢 − (𝐴‘𝑍)))
172	157, 158	resubcld 11699	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 − (𝐴‘𝑍)) ∈ ℝ)
173	172	3adant3 1150	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑤 = (𝑢 − (𝐴‘𝑍))) → (𝑢 − (𝐴‘𝑍)) ∈ ℝ)
174	171, 173	eqeltrd 2860	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑤 = (𝑢 − (𝐴‘𝑍))) → 𝑤 ∈ ℝ)
175	174	3exp 1137	. . . . . . . . 9 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑤 = (𝑢 − (𝐴‘𝑍)) → 𝑤 ∈ ℝ)))
176	175	rexlimdv 3161	. . . . . . . 8 ⊢ (𝜑 → (∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) → 𝑤 ∈ ℝ))
177	176	abssdv 4015	. . . . . . 7 ⊢ (𝜑 → {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ⊆ ℝ)
178	23	eqcomd 2766	. . . . . . . . . 10 ⊢ (𝜑 → 0 = ((𝐴‘𝑍) − (𝐴‘𝑍)))
179		oveq1 7416	. . . . . . . . . . 11 ⊢ (𝑢 = (𝐴‘𝑍) → (𝑢 − (𝐴‘𝑍)) = ((𝐴‘𝑍) − (𝐴‘𝑍)))
180	179	rspceeqv 3599	. . . . . . . . . 10 ⊢ (((𝐴‘𝑍) ∈ 𝑈 ∧ 0 = ((𝐴‘𝑍) − (𝐴‘𝑍))) → ∃𝑢 ∈ 𝑈 0 = (𝑢 − (𝐴‘𝑍)))
181	119, 178, 180	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑢 ∈ 𝑈 0 = (𝑢 − (𝐴‘𝑍)))
182		eqeq1 2764	. . . . . . . . . 10 ⊢ (𝑤 = 0 → (𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ 0 = (𝑢 − (𝐴‘𝑍))))
183	182	rexbidv 3186	. . . . . . . . 9 ⊢ (𝑤 = 0 → (∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ ∃𝑢 ∈ 𝑈 0 = (𝑢 − (𝐴‘𝑍))))
184	46, 181, 183	elabd 3635	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))})
185		ne0i 4287	. . . . . . . 8 ⊢ (0 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} → {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ≠ ∅)
186	184, 185	syl 18	. . . . . . 7 ⊢ (𝜑 → {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ≠ ∅)
187	13, 11	resubcld 11699	. . . . . . . 8 ⊢ (𝜑 → ((𝐵‘𝑍) − (𝐴‘𝑍)) ∈ ℝ)
188		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑠 ∈ V
189		eqeq1 2764	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑠 → (𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ 𝑠 = (𝑢 − (𝐴‘𝑍))))
190	189	rexbidv 3186	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑠 → (∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ ∃𝑢 ∈ 𝑈 𝑠 = (𝑢 − (𝐴‘𝑍))))
191	188, 190	elab 3633	. . . . . . . . . . 11 ⊢ (𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ↔ ∃𝑢 ∈ 𝑈 𝑠 = (𝑢 − (𝐴‘𝑍)))
192	191	bilani 510	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → ∃𝑢 ∈ 𝑈 𝑠 = (𝑢 − (𝐴‘𝑍)))
193		nfcv 2922	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑢𝑠
194	193, 146	nfel 2936	. . . . . . . . . . . 12 ⊢ Ⅎ𝑢 𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}
195	143, 194	nfan 1932	. . . . . . . . . . 11 ⊢ Ⅎ𝑢(𝜑 ∧ 𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))})
196		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑢 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍))
197		simp3 1156	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → 𝑠 = (𝑢 − (𝐴‘𝑍)))
198	158	3adant3 1150	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → (𝐴‘𝑍) ∈ ℝ)
199	13	3ad2ant1 1151	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → (𝐵‘𝑍) ∈ ℝ)
200	154	3adant3 1150	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → 𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
201	21	3ad2ant1 1151	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → (𝐴‘𝑍) ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
202	198, 199, 200, 201	iccsuble 46447	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → (𝑢 − (𝐴‘𝑍)) ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))
203	197, 202	eqbrtrd 5127	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))
204	203	3exp 1137	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑠 = (𝑢 − (𝐴‘𝑍)) → 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))))
205	204	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → (𝑢 ∈ 𝑈 → (𝑠 = (𝑢 − (𝐴‘𝑍)) → 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))))
206	195, 196, 205	rexlimd 3269	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → (∃𝑢 ∈ 𝑈 𝑠 = (𝑢 − (𝐴‘𝑍)) → 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍))))
207	192, 206	mpd 16	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))
208	207	ralrimiva 3154	. . . . . . . 8 ⊢ (𝜑 → ∀𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))
209		brralrspcev 5165	. . . . . . . 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)
210	187, 208, 209	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑠 ≤ 𝑟)
211		eqid 2760	. . . . . . . 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)}
212		biid 264	. . . . . . . 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)))
213	211, 212	supmul1 12241	. . . . . . 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, < ))
214	38, 137, 170, 177, 186, 210, 213	syl33anc 1412	. . . . . 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, < ))
215	125, 134, 214	3eqtrd 2799	. . . . 5 ⊢ (𝜑 → (𝐺 · (𝑆 − (𝐴‘𝑍))) = sup({𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}, ℝ, < ))
216		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑐 ∈ V
217		eqeq1 2764	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑐 → (𝑣 = (𝐺 · 𝑡) ↔ 𝑐 = (𝐺 · 𝑡)))
218	217	rexbidv 3186	. . . . . . . . . . . 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)))
219	216, 218	elab 3633	. . . . . . . . . . 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))
220	219	biimpi 219	. . . . . . . . . 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))
221		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑡∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))
222		vex 3454	. . . . . . . . . . . . . . 15 ⊢ 𝑡 ∈ V
223		eqeq1 2764	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑡 → (𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ 𝑡 = (𝑢 − (𝐴‘𝑍))))
224	223	rexbidv 3186	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑡 → (∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ ∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍))))
225	222, 224	elab 3633	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ↔ ∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)))
226	225	birani 509	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ∧ 𝑐 = (𝐺 · 𝑡)) → ∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)))
227		simpl 488	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 = (𝐺 · 𝑡) ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → 𝑐 = (𝐺 · 𝑡))
228		oveq2 7417	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = (𝑢 − (𝐴‘𝑍)) → (𝐺 · 𝑡) = (𝐺 · (𝑢 − (𝐴‘𝑍))))
229	228	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 = (𝐺 · 𝑡) ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → (𝐺 · 𝑡) = (𝐺 · (𝑢 − (𝐴‘𝑍))))
230	227, 229	eqtrd 2795	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 = (𝐺 · 𝑡) ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))))
231	230	ex 418	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = (𝐺 · 𝑡) → (𝑡 = (𝑢 − (𝐴‘𝑍)) → 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))))
232	231	reximdv 3177	. . . . . . . . . . . . . 14 ⊢ (𝑐 = (𝐺 · 𝑡) → (∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)) → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))))
233	232	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ∧ 𝑐 = (𝐺 · 𝑡)) → (∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)) → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))))
234	226, 233	mpd 16	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ∧ 𝑐 = (𝐺 · 𝑡)) → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))))
235	234	ex 418	. . . . . . . . . . 11 ⊢ (𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} → (𝑐 = (𝐺 · 𝑡) → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))))
236	221, 235	rexlimi 3262	. . . . . . . . . 10 ⊢ (∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑐 = (𝐺 · 𝑡) → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))))
237	220, 236	syl 18	. . . . . . . . 9 ⊢ (𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)} → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))))
238	237	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}) → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))))
239		simp3 1156	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))) → 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))))
240	38	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝐺 ∈ ℝ)
241	240, 172	remulcld 11296	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐺 · (𝑢 − (𝐴‘𝑍))) ∈ ℝ)
242	45	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (1 + 𝐸) ∈ ℝ)
243	52	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ℕ ∈ V)
244	60	adantlr 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
245	64	adantlr 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑊⟶ℝ)
246	157	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → 𝑢 ∈ ℝ)
247	79	adantlr 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑊⟶ℝ)
248	74, 246, 244, 247	hsphoif 47502	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐻‘𝑢)‘(𝐷‘𝑗)):𝑊⟶ℝ)
249	27, 244, 245, 248	hoidmvcl 47508	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
250	54, 249	sselid 3929	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
251	250	fmpttd 7104	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗)))):ℕ⟶(0[,]+∞))
252	243, 251	sge0cl 47307	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ∈ (0[,]+∞))
253	243, 251	sge0xrcl 47311	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ∈ ℝ^*)
254	86	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → +∞ ∈ ℝ^*)
255	89	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ^*)
256		nfv 1947	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑢 ∈ 𝑈)
257	92	adantlr 728	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) ∈ (0[,]+∞))
258	95	adantlr 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → 𝑍 ∈ (𝑊 ∖ 𝑌))
259	27, 244, 258, 9, 246, 74, 245, 247	hsphoidmvle 47512	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))
260	256, 243, 250, 257, 259	sge0lempt 47336	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))
261	98	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) < +∞)
262	253, 255, 254, 260, 261	xrlelttrd 13244	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) < +∞)
263	253, 254, 262	xrltned 46285	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ≠ +∞)
264		ge0xrre 46459	. . . . . . . . . . . . . . . 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)
265	252, 263, 264	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ∈ ℝ)
266	242, 265	remulcld 11296	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗)))))) ∈ ℝ)
267	126, 123	sseldd 3932	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 ∈ ℝ)
268	27, 30, 94, 9, 61, 76, 88, 65, 267	sge0hsphoire 47515	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℝ)
269	45, 268	remulcld 11296	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) ∈ ℝ)
270	269	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) ∈ ℝ)
271	14	eleq2i 2852	. . . . . . . . . . . . . . . . . 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))))))})
272	271	biimpi 219	. . . . . . . . . . . . . . . . 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))))))})
273		oveq1 7416	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑢 → (𝑧 − (𝐴‘𝑍)) = (𝑢 − (𝐴‘𝑍)))
274	273	oveq2d 7425	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑢 → (𝐺 · (𝑧 − (𝐴‘𝑍))) = (𝐺 · (𝑢 − (𝐴‘𝑍))))
275		fveq2 6874	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 = 𝑢 → (𝐻‘𝑧) = (𝐻‘𝑢))
276	275	fveq1d 6876	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = 𝑢 → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘𝑢)‘(𝐷‘𝑗)))
277	276	oveq2d 7425	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑢 → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))
278	277	mpteq2dv 5199	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑢 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗)))))
279	278	fveq2d 6878	. . . . . . . . . . . . . . . . . . . 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))))))
280	279	oveq2d 7425	. . . . . . . . . . . . . . . . . . 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)))))))
281	274, 280	breq12d 5116	. . . . . . . . . . . . . . . . . 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))))))))
282	281	elrab 3645	. . . . . . . . . . . . . . . . 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))))))))
283	272, 282	sylib 221	. . . . . . . . . . . . . . . 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))))))))
284	283	simprd 501	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑈 → (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗)))))))
285	284	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗)))))))
286	268	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℝ)
287	51	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 0 ≤ (1 + 𝐸))
288	267	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑆 ∈ ℝ)
289	74, 288, 60, 79	hsphoif 47502	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐻‘𝑆)‘(𝐷‘𝑗)):𝑊⟶ℝ)
290	27, 60, 64, 289	hoidmvcl 47508	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
291	54, 290	sselid 3929	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
292	291	adantlr 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
293	288	adantlr 728	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → 𝑆 ∈ ℝ)
294	127	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ⊆ ℝ)
295	121	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ≠ ∅)
296	131	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑈 𝑧 ≤ 𝑦)
297		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ 𝑈)
298		suprub 12233	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑈 𝑧 ≤ 𝑦) ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
299	294, 295, 296, 297, 298	syl31anc 1400	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
300	299, 1	breqtrrdi 5147	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ 𝑆)
301	300	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → 𝑢 ≤ 𝑆)
302	27, 244, 258, 9, 246, 293, 301, 74, 245, 247	hsphoidmvle2 47511	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))
303	256, 243, 250, 292, 302	sge0lempt 47336	. . . . . . . . . . . . . . 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))))))
304	265, 286, 242, 287, 303	lemul2ad 12212	. . . . . . . . . . . . . 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)))))))
305	241, 266, 270, 285, 304	letrd 11424	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
306	305	3adant3 1150	. . . . . . . . . . . 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)))))))
307	239, 306	eqbrtrd 5127	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))) → 𝑐 ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
308	307	3exp 1137	. . . . . . . . . 10 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))) → 𝑐 ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))))
309	308	rexlimdv 3161	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))) → 𝑐 ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))))
310	309	adantr 486	. . . . . . . 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))))))))
311	238, 310	mpd 16	. . . . . . 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)))))))
312	311	ralrimiva 3154	. . . . . 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)))))))
313	219	bilani 510	. . . . . . . . . 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))
314		nfv 1947	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡𝜑
315		nfcv 2922	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡𝑐
316		nfre1 3287	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)
317	316	nfab 2928	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡{𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}
318	315, 317	nfel 2936	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}
319	314, 318	nfan 1932	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)})
320		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑡 𝑐 ∈ ℝ
321	225	bilani 510	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → ∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)))
322		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) ∧ 𝑐 = (𝐺 · 𝑡)) → 𝑐 = (𝐺 · 𝑡))
323	240	3adant3 1150	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → 𝐺 ∈ ℝ)
324		simp3 1156	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → 𝑡 = (𝑢 − (𝐴‘𝑍)))
325	172	3adant3 1150	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → (𝑢 − (𝐴‘𝑍)) ∈ ℝ)
326	324, 325	eqeltrd 2860	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → 𝑡 ∈ ℝ)
327	323, 326	remulcld 11296	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → (𝐺 · 𝑡) ∈ ℝ)
328	327	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) ∧ 𝑐 = (𝐺 · 𝑡)) → (𝐺 · 𝑡) ∈ ℝ)
329	322, 328	eqeltrd 2860	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) ∧ 𝑐 = (𝐺 · 𝑡)) → 𝑐 ∈ ℝ)
330	329	ex 418	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → (𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ))
331	330	3exp 1137	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑡 = (𝑢 − (𝐴‘𝑍)) → (𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ))))
332	331	rexlimdv 3161	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)) → (𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ)))
333	332	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → (∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)) → (𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ)))
334	321, 333	mpd 16	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → (𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ))
335	334	ex 418	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} → (𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ)))
336	335	adantr 486	. . . . . . . . . . 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)))
337	319, 320, 336	rexlimd 3269	. . . . . . . . . 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))
338	313, 337	mpd 16	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}) → 𝑐 ∈ ℝ)
339	338	ralrimiva 3154	. . . . . . . 8 ⊢ (𝜑 → ∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ∈ ℝ)
340		dfss3 3920	. . . . . . . 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)
341	339, 340	sylibr 237	. . . . . . 7 ⊢ (𝜑 → {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)} ⊆ ℝ)
342	40	eqcomd 2766	. . . . . . . . . 10 ⊢ (𝜑 → 0 = (𝐺 · 0))
343		oveq2 7417	. . . . . . . . . . 11 ⊢ (𝑡 = 0 → (𝐺 · 𝑡) = (𝐺 · 0))
344	343	rspceeqv 3599	. . . . . . . . . 10 ⊢ ((0 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ∧ 0 = (𝐺 · 0)) → ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}0 = (𝐺 · 𝑡))
345	184, 342, 344	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}0 = (𝐺 · 𝑡))
346		eqeq1 2764	. . . . . . . . . 10 ⊢ (𝑣 = 0 → (𝑣 = (𝐺 · 𝑡) ↔ 0 = (𝐺 · 𝑡)))
347	346	rexbidv 3186	. . . . . . . . 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)))
348	46, 345, 347	elabd 3635	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)})
349		ne0i 4287	. . . . . . . 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)} ≠ ∅)
350	348, 349	syl 18	. . . . . . 7 ⊢ (𝜑 → {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)} ≠ ∅)
351	38, 187	remulcld 11296	. . . . . . . 8 ⊢ (𝜑 → (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))) ∈ ℝ)
352	187	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ((𝐵‘𝑍) − (𝐴‘𝑍)) ∈ ℝ)
353	137	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 0 ≤ 𝐺)
354	13	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐵‘𝑍) ∈ ℝ)
355		iccleub 13487	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴‘𝑍) ∈ ℝ^* ∧ (𝐵‘𝑍) ∈ ℝ^* ∧ 𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → 𝑢 ≤ (𝐵‘𝑍))
356	151, 153, 154, 355	syl3anc 1398	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ (𝐵‘𝑍))
357	157, 354, 158, 356	lesub1dd 11887	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 − (𝐴‘𝑍)) ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))
358	172, 352, 240, 353, 357	lemul2ad 12212	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
359	358	3adant3 1150	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))) → (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
360	239, 359	eqbrtrd 5127	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))) → 𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
361	360	3exp 1137	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))) → 𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))))
362	361	rexlimdv 3161	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))) → 𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍)))))
363	362	adantr 486	. . . . . . . . . 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)))))
364	238, 363	mpd 16	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}) → 𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
365	364	ralrimiva 3154	. . . . . . . 8 ⊢ (𝜑 → ∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
366		brralrspcev 5165	. . . . . . . 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)
367	351, 365, 366	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ≤ 𝑦)
368		suprleub 12238	. . . . . . 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))))))))
369	341, 350, 367, 269, 368	syl31anc 1400	. . . . . 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))))))))
370	312, 369	mpbird 260	. . . . 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)))))))
371	215, 370	eqbrtrd 5127	. . . 4 ⊢ (𝜑 → (𝐺 · (𝑆 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
372	123, 371	jca 521	. . 3 ⊢ (𝜑 → (𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∧ (𝐺 · (𝑆 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))))
373		oveq1 7416	. . . . . 6 ⊢ (𝑧 = 𝑆 → (𝑧 − (𝐴‘𝑍)) = (𝑆 − (𝐴‘𝑍)))
374	373	oveq2d 7425	. . . . 5 ⊢ (𝑧 = 𝑆 → (𝐺 · (𝑧 − (𝐴‘𝑍))) = (𝐺 · (𝑆 − (𝐴‘𝑍))))
375		fveq2 6874	. . . . . . . . . 10 ⊢ (𝑧 = 𝑆 → (𝐻‘𝑧) = (𝐻‘𝑆))
376	375	fveq1d 6876	. . . . . . . . 9 ⊢ (𝑧 = 𝑆 → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘𝑆)‘(𝐷‘𝑗)))
377	376	oveq2d 7425	. . . . . . . 8 ⊢ (𝑧 = 𝑆 → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))
378	377	mpteq2dv 5199	. . . . . . 7 ⊢ (𝑧 = 𝑆 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))
379	378	fveq2d 6878	. . . . . 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))))))
380	379	oveq2d 7425	. . . . 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)))))))
381	374, 380	breq12d 5116	. . . 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))))))))
382	381	elrab 3645	. . 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))))))))
383	372, 382	sylibr 237	. 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))))))})
384	383, 14	eleqtrrdi 2871	1 ⊢ (𝜑 → 𝑆 ∈ 𝑈)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 {cab 2738 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 {crab 3412 Vcvv 3450 ∖ cdif 3896 ∪ cun 3897 ⊆ wss 3899 ∅c0 4279 ifcif 4482 {csn 4584 class class class wbr 5103 ↦ cmpt 5186 ↾ cres 5650 ⟶wf 6524 ‘cfv 6528 (class class class)co 7409 ∈ cmpo 7411 ↑_m cmap 8826 Fincfn 8952 supcsup 9410 ℝcr 11156 0cc0 11157 1c1 11158 + caddc 11160 · cmul 11162 +∞cpnf 11297 ℝ^*cxr 11299 < clt 11300 ≤ cle 11301 − cmin 11498 ℕcn 12290 ℝ⁺crp 13075 [,)cico 13433 [,]cicc 13434 ∏cprod 16025 volcvol 25731 Σ^{^}csumge0 47288

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7735 ax-inf2 9620 ax-cnex 11213 ax-resscn 11214 ax-1cn 11215 ax-icn 11216 ax-addcl 11217 ax-addrcl 11218 ax-mulcl 11219 ax-mulrcl 11220 ax-mulcom 11221 ax-addass 11222 ax-mulass 11223 ax-distr 11224 ax-i2m1 11225 ax-1ne0 11226 ax-1rid 11227 ax-rnegex 11228 ax-rrecex 11229 ax-cnre 11230 ax-pre-lttri 11231 ax-pre-lttrn 11232 ax-pre-ltadd 11233 ax-pre-mulgt0 11234 ax-pre-sup 11235

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-se 5602 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6294 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-isom 6537 df-riota 7366 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7677 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-pm 8829 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-dju 9939 df-card 9977 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11500 df-neg 11501 df-div 11929 df-nn 12291 df-2 12360 df-3 12361 df-n0 12562 df-z 12649 df-uz 12921 df-q 13031 df-rp 13076 df-xneg 13196 df-xadd 13197 df-xmul 13198 df-ioo 13435 df-ico 13437 df-icc 13438 df-fz 13595 df-fzo 13743 df-fl 13886 df-seq 14099 df-exp 14159 df-hash 14428 df-cj 15219 df-re 15220 df-im 15221 df-sqrt 15355 df-abs 15356 df-clim 15608 df-rlim 15609 df-sum 15807 df-prod 16026 df-rest 17540 df-topgen 17561 df-psmet 21617 df-xmet 21618 df-met 21619 df-bl 21620 df-mopn 21621 df-top 23159 df-topon 23176 df-bases 23211 df-cmp 23652 df-ovol 25732 df-vol 25733 df-sumge0 47289

This theorem is used by: hoidmvlelem4 47524

W3C validator