hoidmvlelem1 - Mathbox for Glauco Siliprandi

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Theorem hoidmvlelem1 40809

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: 𝑥,𝑘 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 4206	. . . . . . . . . 10 ⊢ (𝑍 ∈ (𝑋 ∖ 𝑌) → 𝑍 ∈ {𝑍})
6	4, 5	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ {𝑍})
7		elun2 3781	. . . . . . . . 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	syl6eleqr 2712	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
11	3, 10	ffvelrnd 6360	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ)
12		hoidmvlelem1.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑊⟶ℝ)
13	12, 10	ffvelrnd 6360	. . . . . 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 3687	. . . . . . . 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 3635	. . . . . . 7 ⊢ 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍))
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
18	11	leidd 10594	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴‘𝑍) ≤ (𝐴‘𝑍))
19		hoidmvlelem1.ab	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴‘𝑍) < (𝐵‘𝑍))
20	11, 13, 19	ltled 10185	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴‘𝑍) ≤ (𝐵‘𝑍))
21	11, 13, 11, 18, 20	eliccd 39726	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
22	11	recnd 10068	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℂ)
23	22	subidd 10380	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴‘𝑍) − (𝐴‘𝑍)) = 0)
24	23	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 · ((𝐴‘𝑍) − (𝐴‘𝑍))) = (𝐺 · 0))
25		rge0ssre 12280	. . . . . . . . . . . . . . 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 8183	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑌 ∈ Fin)
31		ssun1 3776	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑌 ⊆ (𝑌 ∪ {𝑍})
32	31, 9	sseqtr4i 3638	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑌 ⊆ 𝑊
33	32	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑌 ⊆ 𝑊)
34	3, 33	fssresd 6071	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ↾ 𝑌):𝑌⟶ℝ)
35	12, 33	fssresd 6071	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 ↾ 𝑌):𝑌⟶ℝ)
36	27, 30, 34, 35	hoidmvcl 40796	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ∈ (0[,)+∞))
37	26, 36	syl5eqel 2705	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ∈ (0[,)+∞))
38	25, 37	sseldi 3601	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ ℝ)
39	38	recnd 10068	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ ℂ)
40	39	mul01d 10235	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 · 0) = 0)
41	24, 40	eqtrd 2656	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 · ((𝐴‘𝑍) − (𝐴‘𝑍))) = 0)
42		1red 10055	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℝ)
43		hoidmvlelem1.e	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
44	43	rpred 11872	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ)
45	42, 44	readdcld 10069	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 + 𝐸) ∈ ℝ)
46		0red 10041	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℝ)
47		0lt1 10550	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
48	47	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 1)
49	42, 43	ltaddrpd 11905	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 < (1 + 𝐸))
50	46, 42, 45, 48, 49	lttrd 10198	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < (1 + 𝐸))
51	46, 45, 50	ltled 10185	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (1 + 𝐸))
52		nnex 11026	. . . . . . . . . . . . . . 15 ⊢ ℕ ∈ V
53	52	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℕ ∈ V)
54		icossicc 12260	. . . . . . . . . . . . . . . 16 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
55		snfi 8038	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑍} ∈ Fin
56	55	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → {𝑍} ∈ Fin)
57		unfi 8227	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑌 ∈ Fin ∧ {𝑍} ∈ Fin) → (𝑌 ∪ {𝑍}) ∈ Fin)
58	30, 56, 57	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑌 ∪ {𝑍}) ∈ Fin)
59	9, 58	syl5eqel 2705	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑊 ∈ Fin)
60	59	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
61		hoidmvlelem1.c	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑_𝑚 𝑊))
62	61	ffvelrnda 6359	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑_𝑚 𝑊))
63		elmapi 7879	. . . . . . . . . . . . . . . . . 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		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = ℎ → (𝑗 ∈ 𝑌 ↔ ℎ ∈ 𝑌))
67		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = ℎ → (𝑐‘𝑗) = (𝑐‘ℎ))
68	67	breq1d 4663	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = ℎ → ((𝑐‘𝑗) ≤ 𝑥 ↔ (𝑐‘ℎ) ≤ 𝑥))
69	68, 67	ifbieq1d 4109	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = ℎ → if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥) = if((𝑐‘ℎ) ≤ 𝑥, (𝑐‘ℎ), 𝑥))
70	66, 67, 69	ifbieq12d 4113	. . . . . . . . . . . . . . . . . . . . . 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 4750	. . . . . . . . . . . . . . . . . . . . 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 4741	. . . . . . . . . . . . . . . . . . . 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 4741	. . . . . . . . . . . . . . . . . . 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 2644	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐴‘𝑍) ∈ ℝ)
76		hoidmvlelem1.d	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑_𝑚 𝑊))
77	76	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑_𝑚 𝑊))
78		elmapi 7879	. . . . . . . . . . . . . . . . . . 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 40790	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)):𝑊⟶ℝ)
81	27, 60, 64, 80	hoidmvcl 40796	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))) ∈ (0[,)+∞))
82	54, 81	sseldi 3601	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))) ∈ (0[,]+∞))
83		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))
84	82, 83	fmptd 6385	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)))):ℕ⟶(0[,]+∞))
85	53, 84	sge0cl 40598	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ∈ (0[,]+∞))
86	53, 84	sge0xrcl 40602	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ∈ ℝ^*)
87		pnfxr 10092	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ^*
88	87	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → +∞ ∈ ℝ^*)
89		hoidmvlelem1.r	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
90	89	rexrd 10089	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ^*)
91		nfv 1843	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗𝜑
92	27, 60, 64, 79	hoidmvcl 40796	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) ∈ (0[,)+∞))
93	54, 92	sseldi 3601	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) ∈ (0[,]+∞))
94	4	eldifbd 3587	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
95	10, 94	eldifd 3585	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑍 ∈ (𝑊 ∖ 𝑌))
96	95	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑍 ∈ (𝑊 ∖ 𝑌))
97	27, 60, 96, 9, 75, 74, 64, 79	hsphoidmvle 40800	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))
98	91, 53, 82, 93, 97	sge0lempt 40627	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))
99	89	ltpnfd 11955	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) < +∞)
100	86, 90, 88, 98, 99	xrlelttrd 11991	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) < +∞)
101	86, 88, 100	xrltned 39573	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ≠ +∞)
102		ge0xrre 39758	. . . . . . . . . . . . 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)
103	85, 101, 102	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))) ∈ ℝ)
104	53, 84	sge0ge0 40601	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))))
105		mulge0 10546	. . . . . . . . . . . 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)))))))
106	45, 51, 103, 104, 105	syl22anc 1327	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)))))))
107	41, 106	eqbrtrd 4675	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 · ((𝐴‘𝑍) − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)))))))
108	21, 107	jca 554	. . . . . . . . 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))))))))
109		oveq1 6657	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐴‘𝑍) → (𝑧 − (𝐴‘𝑍)) = ((𝐴‘𝑍) − (𝐴‘𝑍)))
110	109	oveq2d 6666	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐴‘𝑍) → (𝐺 · (𝑧 − (𝐴‘𝑍))) = (𝐺 · ((𝐴‘𝑍) − (𝐴‘𝑍))))
111		fveq2 6191	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐴‘𝑍) → (𝐻‘𝑧) = (𝐻‘(𝐴‘𝑍)))
112	111	fveq1d 6193	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐴‘𝑍) → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗)))
113	112	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐴‘𝑍) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐴‘𝑍))‘(𝐷‘𝑗))))
114	113	mpteq2dv 4745	. . . . . . . . . . . . 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)))))
115	114	fveq2d 6195	. . . . . . . . . . . 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))))))
116	115	oveq2d 6666	. . . . . . . . . . 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)))))))
117	110, 116	breq12d 4666	. . . . . . . . . 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))))))))
118	117	elrab 3363	. . . . . . . . 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))))))))
119	108, 118	sylibr 224	. . . . . . . 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))))))})
120	119, 14	syl6eleqr 2712	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝑍) ∈ 𝑈)
121		ne0i 3921	. . . . . . 7 ⊢ ((𝐴‘𝑍) ∈ 𝑈 → 𝑈 ≠ ∅)
122	120, 121	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈 ≠ ∅)
123	11, 13, 17, 122	supicc 12320	. . . . 5 ⊢ (𝜑 → sup(𝑈, ℝ, < ) ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
124	2, 123	eqeltrd 2701	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
125	2	oveq1d 6665	. . . . . . 7 ⊢ (𝜑 → (𝑆 − (𝐴‘𝑍)) = (sup(𝑈, ℝ, < ) − (𝐴‘𝑍)))
126	125	oveq2d 6666	. . . . . 6 ⊢ (𝜑 → (𝐺 · (𝑆 − (𝐴‘𝑍))) = (𝐺 · (sup(𝑈, ℝ, < ) − (𝐴‘𝑍))))
127	11, 13	iccssred 39727	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴‘𝑍)[,](𝐵‘𝑍)) ⊆ ℝ)
128	17, 127	sstrd 3613	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ⊆ ℝ)
129	11, 13	jca 554	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴‘𝑍) ∈ ℝ ∧ (𝐵‘𝑍) ∈ ℝ))
130		iccsupr 12266	. . . . . . . . . 10 ⊢ ((((𝐴‘𝑍) ∈ ℝ ∧ (𝐵‘𝑍) ∈ ℝ) ∧ 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∧ (𝐴‘𝑍) ∈ 𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑈 𝑧 ≤ 𝑦))
131	129, 17, 120, 130	syl3anc 1326	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑈 𝑧 ≤ 𝑦))
132	131	simp3d 1075	. . . . . . . 8 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑈 𝑧 ≤ 𝑦)
133		eqid 2622	. . . . . . . 8 ⊢ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} = {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}
134	128, 122, 132, 11, 133	supsubc 39569	. . . . . . 7 ⊢ (𝜑 → (sup(𝑈, ℝ, < ) − (𝐴‘𝑍)) = sup({𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}, ℝ, < ))
135	134	oveq2d 6666	. . . . . 6 ⊢ (𝜑 → (𝐺 · (sup(𝑈, ℝ, < ) − (𝐴‘𝑍))) = (𝐺 · sup({𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}, ℝ, < )))
136	46	rexrd 10089	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ^*)
137		icogelb 12225	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐺 ∈ (0[,)+∞)) → 0 ≤ 𝐺)
138	136, 88, 37, 137	syl3anc 1326	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝐺)
139		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑟 ∈ V
140		eqeq1 2626	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑟 → (𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ 𝑟 = (𝑢 − (𝐴‘𝑍))))
141	140	rexbidv 3052	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑟 → (∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ ∃𝑢 ∈ 𝑈 𝑟 = (𝑢 − (𝐴‘𝑍))))
142	139, 141	elab 3350	. . . . . . . . . . 11 ⊢ (𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ↔ ∃𝑢 ∈ 𝑈 𝑟 = (𝑢 − (𝐴‘𝑍)))
143	142	biimpi 206	. . . . . . . . . 10 ⊢ (𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} → ∃𝑢 ∈ 𝑈 𝑟 = (𝑢 − (𝐴‘𝑍)))
144	143	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → ∃𝑢 ∈ 𝑈 𝑟 = (𝑢 − (𝐴‘𝑍)))
145		nfv 1843	. . . . . . . . . . 11 ⊢ Ⅎ𝑢𝜑
146		nfcv 2764	. . . . . . . . . . . 12 ⊢ Ⅎ𝑢𝑟
147		nfre1 3005	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑢∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))
148	147	nfab 2769	. . . . . . . . . . . 12 ⊢ Ⅎ𝑢{𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}
149	146, 148	nfel 2777	. . . . . . . . . . 11 ⊢ Ⅎ𝑢 𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}
150	145, 149	nfan 1828	. . . . . . . . . 10 ⊢ Ⅎ𝑢(𝜑 ∧ 𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))})
151		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑢0 ≤ 𝑟
152	11	rexrd 10089	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ^*)
153	152	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴‘𝑍) ∈ ℝ^*)
154	13	rexrd 10089	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ^*)
155	154	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐵‘𝑍) ∈ ℝ^*)
156	17	sselda 3603	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
157		iccgelb 12230	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴‘𝑍) ∈ ℝ^* ∧ (𝐵‘𝑍) ∈ ℝ^* ∧ 𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → (𝐴‘𝑍) ≤ 𝑢)
158	153, 155, 156, 157	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴‘𝑍) ≤ 𝑢)
159	128	sselda 3603	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ ℝ)
160	11	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴‘𝑍) ∈ ℝ)
161	159, 160	subge0d 10617	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (0 ≤ (𝑢 − (𝐴‘𝑍)) ↔ (𝐴‘𝑍) ≤ 𝑢))
162	158, 161	mpbird 247	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 0 ≤ (𝑢 − (𝐴‘𝑍)))
163	162	3adant3 1081	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑟 = (𝑢 − (𝐴‘𝑍))) → 0 ≤ (𝑢 − (𝐴‘𝑍)))
164		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = (𝑢 − (𝐴‘𝑍)) → 𝑟 = (𝑢 − (𝐴‘𝑍)))
165	164	eqcomd 2628	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑢 − (𝐴‘𝑍)) → (𝑢 − (𝐴‘𝑍)) = 𝑟)
166	165	3ad2ant3 1084	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑟 = (𝑢 − (𝐴‘𝑍))) → (𝑢 − (𝐴‘𝑍)) = 𝑟)
167	163, 166	breqtrd 4679	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑟 = (𝑢 − (𝐴‘𝑍))) → 0 ≤ 𝑟)
168	167	3exp 1264	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑟 = (𝑢 − (𝐴‘𝑍)) → 0 ≤ 𝑟)))
169	168	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → (𝑢 ∈ 𝑈 → (𝑟 = (𝑢 − (𝐴‘𝑍)) → 0 ≤ 𝑟)))
170	150, 151, 169	rexlimd 3026	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → (∃𝑢 ∈ 𝑈 𝑟 = (𝑢 − (𝐴‘𝑍)) → 0 ≤ 𝑟))
171	144, 170	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → 0 ≤ 𝑟)
172	171	ralrimiva 2966	. . . . . . 7 ⊢ (𝜑 → ∀𝑟 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}0 ≤ 𝑟)
173		simp3 1063	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑤 = (𝑢 − (𝐴‘𝑍))) → 𝑤 = (𝑢 − (𝐴‘𝑍)))
174	159, 160	resubcld 10458	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 − (𝐴‘𝑍)) ∈ ℝ)
175	174	3adant3 1081	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑤 = (𝑢 − (𝐴‘𝑍))) → (𝑢 − (𝐴‘𝑍)) ∈ ℝ)
176	173, 175	eqeltrd 2701	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑤 = (𝑢 − (𝐴‘𝑍))) → 𝑤 ∈ ℝ)
177	176	3exp 1264	. . . . . . . . . 10 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑤 = (𝑢 − (𝐴‘𝑍)) → 𝑤 ∈ ℝ)))
178	177	rexlimdv 3030	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) → 𝑤 ∈ ℝ))
179	178	alrimiv 1855	. . . . . . . 8 ⊢ (𝜑 → ∀𝑤(∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) → 𝑤 ∈ ℝ))
180		abss 3671	. . . . . . . 8 ⊢ ({𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ⊆ ℝ ↔ ∀𝑤(∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) → 𝑤 ∈ ℝ))
181	179, 180	sylibr 224	. . . . . . 7 ⊢ (𝜑 → {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ⊆ ℝ)
182	23	eqcomd 2628	. . . . . . . . . 10 ⊢ (𝜑 → 0 = ((𝐴‘𝑍) − (𝐴‘𝑍)))
183		oveq1 6657	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝐴‘𝑍) → (𝑢 − (𝐴‘𝑍)) = ((𝐴‘𝑍) − (𝐴‘𝑍)))
184	183	eqeq2d 2632	. . . . . . . . . . 11 ⊢ (𝑢 = (𝐴‘𝑍) → (0 = (𝑢 − (𝐴‘𝑍)) ↔ 0 = ((𝐴‘𝑍) − (𝐴‘𝑍))))
185	184	rspcev 3309	. . . . . . . . . 10 ⊢ (((𝐴‘𝑍) ∈ 𝑈 ∧ 0 = ((𝐴‘𝑍) − (𝐴‘𝑍))) → ∃𝑢 ∈ 𝑈 0 = (𝑢 − (𝐴‘𝑍)))
186	120, 182, 185	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑢 ∈ 𝑈 0 = (𝑢 − (𝐴‘𝑍)))
187		c0ex 10034	. . . . . . . . . 10 ⊢ 0 ∈ V
188		eqeq1 2626	. . . . . . . . . . 11 ⊢ (𝑤 = 0 → (𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ 0 = (𝑢 − (𝐴‘𝑍))))
189	188	rexbidv 3052	. . . . . . . . . 10 ⊢ (𝑤 = 0 → (∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ ∃𝑢 ∈ 𝑈 0 = (𝑢 − (𝐴‘𝑍))))
190	187, 189	elab 3350	. . . . . . . . 9 ⊢ (0 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ↔ ∃𝑢 ∈ 𝑈 0 = (𝑢 − (𝐴‘𝑍)))
191	186, 190	sylibr 224	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))})
192		ne0i 3921	. . . . . . . 8 ⊢ (0 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} → {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ≠ ∅)
193	191, 192	syl 17	. . . . . . 7 ⊢ (𝜑 → {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ≠ ∅)
194	13, 11	resubcld 10458	. . . . . . . 8 ⊢ (𝜑 → ((𝐵‘𝑍) − (𝐴‘𝑍)) ∈ ℝ)
195		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ V
196		eqeq1 2626	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑠 → (𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ 𝑠 = (𝑢 − (𝐴‘𝑍))))
197	196	rexbidv 3052	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑠 → (∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ ∃𝑢 ∈ 𝑈 𝑠 = (𝑢 − (𝐴‘𝑍))))
198	195, 197	elab 3350	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ↔ ∃𝑢 ∈ 𝑈 𝑠 = (𝑢 − (𝐴‘𝑍)))
199	198	biimpi 206	. . . . . . . . . . 11 ⊢ (𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} → ∃𝑢 ∈ 𝑈 𝑠 = (𝑢 − (𝐴‘𝑍)))
200	199	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → ∃𝑢 ∈ 𝑈 𝑠 = (𝑢 − (𝐴‘𝑍)))
201		nfcv 2764	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑢𝑠
202	201, 148	nfel 2777	. . . . . . . . . . . 12 ⊢ Ⅎ𝑢 𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}
203	145, 202	nfan 1828	. . . . . . . . . . 11 ⊢ Ⅎ𝑢(𝜑 ∧ 𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))})
204		nfv 1843	. . . . . . . . . . 11 ⊢ Ⅎ𝑢 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍))
205		simp3 1063	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → 𝑠 = (𝑢 − (𝐴‘𝑍)))
206	160	3adant3 1081	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → (𝐴‘𝑍) ∈ ℝ)
207	13	3ad2ant1 1082	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → (𝐵‘𝑍) ∈ ℝ)
208	156	3adant3 1081	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → 𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
209	21	3ad2ant1 1082	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → (𝐴‘𝑍) ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
210	206, 207, 208, 209	iccsuble 39745	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → (𝑢 − (𝐴‘𝑍)) ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))
211	205, 210	eqbrtrd 4675	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑠 = (𝑢 − (𝐴‘𝑍))) → 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))
212	211	3exp 1264	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑠 = (𝑢 − (𝐴‘𝑍)) → 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))))
213	212	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → (𝑢 ∈ 𝑈 → (𝑠 = (𝑢 − (𝐴‘𝑍)) → 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))))
214	203, 204, 213	rexlimd 3026	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → (∃𝑢 ∈ 𝑈 𝑠 = (𝑢 − (𝐴‘𝑍)) → 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍))))
215	200, 214	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))
216	215	ralrimiva 2966	. . . . . . . 8 ⊢ (𝜑 → ∀𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))
217		breq2 4657	. . . . . . . . . 10 ⊢ (𝑟 = ((𝐵‘𝑍) − (𝐴‘𝑍)) → (𝑠 ≤ 𝑟 ↔ 𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍))))
218	217	ralbidv 2986	. . . . . . . . 9 ⊢ (𝑟 = ((𝐵‘𝑍) − (𝐴‘𝑍)) → (∀𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑠 ≤ 𝑟 ↔ ∀𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑠 ≤ ((𝐵‘𝑍) − (𝐴‘𝑍))))
219	218	rspcev 3309	. . . . . . . 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)
220	194, 216, 219	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑠 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑠 ≤ 𝑟)
221		eqid 2622	. . . . . . . 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)}
222		biid 251	. . . . . . . 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)))
223	221, 222	supmul1 10992	. . . . . . 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, < ))
224	38, 138, 172, 181, 193, 220, 223	syl33anc 1341	. . . . . 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, < ))
225	126, 135, 224	3eqtrd 2660	. . . . 5 ⊢ (𝜑 → (𝐺 · (𝑆 − (𝐴‘𝑍))) = sup({𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}, ℝ, < ))
226		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑐 ∈ V
227		eqeq1 2626	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑐 → (𝑣 = (𝐺 · 𝑡) ↔ 𝑐 = (𝐺 · 𝑡)))
228	227	rexbidv 3052	. . . . . . . . . . . 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)))
229	226, 228	elab 3350	. . . . . . . . . . 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))
230	229	biimpi 206	. . . . . . . . . 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))
231		nfv 1843	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))
232		vex 3203	. . . . . . . . . . . . . . . . 17 ⊢ 𝑡 ∈ V
233		eqeq1 2626	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑡 → (𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ 𝑡 = (𝑢 − (𝐴‘𝑍))))
234	233	rexbidv 3052	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑡 → (∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍)) ↔ ∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍))))
235	232, 234	elab 3350	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ↔ ∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)))
236	235	biimpi 206	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} → ∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)))
237	236	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ∧ 𝑐 = (𝐺 · 𝑡)) → ∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)))
238		simpl 473	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 = (𝐺 · 𝑡) ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → 𝑐 = (𝐺 · 𝑡))
239		oveq2 6658	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = (𝑢 − (𝐴‘𝑍)) → (𝐺 · 𝑡) = (𝐺 · (𝑢 − (𝐴‘𝑍))))
240	239	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 = (𝐺 · 𝑡) ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → (𝐺 · 𝑡) = (𝐺 · (𝑢 − (𝐴‘𝑍))))
241	238, 240	eqtrd 2656	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 = (𝐺 · 𝑡) ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))))
242	241	ex 450	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝐺 · 𝑡) → (𝑡 = (𝑢 − (𝐴‘𝑍)) → 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))))
243	242	reximdv 3016	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = (𝐺 · 𝑡) → (∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)) → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))))
244	243	adantl 482	. . . . . . . . . . . . . 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)))))
245	237, 244	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ∧ 𝑐 = (𝐺 · 𝑡)) → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))))
246	245	ex 450	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} → (𝑐 = (𝐺 · 𝑡) → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))))
247	231, 246	rexlimi 3024	. . . . . . . . . . 11 ⊢ (∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑐 = (𝐺 · 𝑡) → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))))
248	247	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)))))
249	230, 248	mpd 15	. . . . . . . . 9 ⊢ (𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)} → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))))
250	249	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}) → ∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))))
251		simp3 1063	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))) → 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))))
252	38	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝐺 ∈ ℝ)
253	252, 174	remulcld 10070	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐺 · (𝑢 − (𝐴‘𝑍))) ∈ ℝ)
254	45	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (1 + 𝐸) ∈ ℝ)
255	52	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ℕ ∈ V)
256	60	adantlr 751	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
257	64	adantlr 751	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑊⟶ℝ)
258	159	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → 𝑢 ∈ ℝ)
259	79	adantlr 751	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑊⟶ℝ)
260	74, 258, 256, 259	hsphoif 40790	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐻‘𝑢)‘(𝐷‘𝑗)):𝑊⟶ℝ)
261	27, 256, 257, 260	hoidmvcl 40796	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
262	54, 261	sseldi 3601	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
263		eqid 2622	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))
264	262, 263	fmptd 6385	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗)))):ℕ⟶(0[,]+∞))
265	255, 264	sge0cl 40598	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ∈ (0[,]+∞))
266	255, 264	sge0xrcl 40602	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ∈ ℝ^*)
267	87	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → +∞ ∈ ℝ^*)
268	90	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ^*)
269		nfv 1843	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑢 ∈ 𝑈)
270	93	adantlr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) ∈ (0[,]+∞))
271	96	adantlr 751	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → 𝑍 ∈ (𝑊 ∖ 𝑌))
272	27, 256, 271, 9, 258, 74, 257, 259	hsphoidmvle 40800	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))
273	269, 255, 262, 270, 272	sge0lempt 40627	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))
274	99	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) < +∞)
275	266, 268, 267, 273, 274	xrlelttrd 11991	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) < +∞)
276	266, 267, 275	xrltned 39573	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ≠ +∞)
277		ge0xrre 39758	. . . . . . . . . . . . . . . 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)
278	265, 276, 277	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))) ∈ ℝ)
279	254, 278	remulcld 10070	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗)))))) ∈ ℝ)
280	127, 124	sseldd 3604	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 ∈ ℝ)
281	27, 30, 95, 9, 61, 76, 89, 65, 280	sge0hsphoire 40803	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℝ)
282	45, 281	remulcld 10070	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) ∈ ℝ)
283	282	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) ∈ ℝ)
284	14	eleq2i 2693	. . . . . . . . . . . . . . . . . 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))))))})
285	284	biimpi 206	. . . . . . . . . . . . . . . . 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))))))})
286		oveq1 6657	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑢 → (𝑧 − (𝐴‘𝑍)) = (𝑢 − (𝐴‘𝑍)))
287	286	oveq2d 6666	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑢 → (𝐺 · (𝑧 − (𝐴‘𝑍))) = (𝐺 · (𝑢 − (𝐴‘𝑍))))
288		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 = 𝑢 → (𝐻‘𝑧) = (𝐻‘𝑢))
289	288	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = 𝑢 → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘𝑢)‘(𝐷‘𝑗)))
290	289	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑢 → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))))
291	290	mpteq2dv 4745	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑢 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗)))))
292	291	fveq2d 6195	. . . . . . . . . . . . . . . . . . . 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))))))
293	292	oveq2d 6666	. . . . . . . . . . . . . . . . . . 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)))))))
294	287, 293	breq12d 4666	. . . . . . . . . . . . . . . . . 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))))))))
295	294	elrab 3363	. . . . . . . . . . . . . . . . 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))))))))
296	285, 295	sylib 208	. . . . . . . . . . . . . . . 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))))))))
297	296	simprd 479	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑈 → (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗)))))))
298	297	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗)))))))
299	281	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℝ)
300	51	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 0 ≤ (1 + 𝐸))
301	280	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑆 ∈ ℝ)
302	74, 301, 60, 79	hsphoif 40790	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐻‘𝑆)‘(𝐷‘𝑗)):𝑊⟶ℝ)
303	27, 60, 64, 302	hoidmvcl 40796	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
304	54, 303	sseldi 3601	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
305	304	adantlr 751	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
306	301	adantlr 751	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → 𝑆 ∈ ℝ)
307	128	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ⊆ ℝ)
308	122	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ≠ ∅)
309	132	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑈 𝑧 ≤ 𝑦)
310		simpr 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ 𝑈)
311		suprub 10984	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑈 𝑧 ≤ 𝑦) ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
312	307, 308, 309, 310, 311	syl31anc 1329	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
313	312, 1	syl6breqr 4695	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ 𝑆)
314	313	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → 𝑢 ≤ 𝑆)
315	27, 256, 271, 9, 258, 306, 314, 74, 257, 259	hsphoidmvle2 40799	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑢)‘(𝐷‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))
316	269, 255, 262, 305, 315	sge0lempt 40627	. . . . . . . . . . . . . . 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))))))
317	278, 299, 254, 300, 316	lemul2ad 10964	. . . . . . . . . . . . . 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)))))))
318	253, 279, 283, 298, 317	letrd 10194	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
319	318	3adant3 1081	. . . . . . . . . . . 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)))))))
320	251, 319	eqbrtrd 4675	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))) → 𝑐 ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
321	320	3exp 1264	. . . . . . . . . 10 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))) → 𝑐 ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))))
322	321	rexlimdv 3030	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))) → 𝑐 ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))))
323	322	adantr 481	. . . . . . . 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))))))))
324	250, 323	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)))))))
325	324	ralrimiva 2966	. . . . . 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)))))))
326	230	adantl 482	. . . . . . . . . 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))
327		nfv 1843	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡𝜑
328		nfcv 2764	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡𝑐
329		nfre1 3005	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)
330	329	nfab 2769	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡{𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}
331	328, 330	nfel 2777	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}
332	327, 331	nfan 1828	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)})
333		nfv 1843	. . . . . . . . . . 11 ⊢ Ⅎ𝑡 𝑐 ∈ ℝ
334	236	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → ∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)))
335		simpr 477	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) ∧ 𝑐 = (𝐺 · 𝑡)) → 𝑐 = (𝐺 · 𝑡))
336	252	3adant3 1081	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → 𝐺 ∈ ℝ)
337		simp3 1063	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → 𝑡 = (𝑢 − (𝐴‘𝑍)))
338	174	3adant3 1081	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → (𝑢 − (𝐴‘𝑍)) ∈ ℝ)
339	337, 338	eqeltrd 2701	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → 𝑡 ∈ ℝ)
340	336, 339	remulcld 10070	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → (𝐺 · 𝑡) ∈ ℝ)
341	340	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) ∧ 𝑐 = (𝐺 · 𝑡)) → (𝐺 · 𝑡) ∈ ℝ)
342	335, 341	eqeltrd 2701	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) ∧ 𝑐 = (𝐺 · 𝑡)) → 𝑐 ∈ ℝ)
343	342	ex 450	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑡 = (𝑢 − (𝐴‘𝑍))) → (𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ))
344	343	3exp 1264	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑡 = (𝑢 − (𝐴‘𝑍)) → (𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ))))
345	344	rexlimdv 3030	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)) → (𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ)))
346	345	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → (∃𝑢 ∈ 𝑈 𝑡 = (𝑢 − (𝐴‘𝑍)) → (𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ)))
347	334, 346	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}) → (𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ))
348	347	ex 450	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} → (𝑐 = (𝐺 · 𝑡) → 𝑐 ∈ ℝ)))
349	348	adantr 481	. . . . . . . . . . 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)))
350	332, 333, 349	rexlimd 3026	. . . . . . . . . 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))
351	326, 350	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}) → 𝑐 ∈ ℝ)
352	351	ralrimiva 2966	. . . . . . . 8 ⊢ (𝜑 → ∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ∈ ℝ)
353		dfss3 3592	. . . . . . . 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)
354	352, 353	sylibr 224	. . . . . . 7 ⊢ (𝜑 → {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)} ⊆ ℝ)
355	40	eqcomd 2628	. . . . . . . . . 10 ⊢ (𝜑 → 0 = (𝐺 · 0))
356		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑡 = 0 → (𝐺 · 𝑡) = (𝐺 · 0))
357	356	eqeq2d 2632	. . . . . . . . . . 11 ⊢ (𝑡 = 0 → (0 = (𝐺 · 𝑡) ↔ 0 = (𝐺 · 0)))
358	357	rspcev 3309	. . . . . . . . . 10 ⊢ ((0 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))} ∧ 0 = (𝐺 · 0)) → ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}0 = (𝐺 · 𝑡))
359	191, 355, 358	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}0 = (𝐺 · 𝑡))
360		eqeq1 2626	. . . . . . . . . . 11 ⊢ (𝑣 = 0 → (𝑣 = (𝐺 · 𝑡) ↔ 0 = (𝐺 · 𝑡)))
361	360	rexbidv 3052	. . . . . . . . . 10 ⊢ (𝑣 = 0 → (∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡) ↔ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}0 = (𝐺 · 𝑡)))
362	187, 361	elab 3350	. . . . . . . . 9 ⊢ (0 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)} ↔ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}0 = (𝐺 · 𝑡))
363	359, 362	sylibr 224	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)})
364		ne0i 3921	. . . . . . . 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)} ≠ ∅)
365	363, 364	syl 17	. . . . . . 7 ⊢ (𝜑 → {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)} ≠ ∅)
366	38, 194	remulcld 10070	. . . . . . . 8 ⊢ (𝜑 → (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))) ∈ ℝ)
367	194	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ((𝐵‘𝑍) − (𝐴‘𝑍)) ∈ ℝ)
368	138	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 0 ≤ 𝐺)
369	13	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐵‘𝑍) ∈ ℝ)
370		iccleub 12229	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴‘𝑍) ∈ ℝ^* ∧ (𝐵‘𝑍) ∈ ℝ^* ∧ 𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → 𝑢 ≤ (𝐵‘𝑍))
371	153, 155, 156, 370	syl3anc 1326	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ (𝐵‘𝑍))
372	159, 369, 160, 371	lesub1dd 10643	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 − (𝐴‘𝑍)) ≤ ((𝐵‘𝑍) − (𝐴‘𝑍)))
373	174, 367, 252, 368, 372	lemul2ad 10964	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
374	373	3adant3 1081	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))) → (𝐺 · (𝑢 − (𝐴‘𝑍))) ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
375	251, 374	eqbrtrd 4675	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈 ∧ 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍)))) → 𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
376	375	3exp 1264	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑢 ∈ 𝑈 → (𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))) → 𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))))
377	376	rexlimdv 3030	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑢 ∈ 𝑈 𝑐 = (𝐺 · (𝑢 − (𝐴‘𝑍))) → 𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍)))))
378	377	adantr 481	. . . . . . . . . 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)))))
379	250, 378	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}) → 𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
380	379	ralrimiva 2966	. . . . . . . 8 ⊢ (𝜑 → ∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
381		breq2 4657	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))) → (𝑐 ≤ 𝑦 ↔ 𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍)))))
382	381	ralbidv 2986	. . . . . . . . 9 ⊢ (𝑦 = (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))) → (∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ≤ 𝑦 ↔ ∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ≤ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍)))))
383	382	rspcev 3309	. . . . . . . 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)
384	366, 380, 383	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑐 ∈ {𝑣 ∣ ∃𝑡 ∈ {𝑤 ∣ ∃𝑢 ∈ 𝑈 𝑤 = (𝑢 − (𝐴‘𝑍))}𝑣 = (𝐺 · 𝑡)}𝑐 ≤ 𝑦)
385		suprleub 10989	. . . . . . 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))))))))
386	354, 365, 384, 282, 385	syl31anc 1329	. . . . . 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))))))))
387	325, 386	mpbird 247	. . . . 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)))))))
388	225, 387	eqbrtrd 4675	. . . 4 ⊢ (𝜑 → (𝐺 · (𝑆 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
389	124, 388	jca 554	. . 3 ⊢ (𝜑 → (𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∧ (𝐺 · (𝑆 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))))
390		oveq1 6657	. . . . . 6 ⊢ (𝑧 = 𝑆 → (𝑧 − (𝐴‘𝑍)) = (𝑆 − (𝐴‘𝑍)))
391	390	oveq2d 6666	. . . . 5 ⊢ (𝑧 = 𝑆 → (𝐺 · (𝑧 − (𝐴‘𝑍))) = (𝐺 · (𝑆 − (𝐴‘𝑍))))
392		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑧 = 𝑆 → (𝐻‘𝑧) = (𝐻‘𝑆))
393	392	fveq1d 6193	. . . . . . . . 9 ⊢ (𝑧 = 𝑆 → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘𝑆)‘(𝐷‘𝑗)))
394	393	oveq2d 6666	. . . . . . . 8 ⊢ (𝑧 = 𝑆 → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))
395	394	mpteq2dv 4745	. . . . . . 7 ⊢ (𝑧 = 𝑆 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))
396	395	fveq2d 6195	. . . . . 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))))))
397	396	oveq2d 6666	. . . . 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)))))))
398	391, 397	breq12d 4666	. . . 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))))))))
399	398	elrab 3363	. . 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))))))))
400	389, 399	sylibr 224	. 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))))))})
401	400, 14	syl6eleqr 2712	1 ⊢ (𝜑 → 𝑆 ∈ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 ∀wal 1481 = wceq 1483 ∈ wcel 1990 {cab 2608 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 ⊆ wss 3574 ∅c0 3915 ifcif 4086 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 ↾ cres 5116 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ↑_𝑚 cmap 7857 Fincfn 7955 supcsup 8346 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 +∞cpnf 10071 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 − cmin 10266 ℕcn 11020 ℝ⁺crp 11832 [,)cico 12177 [,]cicc 12178 ∏cprod 14635 volcvol 23232 Σ^{^}csumge0 40579

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 df-prod 14636 df-rest 16083 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 df-cmp 21190 df-ovol 23233 df-vol 23234 df-sumge0 40580

This theorem is referenced by: hoidmvlelem4 40812

W3C validator