hoidmvlelem4 - Mathbox for Glauco Siliprandi

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Theorem hoidmvlelem4 40812

Description: The dimensional volume of a multidimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. Induction step of Lemma 115B of [Fremlin1] p. 29, case nonempty interval and dimension of the space greater than 1. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

**Hypotheses**
Ref	Expression
hoidmvlelem4.l	⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑_𝑚 𝑥), 𝑏 ∈ (ℝ ↑_𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
hoidmvlelem4.x	⊢ (𝜑 → 𝑋 ∈ Fin)
hoidmvlelem4.y	⊢ (𝜑 → 𝑌 ⊆ 𝑋)
hoidmvlelem4.n	⊢ (𝜑 → 𝑌 ≠ ∅)
hoidmvlelem4.z	⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))
hoidmvlelem4.w	⊢ 𝑊 = (𝑌 ∪ {𝑍})
hoidmvlelem4.a	⊢ (𝜑 → 𝐴:𝑊⟶ℝ)
hoidmvlelem4.b	⊢ (𝜑 → 𝐵:𝑊⟶ℝ)
hoidmvlelem4.k	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴‘𝑘) < (𝐵‘𝑘))
hoidmvlelem4.c	⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑_𝑚 𝑊))
hoidmvlelem4.d	⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑_𝑚 𝑊))
hoidmvlelem4.r	⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
hoidmvlelem4.h	⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))
hoidmvlelem4.14	⊢ 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))
hoidmvlelem4.e	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
hoidmvlelem4.u	⊢ 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))}
hoidmvlelem4.s	⊢ 𝑆 = sup(𝑈, ℝ, < )
hoidmvlelem4.i	⊢ (𝜑 → ∀𝑒 ∈ (ℝ ↑_𝑚 𝑌)∀𝑓 ∈ (ℝ ↑_𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑_𝑚 𝑌) ↑_𝑚 ℕ)∀ℎ ∈ ((ℝ ↑_𝑚 𝑌) ↑_𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
hoidmvlelem4.i2	⊢ (𝜑 → X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))

**Assertion**
Ref	Expression
hoidmvlelem4	⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))))

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

Proof of Theorem hoidmvlelem4 Dummy variables 𝑦 𝑢 𝑖 𝑙 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rge0ssre 12280	. . 3 ⊢ (0[,)+∞) ⊆ ℝ
2		hoidmvlelem4.l	. . . 4 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑_𝑚 𝑥), 𝑏 ∈ (ℝ ↑_𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
3		hoidmvlelem4.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin)
4		hoidmvlelem4.w	. . . . . 6 ⊢ 𝑊 = (𝑌 ∪ {𝑍})
5		hoidmvlelem4.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
6		hoidmvlelem4.z	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))
7	6	eldifad 3586	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ 𝑋)
8		snssi 4339	. . . . . . . 8 ⊢ (𝑍 ∈ 𝑋 → {𝑍} ⊆ 𝑋)
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → {𝑍} ⊆ 𝑋)
10	5, 9	unssd 3789	. . . . . 6 ⊢ (𝜑 → (𝑌 ∪ {𝑍}) ⊆ 𝑋)
11	4, 10	syl5eqss 3649	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑋)
12		ssfi 8180	. . . . 5 ⊢ ((𝑋 ∈ Fin ∧ 𝑊 ⊆ 𝑋) → 𝑊 ∈ Fin)
13	3, 11, 12	syl2anc 693	. . . 4 ⊢ (𝜑 → 𝑊 ∈ Fin)
14		hoidmvlelem4.a	. . . 4 ⊢ (𝜑 → 𝐴:𝑊⟶ℝ)
15		hoidmvlelem4.b	. . . 4 ⊢ (𝜑 → 𝐵:𝑊⟶ℝ)
16	2, 13, 14, 15	hoidmvcl 40796	. . 3 ⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) ∈ (0[,)+∞))
17	1, 16	sseldi 3601	. 2 ⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) ∈ ℝ)
18		1red 10055	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
19		hoidmvlelem4.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
20	19	rpred 11872	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ)
21	18, 20	readdcld 10069	. . 3 ⊢ (𝜑 → (1 + 𝐸) ∈ ℝ)
22		nfv 1843	. . . . 5 ⊢ Ⅎ𝑗𝜑
23		nnex 11026	. . . . . 6 ⊢ ℕ ∈ V
24	23	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ∈ V)
25		icossicc 12260	. . . . . 6 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
26	13	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
27		hoidmvlelem4.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑_𝑚 𝑊))
28	27	ffvelrnda 6359	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑_𝑚 𝑊))
29		elmapi 7879	. . . . . . . 8 ⊢ ((𝐶‘𝑗) ∈ (ℝ ↑_𝑚 𝑊) → (𝐶‘𝑗):𝑊⟶ℝ)
30	28, 29	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑊⟶ℝ)
31		hoidmvlelem4.h	. . . . . . . . 9 ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))
32		eleq1 2689	. . . . . . . . . . . . 13 ⊢ (𝑗 = ℎ → (𝑗 ∈ 𝑌 ↔ ℎ ∈ 𝑌))
33		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑗 = ℎ → (𝑐‘𝑗) = (𝑐‘ℎ))
34	33	breq1d 4663	. . . . . . . . . . . . . 14 ⊢ (𝑗 = ℎ → ((𝑐‘𝑗) ≤ 𝑥 ↔ (𝑐‘ℎ) ≤ 𝑥))
35	34, 33	ifbieq1d 4109	. . . . . . . . . . . . 13 ⊢ (𝑗 = ℎ → if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥) = if((𝑐‘ℎ) ≤ 𝑥, (𝑐‘ℎ), 𝑥))
36	32, 33, 35	ifbieq12d 4113	. . . . . . . . . . . 12 ⊢ (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)))
37	36	cbvmptv 4750	. . . . . . . . . . 11 ⊢ (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)))
38	37	mpteq2i 4741	. . . . . . . . . 10 ⊢ (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))))
39	38	mpteq2i 4741	. . . . . . . . 9 ⊢ (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)))))
40	31, 39	eqtri 2644	. . . . . . . 8 ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_𝑚 𝑊) ↦ (ℎ ∈ 𝑊 ↦ if(ℎ ∈ 𝑌, (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑥, (𝑐‘ℎ), 𝑥)))))
41		snidg 4206	. . . . . . . . . . . . 13 ⊢ (𝑍 ∈ (𝑋 ∖ 𝑌) → 𝑍 ∈ {𝑍})
42	6, 41	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ∈ {𝑍})
43		elun2 3781	. . . . . . . . . . . 12 ⊢ (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑌 ∪ {𝑍}))
44	42, 43	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ (𝑌 ∪ {𝑍}))
45	4	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 = (𝑌 ∪ {𝑍}))
46	45	eqcomd 2628	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 ∪ {𝑍}) = 𝑊)
47	44, 46	eleqtrd 2703	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
48	15, 47	ffvelrnd 6360	. . . . . . . . 9 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ)
49	48	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐵‘𝑍) ∈ ℝ)
50		hoidmvlelem4.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑_𝑚 𝑊))
51	50	ffvelrnda 6359	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑_𝑚 𝑊))
52		elmapi 7879	. . . . . . . . 9 ⊢ ((𝐷‘𝑗) ∈ (ℝ ↑_𝑚 𝑊) → (𝐷‘𝑗):𝑊⟶ℝ)
53	51, 52	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑊⟶ℝ)
54	40, 49, 26, 53	hsphoif 40790	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗)):𝑊⟶ℝ)
55	2, 26, 30, 54	hoidmvcl 40796	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))) ∈ (0[,)+∞))
56	25, 55	sseldi 3601	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))) ∈ (0[,]+∞))
57	22, 24, 56	sge0clmpt 40642	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))) ∈ (0[,]+∞))
58	22, 24, 56	sge0xrclmpt 40645	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))) ∈ ℝ^*)
59		pnfxr 10092	. . . . . 6 ⊢ +∞ ∈ ℝ^*
60	59	a1i 11	. . . . 5 ⊢ (𝜑 → +∞ ∈ ℝ^*)
61		hoidmvlelem4.r	. . . . . . 7 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
62	61	rexrd 10089	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ^*)
63	2, 26, 30, 53	hoidmvcl 40796	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) ∈ (0[,)+∞))
64	25, 63	sseldi 3601	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) ∈ (0[,]+∞))
65	6	eldifbd 3587	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
66	47, 65	eldifd 3585	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ (𝑊 ∖ 𝑌))
67	66	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑍 ∈ (𝑊 ∖ 𝑌))
68	2, 26, 67, 4, 49, 40, 30, 53	hsphoidmvle 40800	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))
69	22, 24, 56, 64, 68	sge0lempt 40627	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))
70	61	ltpnfd 11955	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) < +∞)
71	58, 62, 60, 69, 70	xrlelttrd 11991	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))) < +∞)
72	58, 60, 71	xrltned 39573	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))) ≠ +∞)
73		ge0xrre 39758	. . . 4 ⊢ (((Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))) ∈ (0[,]+∞) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))) ≠ +∞) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))) ∈ ℝ)
74	57, 72, 73	syl2anc 693	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))) ∈ ℝ)
75	21, 74	remulcld 10070	. 2 ⊢ (𝜑 → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗)))))) ∈ ℝ)
76	21, 61	remulcld 10070	. 2 ⊢ (𝜑 → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))) ∈ ℝ)
77		hoidmvlelem4.14	. . . . . . 7 ⊢ 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))
78		hoidmvlelem4.u	. . . . . . 7 ⊢ 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))}
79		hoidmvlelem4.s	. . . . . . 7 ⊢ 𝑆 = sup(𝑈, ℝ, < )
80	47	ancli 574	. . . . . . . 8 ⊢ (𝜑 → (𝜑 ∧ 𝑍 ∈ 𝑊))
81		eleq1 2689	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑍 → (𝑘 ∈ 𝑊 ↔ 𝑍 ∈ 𝑊))
82	81	anbi2d 740	. . . . . . . . . 10 ⊢ (𝑘 = 𝑍 → ((𝜑 ∧ 𝑘 ∈ 𝑊) ↔ (𝜑 ∧ 𝑍 ∈ 𝑊)))
83		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑍 → (𝐴‘𝑘) = (𝐴‘𝑍))
84		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑍 → (𝐵‘𝑘) = (𝐵‘𝑍))
85	83, 84	breq12d 4666	. . . . . . . . . 10 ⊢ (𝑘 = 𝑍 → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ (𝐴‘𝑍) < (𝐵‘𝑍)))
86	82, 85	imbi12d 334	. . . . . . . . 9 ⊢ (𝑘 = 𝑍 → (((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴‘𝑘) < (𝐵‘𝑘)) ↔ ((𝜑 ∧ 𝑍 ∈ 𝑊) → (𝐴‘𝑍) < (𝐵‘𝑍))))
87		hoidmvlelem4.k	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴‘𝑘) < (𝐵‘𝑘))
88	86, 87	vtoclg 3266	. . . . . . . 8 ⊢ (𝑍 ∈ 𝑊 → ((𝜑 ∧ 𝑍 ∈ 𝑊) → (𝐴‘𝑍) < (𝐵‘𝑍)))
89	47, 80, 88	sylc 65	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝑍) < (𝐵‘𝑍))
90	2, 3, 5, 6, 4, 14, 15, 27, 50, 61, 31, 77, 19, 78, 79, 89	hoidmvlelem1 40809	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝑈)
91	48	rexrd 10089	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ^*)
92		iccssxr 12256	. . . . . . . . 9 ⊢ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ⊆ ℝ^*
93		ssrab2 3687	. . . . . . . . . . 11 ⊢ {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))
94	78, 93	eqsstri 3635	. . . . . . . . . 10 ⊢ 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍))
95	94, 90	sseldi 3601	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
96	92, 95	sseldi 3601	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℝ^*)
97		simpl 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝑆) → 𝜑)
98		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝑆) → ¬ (𝐵‘𝑍) ≤ 𝑆)
99	14, 47	ffvelrnd 6360	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ)
100	99, 48	iccssred 39727	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐴‘𝑍)[,](𝐵‘𝑍)) ⊆ ℝ)
101	100, 95	sseldd 3604	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ ℝ)
102	101	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝑆) → 𝑆 ∈ ℝ)
103	97, 48	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝑆) → (𝐵‘𝑍) ∈ ℝ)
104	102, 103	ltnled 10184	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝑆) → (𝑆 < (𝐵‘𝑍) ↔ ¬ (𝐵‘𝑍) ≤ 𝑆))
105	98, 104	mpbird 247	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝑆) → 𝑆 < (𝐵‘𝑍))
106	3	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) → 𝑋 ∈ Fin)
107	5	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) → 𝑌 ⊆ 𝑋)
108	6	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) → 𝑍 ∈ (𝑋 ∖ 𝑌))
109	14	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) → 𝐴:𝑊⟶ℝ)
110	15	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) → 𝐵:𝑊⟶ℝ)
111	87	adantlr 751	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) ∧ 𝑘 ∈ 𝑊) → (𝐴‘𝑘) < (𝐵‘𝑘))
112		eqid 2622	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑌 ↦ 0) = (𝑦 ∈ 𝑌 ↦ 0)
113	27	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) → 𝐶:ℕ⟶(ℝ ↑_𝑚 𝑊))
114		fveq2 6191	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑗 → (𝐶‘𝑖) = (𝐶‘𝑗))
115	114	fveq1d 6193	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)‘𝑍) = ((𝐶‘𝑗)‘𝑍))
116		fveq2 6191	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗))
117	116	fveq1d 6193	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑗 → ((𝐷‘𝑖)‘𝑍) = ((𝐷‘𝑗)‘𝑍))
118	115, 117	oveq12d 6668	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)) = (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
119	118	eleq2d 2687	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))))
120	114	reseq1d 5395	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖) ↾ 𝑌) = ((𝐶‘𝑗) ↾ 𝑌))
121	119, 120	ifbieq1d 4109	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))
122	121	cbvmptv 4750	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0))) = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))
123	50	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) → 𝐷:ℕ⟶(ℝ ↑_𝑚 𝑊))
124	116	reseq1d 5395	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → ((𝐷‘𝑖) ↾ 𝑌) = ((𝐷‘𝑗) ↾ 𝑌))
125	119, 124	ifbieq1d 4109	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))
126	125	cbvmptv 4750	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0))) = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))
127	61	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
128	19	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) → 𝐸 ∈ ℝ⁺)
129	90	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) → 𝑆 ∈ 𝑈)
130		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) → 𝑆 < (𝐵‘𝑍))
131		biid 251	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
132		eqidd 2623	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑦 → 0 = 0)
133	132	cbvmptv 4750	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ 𝑌 ↦ 0) = (𝑦 ∈ 𝑌 ↦ 0)
134	131, 133	ifbieq2i 4110	. . . . . . . . . . . . . . . 16 ⊢ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑤 ∈ 𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0))
135	134	mpteq2i 4741	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑤 ∈ 𝑌 ↦ 0))) = (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))
136	135	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑗 → (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑤 ∈ 𝑌 ↦ 0))) = (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0))))
137		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑗 → 𝑙 = 𝑗)
138	136, 137	fveq12d 6197	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑗 → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑤 ∈ 𝑌 ↦ 0)))‘𝑙) = ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑗))
139	131, 133	ifbieq2i 4110	. . . . . . . . . . . . . . . 16 ⊢ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑤 ∈ 𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0))
140	139	mpteq2i 4741	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑤 ∈ 𝑌 ↦ 0))) = (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))
141	140	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑗 → (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑤 ∈ 𝑌 ↦ 0))) = (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0))))
142	141, 137	fveq12d 6197	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑗 → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑤 ∈ 𝑌 ↦ 0)))‘𝑙) = ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑗))
143	138, 142	oveq12d 6668	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑤 ∈ 𝑌 ↦ 0)))‘𝑙)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑤 ∈ 𝑌 ↦ 0)))‘𝑙)) = (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑗)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑗)))
144	143	cbvmptv 4750	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑤 ∈ 𝑌 ↦ 0)))‘𝑙)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑤 ∈ 𝑌 ↦ 0)))‘𝑙))) = (𝑗 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑗)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑗)))
145		hoidmvlelem4.i	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑒 ∈ (ℝ ↑_𝑚 𝑌)∀𝑓 ∈ (ℝ ↑_𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑_𝑚 𝑌) ↑_𝑚 ℕ)∀ℎ ∈ ((ℝ ↑_𝑚 𝑌) ↑_𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
146	145	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) → ∀𝑒 ∈ (ℝ ↑_𝑚 𝑌)∀𝑓 ∈ (ℝ ↑_𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑_𝑚 𝑌) ↑_𝑚 ℕ)∀ℎ ∈ ((ℝ ↑_𝑚 𝑌) ↑_𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
147		hoidmvlelem4.i2	. . . . . . . . . . . 12 ⊢ (𝜑 → X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
148	147	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) → X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
149		eqid 2622	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ X𝑦 ∈ 𝑌 ((𝐴‘𝑦)[,)(𝐵‘𝑦)) ↦ (𝑦 ∈ 𝑊 ↦ if(𝑦 ∈ 𝑌, (𝑥‘𝑦), 𝑆))) = (𝑥 ∈ X𝑦 ∈ 𝑌 ((𝐴‘𝑦)[,)(𝐵‘𝑦)) ↦ (𝑦 ∈ 𝑊 ↦ if(𝑦 ∈ 𝑌, (𝑥‘𝑦), 𝑆)))
150		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑘 → (𝐴‘𝑦) = (𝐴‘𝑘))
151		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑘 → (𝐵‘𝑦) = (𝐵‘𝑘))
152	150, 151	oveq12d 6668	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑘 → ((𝐴‘𝑦)[,)(𝐵‘𝑦)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
153	152	cbvixpv 7926	. . . . . . . . . . . . 13 ⊢ X𝑦 ∈ 𝑌 ((𝐴‘𝑦)[,)(𝐵‘𝑦)) = X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
154		eleq1 2689	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑘 → (𝑦 ∈ 𝑌 ↔ 𝑘 ∈ 𝑌))
155		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑘 → (𝑥‘𝑦) = (𝑥‘𝑘))
156	154, 155	ifbieq1d 4109	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑘 → if(𝑦 ∈ 𝑌, (𝑥‘𝑦), 𝑆) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
157	156	cbvmptv 4750	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑊 ↦ if(𝑦 ∈ 𝑌, (𝑥‘𝑦), 𝑆)) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
158	153, 157	mpteq12i 4742	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ X𝑦 ∈ 𝑌 ((𝐴‘𝑦)[,)(𝐵‘𝑦)) ↦ (𝑦 ∈ 𝑊 ↦ if(𝑦 ∈ 𝑌, (𝑥‘𝑦), 𝑆))) = (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↦ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))
159	149, 158	eqtri 2644	. . . . . . . . . . 11 ⊢ (𝑥 ∈ X𝑦 ∈ 𝑌 ((𝐴‘𝑦)[,)(𝐵‘𝑦)) ↦ (𝑦 ∈ 𝑊 ↦ if(𝑦 ∈ 𝑌, (𝑥‘𝑦), 𝑆))) = (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↦ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))
160	2, 106, 107, 108, 4, 109, 110, 111, 112, 113, 122, 123, 126, 127, 31, 77, 128, 78, 129, 130, 144, 146, 148, 159	hoidmvlelem3 40811	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
161	97, 105, 160	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝑆) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
162	94	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
163	162, 100	sstrd 3613	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑈 ⊆ ℝ)
164	163	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ⊆ ℝ)
165		ne0i 3921	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ 𝑈 → 𝑈 ≠ ∅)
166	165	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ≠ ∅)
167	99	rexrd 10089	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ^*)
168	167	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴‘𝑍) ∈ ℝ^*)
169	91	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐵‘𝑍) ∈ ℝ^*)
170	162	sselda 3603	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
171		iccleub 12229	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴‘𝑍) ∈ ℝ^* ∧ (𝐵‘𝑍) ∈ ℝ^* ∧ 𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → 𝑢 ≤ (𝐵‘𝑍))
172	168, 169, 170, 171	syl3anc 1326	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ (𝐵‘𝑍))
173	172	ralrimiva 2966	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑢 ∈ 𝑈 𝑢 ≤ (𝐵‘𝑍))
174		breq2 4657	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝐵‘𝑍) → (𝑢 ≤ 𝑦 ↔ 𝑢 ≤ (𝐵‘𝑍)))
175	174	ralbidv 2986	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝐵‘𝑍) → (∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑦 ↔ ∀𝑢 ∈ 𝑈 𝑢 ≤ (𝐵‘𝑍)))
176	175	rspcev 3309	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵‘𝑍) ∈ ℝ ∧ ∀𝑢 ∈ 𝑈 𝑢 ≤ (𝐵‘𝑍)) → ∃𝑦 ∈ ℝ ∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑦)
177	48, 173, 176	syl2anc 693	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑦)
178	177	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ∃𝑦 ∈ ℝ ∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑦)
179		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ 𝑈)
180		suprub 10984	. . . . . . . . . . . . . . . 16 ⊢ (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑦) ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
181	164, 166, 178, 179, 180	syl31anc 1329	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
182	181, 79	syl6breqr 4695	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ 𝑆)
183	182	ralrimiva 2966	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑆)
184	164, 179	sseldd 3604	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ ℝ)
185	101	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑆 ∈ ℝ)
186	184, 185	lenltd 10183	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 ≤ 𝑆 ↔ ¬ 𝑆 < 𝑢))
187	186	ralbidva 2985	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑆 ↔ ∀𝑢 ∈ 𝑈 ¬ 𝑆 < 𝑢))
188	183, 187	mpbid 222	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑢 ∈ 𝑈 ¬ 𝑆 < 𝑢)
189		ralnex 2992	. . . . . . . . . . . 12 ⊢ (∀𝑢 ∈ 𝑈 ¬ 𝑆 < 𝑢 ↔ ¬ ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
190	188, 189	sylib 208	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
191	190	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) → ¬ ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
192	97, 105, 191	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝑆) → ¬ ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
193	161, 192	condan 835	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘𝑍) ≤ 𝑆)
194		iccleub 12229	. . . . . . . . 9 ⊢ (((𝐴‘𝑍) ∈ ℝ^* ∧ (𝐵‘𝑍) ∈ ℝ^* ∧ 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → 𝑆 ≤ (𝐵‘𝑍))
195	167, 91, 95, 194	syl3anc 1326	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ≤ (𝐵‘𝑍))
196	91, 96, 193, 195	xrletrid 11986	. . . . . . 7 ⊢ (𝜑 → (𝐵‘𝑍) = 𝑆)
197	78	eqcomi 2631	. . . . . . . 8 ⊢ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} = 𝑈
198	197	a1i 11	. . . . . . 7 ⊢ (𝜑 → {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} = 𝑈)
199	196, 198	eleq12d 2695	. . . . . 6 ⊢ (𝜑 → ((𝐵‘𝑍) ∈ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} ↔ 𝑆 ∈ 𝑈))
200	90, 199	mpbird 247	. . . . 5 ⊢ (𝜑 → (𝐵‘𝑍) ∈ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))})
201		oveq1 6657	. . . . . . . 8 ⊢ (𝑧 = (𝐵‘𝑍) → (𝑧 − (𝐴‘𝑍)) = ((𝐵‘𝑍) − (𝐴‘𝑍)))
202	201	oveq2d 6666	. . . . . . 7 ⊢ (𝑧 = (𝐵‘𝑍) → (𝐺 · (𝑧 − (𝐴‘𝑍))) = (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
203		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐵‘𝑍) → (𝐻‘𝑧) = (𝐻‘(𝐵‘𝑍)))
204	203	fveq1d 6193	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐵‘𝑍) → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗)))
205	204	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑧 = (𝐵‘𝑍) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))
206	205	mpteq2dv 4745	. . . . . . . . 9 ⊢ (𝑧 = (𝐵‘𝑍) → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗)))))
207	206	fveq2d 6195	. . . . . . . 8 ⊢ (𝑧 = (𝐵‘𝑍) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))))
208	207	oveq2d 6666	. . . . . . 7 ⊢ (𝑧 = (𝐵‘𝑍) → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) = ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗)))))))
209	202, 208	breq12d 4666	. . . . . 6 ⊢ (𝑧 = (𝐵‘𝑍) → ((𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) ↔ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))))))
210	209	elrab 3363	. . . . 5 ⊢ ((𝐵‘𝑍) ∈ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} ↔ ((𝐵‘𝑍) ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∧ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))))))
211	200, 210	sylib 208	. . . 4 ⊢ (𝜑 → ((𝐵‘𝑍) ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∧ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))))))
212	211	simprd 479	. . 3 ⊢ (𝜑 → (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗)))))))
213	3, 5	ssfid 8183	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ Fin)
214		eqid 2622	. . . . . 6 ⊢ ∏𝑘 ∈ 𝑌 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑌 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))
215	2, 213, 6, 65, 4, 14, 15, 214	hoiprodp1 40802	. . . . 5 ⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) = (∏𝑘 ∈ 𝑌 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) · (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍)))))
216		eqidd 2623	. . . . . . 7 ⊢ (𝜑 → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) = ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)))
217	14	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐴:𝑊⟶ℝ)
218		ssun1 3776	. . . . . . . . . . . 12 ⊢ 𝑌 ⊆ (𝑌 ∪ {𝑍})
219	4	eqcomi 2631	. . . . . . . . . . . 12 ⊢ (𝑌 ∪ {𝑍}) = 𝑊
220	218, 219	sseqtri 3637	. . . . . . . . . . 11 ⊢ 𝑌 ⊆ 𝑊
221		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑌)
222	220, 221	sseldi 3601	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑊)
223	217, 222	ffvelrnd 6360	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐴‘𝑘) ∈ ℝ)
224	15	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐵:𝑊⟶ℝ)
225	224, 222	ffvelrnd 6360	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐵‘𝑘) ∈ ℝ)
226	222, 87	syldan 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐴‘𝑘) < (𝐵‘𝑘))
227	223, 225, 226	volicon0 40789	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
228	227	prodeq2dv 14653	. . . . . . 7 ⊢ (𝜑 → ∏𝑘 ∈ 𝑌 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)))
229	77	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)))
230		hoidmvlelem4.n	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ≠ ∅)
231	220	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ⊆ 𝑊)
232	14, 231	fssresd 6071	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ↾ 𝑌):𝑌⟶ℝ)
233	15, 231	fssresd 6071	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ↾ 𝑌):𝑌⟶ℝ)
234	2, 213, 230, 232, 233	hoidmvn0val 40798	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))))
235		fvres 6207	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑌 → ((𝐴 ↾ 𝑌)‘𝑘) = (𝐴‘𝑘))
236		fvres 6207	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑌 → ((𝐵 ↾ 𝑌)‘𝑘) = (𝐵‘𝑘))
237	235, 236	oveq12d 6668	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑌 → (((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
238	237	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑌 → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
239	238	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
240		volico 40200	. . . . . . . . . . 11 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0))
241	223, 225, 240	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0))
242	241, 227	eqtr3d 2658	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
243	239, 241, 242	3eqtrd 2660	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
244	243	prodeq2dv 14653	. . . . . . . 8 ⊢ (𝜑 → ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)))
245	229, 234, 244	3eqtrd 2660	. . . . . . 7 ⊢ (𝜑 → 𝐺 = ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)))
246	216, 228, 245	3eqtr4d 2666	. . . . . 6 ⊢ (𝜑 → ∏𝑘 ∈ 𝑌 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = 𝐺)
247	99, 48, 89	volicon0 40789	. . . . . 6 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))) = ((𝐵‘𝑍) − (𝐴‘𝑍)))
248	246, 247	oveq12d 6668	. . . . 5 ⊢ (𝜑 → (∏𝑘 ∈ 𝑌 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) · (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍)))) = (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
249	215, 248	eqtrd 2656	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) = (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
250	249	breq1d 4663	. . 3 ⊢ (𝜑 → ((𝐴(𝐿‘𝑊)𝐵) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗)))))) ↔ (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))))))
251	212, 250	mpbird 247	. 2 ⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗)))))))
252		0le1 10551	. . . . 5 ⊢ 0 ≤ 1
253	252	a1i 11	. . . 4 ⊢ (𝜑 → 0 ≤ 1)
254	19	rpge0d 11876	. . . 4 ⊢ (𝜑 → 0 ≤ 𝐸)
255	18, 20, 253, 254	addge0d 10603	. . 3 ⊢ (𝜑 → 0 ≤ (1 + 𝐸))
256	74, 61, 21, 255, 69	lemul2ad 10964	. 2 ⊢ (𝜑 → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗)))))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))))
257	17, 75, 76, 251, 256	letrd 10194	1 ⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 ⊆ wss 3574 ∅c0 3915 ifcif 4086 {csn 4177 ∪ ciun 4520 class class class wbr 4653 ↦ cmpt 4729 ↾ cres 5116 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ↑_𝑚 cmap 7857 Xcixp 7908 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-ixp 7909 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: hoidmvlelem5 40813

W3C validator