hoidmvlelem4 - Mathbox for Glauco Siliprandi

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Theorem hoidmvlelem4 45304

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: 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 13432	. . 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 3960	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ 𝑋)
8		snssi 4811	. . . . . . . 8 ⊢ (𝑍 ∈ 𝑋 → {𝑍} ⊆ 𝑋)
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → {𝑍} ⊆ 𝑋)
10	5, 9	unssd 4186	. . . . . 6 ⊢ (𝜑 → (𝑌 ∪ {𝑍}) ⊆ 𝑋)
11	4, 10	eqsstrid 4030	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑋)
12		ssfi 9172	. . . . 5 ⊢ ((𝑋 ∈ Fin ∧ 𝑊 ⊆ 𝑋) → 𝑊 ∈ Fin)
13	3, 11, 12	syl2anc 584	. . . 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 45288	. . 3 ⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) ∈ (0[,)+∞))
17	1, 16	sselid 3980	. 2 ⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) ∈ ℝ)
18		1red 11214	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
19		hoidmvlelem4.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
20	19	rpred 13015	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ)
21	18, 20	readdcld 11242	. . 3 ⊢ (𝜑 → (1 + 𝐸) ∈ ℝ)
22		nfv 1917	. . . . 5 ⊢ Ⅎ𝑗𝜑
23		nnex 12217	. . . . . 6 ⊢ ℕ ∈ V
24	23	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ∈ V)
25		icossicc 13412	. . . . . 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	ffvelcdmda 7086	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑_m 𝑊))
29		elmapi 8842	. . . . . . . 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 2821	. . . . . . . . . . . . 13 ⊢ (𝑗 = ℎ → (𝑗 ∈ 𝑌 ↔ ℎ ∈ 𝑌))
33		fveq2 6891	. . . . . . . . . . . . 13 ⊢ (𝑗 = ℎ → (𝑐‘𝑗) = (𝑐‘ℎ))
34	33	breq1d 5158	. . . . . . . . . . . . . 14 ⊢ (𝑗 = ℎ → ((𝑐‘𝑗) ≤ 𝑥 ↔ (𝑐‘ℎ) ≤ 𝑥))
35	34, 33	ifbieq1d 4552	. . . . . . . . . . . . 13 ⊢ (𝑗 = ℎ → if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥) = if((𝑐‘ℎ) ≤ 𝑥, (𝑐‘ℎ), 𝑥))
36	32, 33, 35	ifbieq12d 4556	. . . . . . . . . . . 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 5261	. . . . . . . . . . 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 5253	. . . . . . . . . 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 5253	. . . . . . . . 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 2760	. . . . . . . 8 ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑊) ↦ (ℎ ∈ 𝑊 ↦ if(ℎ ∈ 𝑌, (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑥, (𝑐‘ℎ), 𝑥)))))
41		snidg 4662	. . . . . . . . . . . . 13 ⊢ (𝑍 ∈ (𝑋 ∖ 𝑌) → 𝑍 ∈ {𝑍})
42	6, 41	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ∈ {𝑍})
43		elun2 4177	. . . . . . . . . . . 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 2738	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 ∪ {𝑍}) = 𝑊)
47	44, 46	eleqtrd 2835	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
48	15, 47	ffvelcdmd 7087	. . . . . . . . 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	ffvelcdmda 7086	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑_m 𝑊))
52		elmapi 8842	. . . . . . . . 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 45282	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗)):𝑊⟶ℝ)
55	2, 26, 30, 54	hoidmvcl 45288	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))) ∈ (0[,)+∞))
56	25, 55	sselid 3980	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))) ∈ (0[,]+∞))
57	22, 24, 56	sge0clmpt 45131	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))) ∈ (0[,]+∞))
58	22, 24, 56	sge0xrclmpt 45134	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))) ∈ ℝ^*)
59		pnfxr 11267	. . . . . 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 11263	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ^*)
63	2, 26, 30, 53	hoidmvcl 45288	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) ∈ (0[,)+∞))
64	25, 63	sselid 3980	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) ∈ (0[,]+∞))
65	6	eldifbd 3961	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
66	47, 65	eldifd 3959	. . . . . . . . 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 45292	. . . . . . 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 45116	. . . . . 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 13100	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) < +∞)
71	58, 62, 60, 69, 70	xrlelttrd 13138	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))) < +∞)
72	58, 60, 71	xrltned 44057	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))) ≠ +∞)
73		ge0xrre 44234	. . . 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 584	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))) ∈ ℝ)
75	21, 74	remulcld 11243	. 2 ⊢ (𝜑 → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗)))))) ∈ ℝ)
76	21, 61	remulcld 11243	. 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 549	. . . . . . . 8 ⊢ (𝜑 → (𝜑 ∧ 𝑍 ∈ 𝑊))
81		eleq1 2821	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑍 → (𝑘 ∈ 𝑊 ↔ 𝑍 ∈ 𝑊))
82	81	anbi2d 629	. . . . . . . . . 10 ⊢ (𝑘 = 𝑍 → ((𝜑 ∧ 𝑘 ∈ 𝑊) ↔ (𝜑 ∧ 𝑍 ∈ 𝑊)))
83		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑍 → (𝐴‘𝑘) = (𝐴‘𝑍))
84		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑍 → (𝐵‘𝑘) = (𝐵‘𝑍))
85	83, 84	breq12d 5161	. . . . . . . . . 10 ⊢ (𝑘 = 𝑍 → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ (𝐴‘𝑍) < (𝐵‘𝑍)))
86	82, 85	imbi12d 344	. . . . . . . . 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 3556	. . . . . . . 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 45301	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝑈)
91	48	rexrd 11263	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ^*)
92		iccssxr 13406	. . . . . . . . 9 ⊢ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ⊆ ℝ^*
93		ssrab2 4077	. . . . . . . . . . 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 4016	. . . . . . . . . 10 ⊢ 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍))
95	94, 90	sselid 3980	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
96	92, 95	sselid 3980	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℝ^*)
97		simpl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝑆) → 𝜑)
98		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝑆) → ¬ (𝐵‘𝑍) ≤ 𝑆)
99	14, 47	ffvelcdmd 7087	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ)
100	99, 48	iccssred 13410	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐴‘𝑍)[,](𝐵‘𝑍)) ⊆ ℝ)
101	100, 95	sseldd 3983	. . . . . . . . . . . . 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 11360	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝑆) → (𝑆 < (𝐵‘𝑍) ↔ ¬ (𝐵‘𝑍) ≤ 𝑆))
105	98, 104	mpbird 256	. . . . . . . . . 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 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) ∧ 𝑘 ∈ 𝑊) → (𝐴‘𝑘) < (𝐵‘𝑘))
112		eqid 2732	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑌 ↦ 0) = (𝑦 ∈ 𝑌 ↦ 0)
113	27	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) → 𝐶:ℕ⟶(ℝ ↑_m 𝑊))
114		fveq2 6891	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑗 → (𝐶‘𝑖) = (𝐶‘𝑗))
115	114	fveq1d 6893	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)‘𝑍) = ((𝐶‘𝑗)‘𝑍))
116		fveq2 6891	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗))
117	116	fveq1d 6893	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑗 → ((𝐷‘𝑖)‘𝑍) = ((𝐷‘𝑗)‘𝑍))
118	115, 117	oveq12d 7426	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)) = (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
119	118	eleq2d 2819	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))))
120	114	reseq1d 5980	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖) ↾ 𝑌) = ((𝐶‘𝑗) ↾ 𝑌))
121	119, 120	ifbieq1d 4552	. . . . . . . . . . . 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 5261	. . . . . . . . . . 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 5980	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → ((𝐷‘𝑖) ↾ 𝑌) = ((𝐷‘𝑗) ↾ 𝑌))
125	119, 124	ifbieq1d 4552	. . . . . . . . . . . 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 5261	. . . . . . . . . . 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 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) → 𝑆 < (𝐵‘𝑍))
131		biid 260	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
132		eqidd 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑦 → 0 = 0)
133	132	cbvmptv 5261	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ 𝑌 ↦ 0) = (𝑦 ∈ 𝑌 ↦ 0)
134	131, 133	ifbieq2i 4553	. . . . . . . . . . . . . . . 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 5253	. . . . . . . . . . . . . . 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 6898	. . . . . . . . . . . . 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 4553	. . . . . . . . . . . . . . . 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 5253	. . . . . . . . . . . . . . 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 6898	. . . . . . . . . . . . 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 7426	. . . . . . . . . . . 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 5261	. . . . . . . . . . 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 2732	. . . . . . . . . . . 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 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑘 → (𝐴‘𝑦) = (𝐴‘𝑘))
151		fveq2 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑘 → (𝐵‘𝑦) = (𝐵‘𝑘))
152	150, 151	oveq12d 7426	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑘 → ((𝐴‘𝑦)[,)(𝐵‘𝑦)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
153	152	cbvixpv 8908	. . . . . . . . . . . . 13 ⊢ X𝑦 ∈ 𝑌 ((𝐴‘𝑦)[,)(𝐵‘𝑦)) = X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
154		eleq1 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑘 → (𝑦 ∈ 𝑌 ↔ 𝑘 ∈ 𝑌))
155		fveq2 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑘 → (𝑥‘𝑦) = (𝑥‘𝑘))
156	154, 155	ifbieq1d 4552	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑘 → if(𝑦 ∈ 𝑌, (𝑥‘𝑦), 𝑆) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
157	156	cbvmptv 5261	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑊 ↦ if(𝑦 ∈ 𝑌, (𝑥‘𝑦), 𝑆)) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
158	153, 157	mpteq12i 5254	. . . . . . . . . . . 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 2760	. . . . . . . . . . 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 45303	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
161	97, 105, 160	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝑆) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
162	94	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
163	162, 100	sstrd 3992	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑈 ⊆ ℝ)
164	163	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ⊆ ℝ)
165		ne0i 4334	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ 𝑈 → 𝑈 ≠ ∅)
166	165	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ≠ ∅)
167	99	rexrd 11263	. . . . . . . . . . . . . . . . . . . . 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 3982	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
171		iccleub 13378	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴‘𝑍) ∈ ℝ^* ∧ (𝐵‘𝑍) ∈ ℝ^* ∧ 𝑢 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → 𝑢 ≤ (𝐵‘𝑍))
172	168, 169, 170, 171	syl3anc 1371	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ (𝐵‘𝑍))
173	172	ralrimiva 3146	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑢 ∈ 𝑈 𝑢 ≤ (𝐵‘𝑍))
174		brralrspcev 5208	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵‘𝑍) ∈ ℝ ∧ ∀𝑢 ∈ 𝑈 𝑢 ≤ (𝐵‘𝑍)) → ∃𝑦 ∈ ℝ ∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑦)
175	48, 173, 174	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑦)
176	175	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ∃𝑦 ∈ ℝ ∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑦)
177		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ 𝑈)
178		suprub 12174	. . . . . . . . . . . . . . . 16 ⊢ (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑦) ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
179	164, 166, 176, 177, 178	syl31anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
180	179, 79	breqtrrdi 5190	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ 𝑆)
181	180	ralrimiva 3146	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑆)
182	164, 177	sseldd 3983	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ ℝ)
183	101	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑆 ∈ ℝ)
184	182, 183	lenltd 11359	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 ≤ 𝑆 ↔ ¬ 𝑆 < 𝑢))
185	184	ralbidva 3175	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑆 ↔ ∀𝑢 ∈ 𝑈 ¬ 𝑆 < 𝑢))
186	181, 185	mpbid 231	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑢 ∈ 𝑈 ¬ 𝑆 < 𝑢)
187		ralnex 3072	. . . . . . . . . . . 12 ⊢ (∀𝑢 ∈ 𝑈 ¬ 𝑆 < 𝑢 ↔ ¬ ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
188	186, 187	sylib 217	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
189	188	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑆 < (𝐵‘𝑍)) → ¬ ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
190	97, 105, 189	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝐵‘𝑍) ≤ 𝑆) → ¬ ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
191	161, 190	condan 816	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘𝑍) ≤ 𝑆)
192		iccleub 13378	. . . . . . . . 9 ⊢ (((𝐴‘𝑍) ∈ ℝ^* ∧ (𝐵‘𝑍) ∈ ℝ^* ∧ 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → 𝑆 ≤ (𝐵‘𝑍))
193	167, 91, 95, 192	syl3anc 1371	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ≤ (𝐵‘𝑍))
194	91, 96, 191, 193	xrletrid 13133	. . . . . . 7 ⊢ (𝜑 → (𝐵‘𝑍) = 𝑆)
195	78	eqcomi 2741	. . . . . . . 8 ⊢ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} = 𝑈
196	195	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)
197	194, 196	eleq12d 2827	. . . . . 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))
198	90, 197	mpbird 256	. . . . 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))))))})
199		oveq1 7415	. . . . . . . 8 ⊢ (𝑧 = (𝐵‘𝑍) → (𝑧 − (𝐴‘𝑍)) = ((𝐵‘𝑍) − (𝐴‘𝑍)))
200	199	oveq2d 7424	. . . . . . 7 ⊢ (𝑧 = (𝐵‘𝑍) → (𝐺 · (𝑧 − (𝐴‘𝑍))) = (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
201		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐵‘𝑍) → (𝐻‘𝑧) = (𝐻‘(𝐵‘𝑍)))
202	201	fveq1d 6893	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐵‘𝑍) → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗)))
203	202	oveq2d 7424	. . . . . . . . . 10 ⊢ (𝑧 = (𝐵‘𝑍) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗))))
204	203	mpteq2dv 5250	. . . . . . . . 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)))))
205	204	fveq2d 6895	. . . . . . . 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))))))
206	205	oveq2d 7424	. . . . . . 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)))))))
207	200, 206	breq12d 5161	. . . . . 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))))))))
208	207	elrab 3683	. . . . 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))))))))
209	198, 208	sylib 217	. . . 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))))))))
210	209	simprd 496	. . 3 ⊢ (𝜑 → (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗)))))))
211	3, 5	ssfid 9266	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ Fin)
212		eqid 2732	. . . . . 6 ⊢ ∏𝑘 ∈ 𝑌 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑌 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))
213	2, 211, 6, 65, 4, 14, 15, 212	hoiprodp1 45294	. . . . 5 ⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) = (∏𝑘 ∈ 𝑌 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) · (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍)))))
214		eqidd 2733	. . . . . . 7 ⊢ (𝜑 → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) = ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)))
215	14	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐴:𝑊⟶ℝ)
216		ssun1 4172	. . . . . . . . . . . 12 ⊢ 𝑌 ⊆ (𝑌 ∪ {𝑍})
217	4	eqcomi 2741	. . . . . . . . . . . 12 ⊢ (𝑌 ∪ {𝑍}) = 𝑊
218	216, 217	sseqtri 4018	. . . . . . . . . . 11 ⊢ 𝑌 ⊆ 𝑊
219		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑌)
220	218, 219	sselid 3980	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑊)
221	215, 220	ffvelcdmd 7087	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐴‘𝑘) ∈ ℝ)
222	15	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐵:𝑊⟶ℝ)
223	222, 220	ffvelcdmd 7087	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐵‘𝑘) ∈ ℝ)
224	220, 87	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐴‘𝑘) < (𝐵‘𝑘))
225	221, 223, 224	volicon0 45281	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
226	225	prodeq2dv 15866	. . . . . . 7 ⊢ (𝜑 → ∏𝑘 ∈ 𝑌 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)))
227	77	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)))
228		hoidmvlelem4.n	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ≠ ∅)
229	218	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ⊆ 𝑊)
230	14, 229	fssresd 6758	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ↾ 𝑌):𝑌⟶ℝ)
231	15, 229	fssresd 6758	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ↾ 𝑌):𝑌⟶ℝ)
232	2, 211, 228, 230, 231	hoidmvn0val 45290	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))))
233		fvres 6910	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑌 → ((𝐴 ↾ 𝑌)‘𝑘) = (𝐴‘𝑘))
234		fvres 6910	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑌 → ((𝐵 ↾ 𝑌)‘𝑘) = (𝐵‘𝑘))
235	233, 234	oveq12d 7426	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑌 → (((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
236	235	fveq2d 6895	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑌 → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
237	236	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
238		volico 44689	. . . . . . . . . . 11 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0))
239	221, 223, 238	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0))
240	239, 225	eqtr3d 2774	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
241	237, 239, 240	3eqtrd 2776	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
242	241	prodeq2dv 15866	. . . . . . . 8 ⊢ (𝜑 → ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)))
243	227, 232, 242	3eqtrd 2776	. . . . . . 7 ⊢ (𝜑 → 𝐺 = ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)))
244	214, 226, 243	3eqtr4d 2782	. . . . . 6 ⊢ (𝜑 → ∏𝑘 ∈ 𝑌 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = 𝐺)
245	99, 48, 89	volicon0 45281	. . . . . 6 ⊢ (𝜑 → (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍))) = ((𝐵‘𝑍) − (𝐴‘𝑍)))
246	244, 245	oveq12d 7426	. . . . 5 ⊢ (𝜑 → (∏𝑘 ∈ 𝑌 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) · (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍)))) = (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
247	213, 246	eqtrd 2772	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) = (𝐺 · ((𝐵‘𝑍) − (𝐴‘𝑍))))
248	247	breq1d 5158	. . 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))))))))
249	210, 248	mpbird 256	. 2 ⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘(𝐵‘𝑍))‘(𝐷‘𝑗)))))))
250		0le1 11736	. . . . 5 ⊢ 0 ≤ 1
251	250	a1i 11	. . . 4 ⊢ (𝜑 → 0 ≤ 1)
252	19	rpge0d 13019	. . . 4 ⊢ (𝜑 → 0 ≤ 𝐸)
253	18, 20, 251, 252	addge0d 11789	. . 3 ⊢ (𝜑 → 0 ≤ (1 + 𝐸))
254	74, 61, 21, 253, 69	lemul2ad 12153	. 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))))))
255	17, 75, 76, 249, 254	letrd 11370	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 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3432 Vcvv 3474 ∖ cdif 3945 ∪ cun 3946 ⊆ wss 3948 ∅c0 4322 ifcif 4528 {csn 4628 ∪ ciun 4997 class class class wbr 5148 ↦ cmpt 5231 ↾ cres 5678 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 ∈ cmpo 7410 ↑_m cmap 8819 Xcixp 8890 Fincfn 8938 supcsup 9434 ℝcr 11108 0cc0 11109 1c1 11110 + caddc 11112 · cmul 11114 +∞cpnf 11244 ℝ^*cxr 11246 < clt 11247 ≤ cle 11248 − cmin 11443 ℕcn 12211 ℝ⁺crp 12973 [,)cico 13325 [,]cicc 13326 ∏cprod 15848 volcvol 24979 Σ^{^}csumge0 45068

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-ioo 13327 df-ico 13329 df-icc 13330 df-fz 13484 df-fzo 13627 df-fl 13756 df-seq 13966 df-exp 14027 df-hash 14290 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-clim 15431 df-rlim 15432 df-sum 15632 df-prod 15849 df-rest 17367 df-topgen 17388 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-top 22395 df-topon 22412 df-bases 22448 df-cmp 22890 df-ovol 24980 df-vol 24981 df-sumge0 45069

This theorem is referenced by: hoidmvlelem5 45305

W3C validator