vonn0icc2 - Mathbox for Glauco Siliprandi

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Theorem vonn0icc2 46688

Description: The n-dimensional Lebesgue measure of a closed interval, when the dimension of the space is nonzero. This is the second statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

**Hypotheses**
Ref	Expression
vonn0icc2.k	⊢ Ⅎ𝑘𝜑
vonn0icc2.x	⊢ (𝜑 → 𝑋 ∈ Fin)
vonn0icc2.n	⊢ (𝜑 → 𝑋 ≠ ∅)
vonn0icc2.a	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)
vonn0icc2.b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ)
vonn0icc2.i	⊢ 𝐼 = X𝑘 ∈ 𝑋 (𝐴[,]𝐵)

**Assertion**
Ref	Expression
vonn0icc2	⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 (vol‘(𝐴[,]𝐵)))

Distinct variable group: 𝑘,𝑋 Allowed substitution hints: 𝜑(𝑘) 𝐴(𝑘) 𝐵(𝑘) 𝐼(𝑘)

Proof of Theorem vonn0icc2 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vonn0icc2.i	. . . . 5 ⊢ 𝐼 = X𝑘 ∈ 𝑋 (𝐴[,]𝐵)
2	1	a1i 11	. . . 4 ⊢ (𝜑 → 𝐼 = X𝑘 ∈ 𝑋 (𝐴[,]𝐵))
3		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝑗 ∈ 𝑋)
4		vonn0icc2.k	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝜑
5		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑋
6	4, 5	nfan 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑋)
7		nfcsb1v 3903	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴
8		nfcv 2899	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℝ
9	7, 8	nfel 2914	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ
10	6, 9	nfim 1896	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ)
11		eleq1w 2818	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑋 ↔ 𝑗 ∈ 𝑋))
12	11	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑋) ↔ (𝜑 ∧ 𝑗 ∈ 𝑋)))
13		csbeq1a 3893	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → 𝐴 = ⦋𝑗 / 𝑘⦌𝐴)
14	13	eleq1d 2820	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐴 ∈ ℝ ↔ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ))
15	12, 14	imbi12d 344	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ)))
16		vonn0icc2.a	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)
17	10, 15, 16	chvarfv 2241	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ)
18		eqid 2736	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝑋 ↦ 𝐴) = (𝑘 ∈ 𝑋 ↦ 𝐴)
19	18	fvmpts 6994	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝑋 ∧ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ) → ((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴)
20	3, 17, 19	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴)
21		nfcsb1v 3903	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵
22	21, 8	nfel 2914	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ
23	6, 22	nfim 1896	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ)
24		csbeq1a 3893	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵)
25	24	eleq1d 2820	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐵 ∈ ℝ ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ))
26	12, 25	imbi12d 344	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ)))
27		vonn0icc2.b	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ)
28	23, 26, 27	chvarfv 2241	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ)
29		eqid 2736	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝑋 ↦ 𝐵) = (𝑘 ∈ 𝑋 ↦ 𝐵)
30	29	fvmpts 6994	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝑋 ∧ ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ) → ((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵)
31	3, 28, 30	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵)
32	20, 31	oveq12d 7428	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = (⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵))
33	32	ixpeq2dva 8931	. . . . 5 ⊢ (𝜑 → X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = X𝑗 ∈ 𝑋 (⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵))
34		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑘[,]
35	7, 34, 21	nfov 7440	. . . . . . 7 ⊢ Ⅎ𝑘(⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵)
36		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑗(𝐴[,]𝐵)
37	13	equcoms 2020	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → 𝐴 = ⦋𝑗 / 𝑘⦌𝐴)
38	37	eqcomd 2742	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐴 = 𝐴)
39		eqidd 2737	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → 𝐴 = 𝐴)
40	38, 39	eqtrd 2771	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐴 = 𝐴)
41	24	equcoms 2020	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵)
42	41	eqcomd 2742	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐵 = 𝐵)
43	40, 42	oveq12d 7428	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵) = (𝐴[,]𝐵))
44	35, 36, 43	cbvixp 8933	. . . . . 6 ⊢ X𝑗 ∈ 𝑋 (⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵) = X𝑘 ∈ 𝑋 (𝐴[,]𝐵)
45	44	a1i 11	. . . . 5 ⊢ (𝜑 → X𝑗 ∈ 𝑋 (⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵) = X𝑘 ∈ 𝑋 (𝐴[,]𝐵))
46	33, 45	eqtrd 2771	. . . 4 ⊢ (𝜑 → X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = X𝑘 ∈ 𝑋 (𝐴[,]𝐵))
47	2, 46	eqtr4d 2774	. . 3 ⊢ (𝜑 → 𝐼 = X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)))
48	47	fveq2d 6885	. 2 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))))
49		vonn0icc2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
50		vonn0icc2.n	. . 3 ⊢ (𝜑 → 𝑋 ≠ ∅)
51	4, 16, 18	fmptdf 7112	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℝ)
52	4, 27, 29	fmptdf 7112	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℝ)
53		eqid 2736	. . 3 ⊢ X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))
54	49, 50, 51, 52, 53	vonn0icc 46684	. 2 ⊢ (𝜑 → ((voln‘𝑋)‘X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))) = ∏𝑗 ∈ 𝑋 (vol‘(((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))))
55	32	fveq2d 6885	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (vol‘(((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))) = (vol‘(⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵)))
56	55	prodeq2dv 15943	. . 3 ⊢ (𝜑 → ∏𝑗 ∈ 𝑋 (vol‘(((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))) = ∏𝑗 ∈ 𝑋 (vol‘(⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵)))
57	43	fveq2d 6885	. . . . 5 ⊢ (𝑗 = 𝑘 → (vol‘(⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵)) = (vol‘(𝐴[,]𝐵)))
58		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑘𝑋
59		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑗𝑋
60		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑘vol
61	60, 35	nffv 6891	. . . . 5 ⊢ Ⅎ𝑘(vol‘(⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵))
62		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑗(vol‘(𝐴[,]𝐵))
63	57, 58, 59, 61, 62	cbvprod 15934	. . . 4 ⊢ ∏𝑗 ∈ 𝑋 (vol‘(⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵)) = ∏𝑘 ∈ 𝑋 (vol‘(𝐴[,]𝐵))
64	63	a1i 11	. . 3 ⊢ (𝜑 → ∏𝑗 ∈ 𝑋 (vol‘(⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵)) = ∏𝑘 ∈ 𝑋 (vol‘(𝐴[,]𝐵)))
65	56, 64	eqtrd 2771	. 2 ⊢ (𝜑 → ∏𝑗 ∈ 𝑋 (vol‘(((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))) = ∏𝑘 ∈ 𝑋 (vol‘(𝐴[,]𝐵)))
66	48, 54, 65	3eqtrd 2775	1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 (vol‘(𝐴[,]𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2109 ≠ wne 2933 ⦋csb 3879 ∅c0 4313 ↦ cmpt 5206 ‘cfv 6536 (class class class)co 7410 Xcixp 8916 Fincfn 8964 ℝcr 11133 [,]cicc 13370 ∏cprod 15924 volcvol 25421 volncvoln 46534

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cc 10454 ax-ac2 10482 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213 ax-mulf 11214

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-disj 5092 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-omul 8490 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-dju 9920 df-card 9958 df-acn 9961 df-ac 10135 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13371 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-seq 14025 df-exp 14085 df-hash 14354 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-clim 15509 df-rlim 15510 df-sum 15708 df-prod 15925 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-pws 17468 df-xrs 17521 df-qtop 17526 df-imas 17527 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-mhm 18766 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mulg 19056 df-subg 19111 df-ghm 19201 df-cntz 19305 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-ring 20200 df-cring 20201 df-oppr 20302 df-dvdsr 20322 df-unit 20323 df-invr 20353 df-dvr 20366 df-rhm 20437 df-subrng 20511 df-subrg 20535 df-drng 20696 df-field 20697 df-abv 20774 df-staf 20804 df-srng 20805 df-lmod 20824 df-lss 20894 df-lmhm 20985 df-lvec 21066 df-sra 21136 df-rgmod 21137 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-cnfld 21321 df-refld 21570 df-phl 21591 df-dsmm 21697 df-frlm 21712 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cn 23170 df-cnp 23171 df-cmp 23330 df-tx 23505 df-hmeo 23698 df-xms 24264 df-ms 24265 df-tms 24266 df-nm 24526 df-ngp 24527 df-tng 24528 df-nrg 24529 df-nlm 24530 df-cncf 24827 df-clm 25019 df-cph 25125 df-tcph 25126 df-rrx 25342 df-ovol 25422 df-vol 25423 df-salg 46305 df-sumge0 46359 df-mea 46446 df-ome 46486 df-caragen 46488 df-ovoln 46533 df-voln 46535

This theorem is referenced by: (None)

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