vonn0icc2 - Mathbox for Glauco Siliprandi

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Theorem vonn0icc2 47149

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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝑗 ∈ 𝑋)
4		vonn0icc2.k	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝜑
5		nfv 1922	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑋
6	4, 5	nfan 1907	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑋)
7		nfcsb1v 3857	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴
8		nfcv 2903	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℝ
9	7, 8	nfel 2917	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ
10	6, 9	nfim 1904	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ)
11		eleq1w 2824	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑋 ↔ 𝑗 ∈ 𝑋))
12	11	anbi2d 637	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑋) ↔ (𝜑 ∧ 𝑗 ∈ 𝑋)))
13		csbeq1a 3847	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → 𝐴 = ⦋𝑗 / 𝑘⦌𝐴)
14	13	eleq1d 2826	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐴 ∈ ℝ ↔ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ))
15	12, 14	imbi12d 346	. . . . . . . . 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 2254	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ)
18		eqid 2741	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝑋 ↦ 𝐴) = (𝑘 ∈ 𝑋 ↦ 𝐴)
19	18	fvmpts 6943	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝑋 ∧ ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ) → ((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴)
20	3, 17, 19	syl2anc 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴)
21		nfcsb1v 3857	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵
22	21, 8	nfel 2917	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ
23	6, 22	nfim 1904	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ)
24		csbeq1a 3847	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵)
25	24	eleq1d 2826	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐵 ∈ ℝ ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ))
26	12, 25	imbi12d 346	. . . . . . . . 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 2254	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ)
29		eqid 2741	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝑋 ↦ 𝐵) = (𝑘 ∈ 𝑋 ↦ 𝐵)
30	29	fvmpts 6943	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝑋 ∧ ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ) → ((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵)
31	3, 28, 30	syl2anc 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵)
32	20, 31	oveq12d 7378	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = (⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵))
33	32	ixpeq2dva 8854	. . . . 5 ⊢ (𝜑 → X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = X𝑗 ∈ 𝑋 (⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵))
34		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑘[,]
35	7, 34, 21	nfov 7390	. . . . . . 7 ⊢ Ⅎ𝑘(⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵)
36		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑗(𝐴[,]𝐵)
37	13	equcoms 2028	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → 𝐴 = ⦋𝑗 / 𝑘⦌𝐴)
38	37	eqcomd 2747	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐴 = 𝐴)
39		eqidd 2742	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → 𝐴 = 𝐴)
40	38, 39	eqtrd 2776	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐴 = 𝐴)
41	24	equcoms 2028	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵)
42	41	eqcomd 2747	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐵 = 𝐵)
43	40, 42	oveq12d 7378	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵) = (𝐴[,]𝐵))
44	35, 36, 43	cbvixp 8856	. . . . . 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 2776	. . . 4 ⊢ (𝜑 → X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = X𝑘 ∈ 𝑋 (𝐴[,]𝐵))
47	2, 46	eqtr4d 2779	. . 3 ⊢ (𝜑 → 𝐼 = X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)))
48	47	fveq2d 6835	. 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 7062	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℝ)
52	4, 27, 29	fmptdf 7062	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℝ)
53		eqid 2741	. . 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 47145	. 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 6835	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (vol‘(((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))) = (vol‘(⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵)))
56	55	prodeq2dv 15882	. . 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 6835	. . . . 5 ⊢ (𝑗 = 𝑘 → (vol‘(⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵)) = (vol‘(𝐴[,]𝐵)))
58		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑘𝑋
59		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑗𝑋
60		nfcv 2903	. . . . . 6 ⊢ Ⅎ𝑘vol
61	60, 35	nffv 6841	. . . . 5 ⊢ Ⅎ𝑘(vol‘(⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵))
62		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑗(vol‘(𝐴[,]𝐵))
63	57, 58, 59, 61, 62	cbvprod 15873	. . . 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 2776	. 2 ⊢ (𝜑 → ∏𝑗 ∈ 𝑋 (vol‘(((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))) = ∏𝑘 ∈ 𝑋 (vol‘(𝐴[,]𝐵)))
66	48, 54, 65	3eqtrd 2780	1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 (vol‘(𝐴[,]𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1548 Ⅎwnf 1791 ∈ wcel 2121 ≠ wne 2936 ⦋csb 3833 ∅c0 4264 ↦ cmpt 5156 ‘cfv 6489 (class class class)co 7360 Xcixp 8839 Fincfn 8887 ℝcr 11032 [,]cicc 13296 ∏cprod 15863 volcvol 25452 volncvoln 46995

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-inf2 9557 ax-cc 10352 ax-ac2 10380 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112 ax-mulf 11113

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-iin 4927 df-disj 5043 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-omul 8404 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9820 df-card 9858 df-acn 9861 df-ac 10033 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-ioo 13297 df-ico 13299 df-icc 13300 df-fz 13457 df-fzo 13604 df-fl 13746 df-seq 13959 df-exp 14019 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-clim 15445 df-rlim 15446 df-sum 15644 df-prod 15864 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-rest 17380 df-topn 17381 df-0g 17399 df-gsum 17400 df-topgen 17401 df-pt 17402 df-prds 17405 df-pws 17407 df-xrs 17461 df-qtop 17466 df-imas 17467 df-xps 17469 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18746 df-submnd 18747 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19039 df-subg 19094 df-ghm 19183 df-cntz 19287 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-ring 20211 df-cring 20212 df-oppr 20312 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-dvr 20376 df-rhm 20447 df-subrng 20522 df-subrg 20546 df-drng 20707 df-field 20708 df-abv 20785 df-staf 20815 df-srng 20816 df-lmod 20856 df-lss 20926 df-lmhm 21016 df-lvec 21097 df-sra 21167 df-rgmod 21168 df-psmet 21343 df-xmet 21344 df-met 21345 df-bl 21346 df-mopn 21347 df-cnfld 21352 df-refld 21584 df-phl 21605 df-dsmm 21711 df-frlm 21726 df-top 22881 df-topon 22898 df-topsp 22920 df-bases 22933 df-cn 23214 df-cnp 23215 df-cmp 23374 df-tx 23549 df-hmeo 23742 df-xms 24307 df-ms 24308 df-tms 24309 df-nm 24569 df-ngp 24570 df-tng 24571 df-nrg 24572 df-nlm 24573 df-cncf 24867 df-clm 25052 df-cph 25157 df-tcph 25158 df-rrx 25374 df-ovol 25453 df-vol 25454 df-salg 46766 df-sumge0 46820 df-mea 46907 df-ome 46947 df-caragen 46949 df-ovoln 46994 df-voln 46996

This theorem is referenced by: (None)

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