vonn0icc2 - Mathbox for Glauco Siliprandi

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Theorem vonn0icc2 46683

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 3875	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴
8		nfcv 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℝ
9	7, 8	nfel 2906	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ
10	6, 9	nfim 1896	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℝ)
11		eleq1w 2811	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑋 ↔ 𝑗 ∈ 𝑋))
12	11	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑋) ↔ (𝜑 ∧ 𝑗 ∈ 𝑋)))
13		csbeq1a 3865	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → 𝐴 = ⦋𝑗 / 𝑘⦌𝐴)
14	13	eleq1d 2813	. . . . . . . . . 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 2729	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝑋 ↦ 𝐴) = (𝑘 ∈ 𝑋 ↦ 𝐴)
19	18	fvmpts 6933	. . . . . . . 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 3875	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵
22	21, 8	nfel 2906	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ
23	6, 22	nfim 1896	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑋) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ)
24		csbeq1a 3865	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵)
25	24	eleq1d 2813	. . . . . . . . . 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 2729	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝑋 ↦ 𝐵) = (𝑘 ∈ 𝑋 ↦ 𝐵)
30	29	fvmpts 6933	. . . . . . . 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 7367	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = (⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵))
33	32	ixpeq2dva 8839	. . . . 5 ⊢ (𝜑 → X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = X𝑗 ∈ 𝑋 (⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵))
34		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑘[,]
35	7, 34, 21	nfov 7379	. . . . . . 7 ⊢ Ⅎ𝑘(⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵)
36		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑗(𝐴[,]𝐵)
37	13	equcoms 2020	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → 𝐴 = ⦋𝑗 / 𝑘⦌𝐴)
38	37	eqcomd 2735	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐴 = 𝐴)
39		eqidd 2730	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → 𝐴 = 𝐴)
40	38, 39	eqtrd 2764	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐴 = 𝐴)
41	24	equcoms 2020	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵)
42	41	eqcomd 2735	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐵 = 𝐵)
43	40, 42	oveq12d 7367	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵) = (𝐴[,]𝐵))
44	35, 36, 43	cbvixp 8841	. . . . . 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 2764	. . . 4 ⊢ (𝜑 → X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)) = X𝑘 ∈ 𝑋 (𝐴[,]𝐵))
47	2, 46	eqtr4d 2767	. . 3 ⊢ (𝜑 → 𝐼 = X𝑗 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗)))
48	47	fveq2d 6826	. 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 7051	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℝ)
52	4, 27, 29	fmptdf 7051	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℝ)
53		eqid 2729	. . 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 46679	. 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 6826	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (vol‘(((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))) = (vol‘(⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵)))
56	55	prodeq2dv 15829	. . 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 6826	. . . . 5 ⊢ (𝑗 = 𝑘 → (vol‘(⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵)) = (vol‘(𝐴[,]𝐵)))
58		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑘𝑋
59		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑗𝑋
60		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑘vol
61	60, 35	nffv 6832	. . . . 5 ⊢ Ⅎ𝑘(vol‘(⦋𝑗 / 𝑘⦌𝐴[,]⦋𝑗 / 𝑘⦌𝐵))
62		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑗(vol‘(𝐴[,]𝐵))
63	57, 58, 59, 61, 62	cbvprod 15820	. . . 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 2764	. 2 ⊢ (𝜑 → ∏𝑗 ∈ 𝑋 (vol‘(((𝑘 ∈ 𝑋 ↦ 𝐴)‘𝑗)[,]((𝑘 ∈ 𝑋 ↦ 𝐵)‘𝑗))) = ∏𝑘 ∈ 𝑋 (vol‘(𝐴[,]𝐵)))
66	48, 54, 65	3eqtrd 2768	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 2925 ⦋csb 3851 ∅c0 4284 ↦ cmpt 5173 ‘cfv 6482 (class class class)co 7349 Xcixp 8824 Fincfn 8872 ℝcr 11008 [,]cicc 13251 ∏cprod 15810 volcvol 25362 volncvoln 46529

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 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cc 10329 ax-ac2 10357 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-disj 5060 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-tpos 8159 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-oadd 8392 df-omul 8393 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-dju 9797 df-card 9835 df-acn 9838 df-ac 10010 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-rlim 15396 df-sum 15594 df-prod 15811 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-pws 17353 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-mhm 18657 df-submnd 18658 df-grp 18815 df-minusg 18816 df-sbg 18817 df-mulg 18947 df-subg 19002 df-ghm 19092 df-cntz 19196 df-cmn 19661 df-abl 19662 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-cring 20121 df-oppr 20222 df-dvdsr 20242 df-unit 20243 df-invr 20273 df-dvr 20286 df-rhm 20357 df-subrng 20431 df-subrg 20455 df-drng 20616 df-field 20617 df-abv 20694 df-staf 20724 df-srng 20725 df-lmod 20765 df-lss 20835 df-lmhm 20926 df-lvec 21007 df-sra 21077 df-rgmod 21078 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-cnfld 21262 df-refld 21512 df-phl 21533 df-dsmm 21639 df-frlm 21654 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cn 23112 df-cnp 23113 df-cmp 23272 df-tx 23447 df-hmeo 23640 df-xms 24206 df-ms 24207 df-tms 24208 df-nm 24468 df-ngp 24469 df-tng 24470 df-nrg 24471 df-nlm 24472 df-cncf 24769 df-clm 24961 df-cph 25066 df-tcph 25067 df-rrx 25283 df-ovol 25363 df-vol 25364 df-salg 46300 df-sumge0 46354 df-mea 46441 df-ome 46481 df-caragen 46483 df-ovoln 46528 df-voln 46530

This theorem is referenced by: (None)

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