volioof - Mathbox for Glauco Siliprandi

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Theorem volioof 46231

Description: The function that assigns the Lebesgue measure to open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Assertion**
Ref	Expression
volioof	⊢ (vol ∘ (,)):(ℝ^* × ℝ^*)⟶(0[,]+∞)

**Proof of Theorem volioof**
Step	Hyp	Ref	Expression
1		volf 25486	. 2 ⊢ vol:dom vol⟶(0[,]+∞)
2		ioof 13363	. . . . 5 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
3		ffn 6662	. . . . 5 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
4	2, 3	ax-mp 5	. . . 4 ⊢ (,) Fn (ℝ^* × ℝ^*)
5		df-ov 7361	. . . . . . . 8 ⊢ ((1^st ‘𝑥)(,)(2^nd ‘𝑥)) = ((,)‘⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
6	5	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → ((1^st ‘𝑥)(,)(2^nd ‘𝑥)) = ((,)‘⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩))
7		1st2nd2 7972	. . . . . . . . 9 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
8	7	eqcomd 2742	. . . . . . . 8 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ = 𝑥)
9	8	fveq2d 6838	. . . . . . 7 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → ((,)‘⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) = ((,)‘𝑥))
10	6, 9	eqtr2d 2772	. . . . . 6 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → ((,)‘𝑥) = ((1^st ‘𝑥)(,)(2^nd ‘𝑥)))
11		ioombl 25522	. . . . . 6 ⊢ ((1^st ‘𝑥)(,)(2^nd ‘𝑥)) ∈ dom vol
12	10, 11	eqeltrdi 2844	. . . . 5 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → ((,)‘𝑥) ∈ dom vol)
13	12	rgen 3053	. . . 4 ⊢ ∀𝑥 ∈ (ℝ^* × ℝ^*)((,)‘𝑥) ∈ dom vol
14	4, 13	pm3.2i 470	. . 3 ⊢ ((,) Fn (ℝ^* × ℝ^) ∧ ∀𝑥 ∈ (ℝ^ × ℝ^*)((,)‘𝑥) ∈ dom vol)
15		ffnfv 7064	. . 3 ⊢ ((,):(ℝ^* × ℝ^)⟶dom vol ↔ ((,) Fn (ℝ^ × ℝ^) ∧ ∀𝑥 ∈ (ℝ^ × ℝ^*)((,)‘𝑥) ∈ dom vol))
16	14, 15	mpbir 231	. 2 ⊢ (,):(ℝ^* × ℝ^*)⟶dom vol
17		fco 6686	. 2 ⊢ ((vol:dom vol⟶(0[,]+∞) ∧ (,):(ℝ^* × ℝ^)⟶dom vol) → (vol ∘ (,)):(ℝ^ × ℝ^*)⟶(0[,]+∞))
18	1, 16, 17	mp2an 692	1 ⊢ (vol ∘ (,)):(ℝ^* × ℝ^*)⟶(0[,]+∞)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3051 𝒫 cpw 4554 ⟨cop 4586 × cxp 5622 dom cdm 5624 ∘ ccom 5628 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 1^st c1st 7931 2^nd c2nd 7932 ℝcr 11025 0cc0 11026 +∞cpnf 11163 ℝ^*cxr 11165 (,)cioo 13261 [,]cicc 13264 volcvol 25420

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9813 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-n0 12402 df-z 12489 df-uz 12752 df-q 12862 df-rp 12906 df-xadd 13027 df-ioo 13265 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-rlim 15412 df-sum 15610 df-xmet 21302 df-met 21303 df-ovol 25421 df-vol 25422

This theorem is referenced by: volioofmpt 46238 voliooicof 46240 ovolval3 46891 ovolval5lem2 46897

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