volioof - Mathbox for Glauco Siliprandi

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Theorem volioof 45256

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 25409	. 2 ⊢ vol:dom vol⟶(0[,]+∞)
2		ioof 13427	. . . . 5 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
3		ffn 6710	. . . . 5 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
4	2, 3	ax-mp 5	. . . 4 ⊢ (,) Fn (ℝ^* × ℝ^*)
5		df-ov 7407	. . . . . . . 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 8010	. . . . . . . . 9 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
8	7	eqcomd 2732	. . . . . . . 8 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ = 𝑥)
9	8	fveq2d 6888	. . . . . . 7 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → ((,)‘⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) = ((,)‘𝑥))
10	6, 9	eqtr2d 2767	. . . . . 6 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → ((,)‘𝑥) = ((1^st ‘𝑥)(,)(2^nd ‘𝑥)))
11		ioombl 25445	. . . . . 6 ⊢ ((1^st ‘𝑥)(,)(2^nd ‘𝑥)) ∈ dom vol
12	10, 11	eqeltrdi 2835	. . . . 5 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → ((,)‘𝑥) ∈ dom vol)
13	12	rgen 3057	. . . 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 7113	. . 3 ⊢ ((,):(ℝ^* × ℝ^)⟶dom vol ↔ ((,) Fn (ℝ^ × ℝ^) ∧ ∀𝑥 ∈ (ℝ^ × ℝ^*)((,)‘𝑥) ∈ dom vol))
16	14, 15	mpbir 230	. 2 ⊢ (,):(ℝ^* × ℝ^*)⟶dom vol
17		fco 6734	. 2 ⊢ ((vol:dom vol⟶(0[,]+∞) ∧ (,):(ℝ^* × ℝ^)⟶dom vol) → (vol ∘ (,)):(ℝ^ × ℝ^*)⟶(0[,]+∞))
18	1, 16, 17	mp2an 689	1 ⊢ (vol ∘ (,)):(ℝ^* × ℝ^*)⟶(0[,]+∞)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 𝒫 cpw 4597 ⟨cop 4629 × cxp 5667 dom cdm 5669 ∘ ccom 5673 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 1^st c1st 7969 2^nd c2nd 7970 ℝcr 11108 0cc0 11109 +∞cpnf 11246 ℝ^*cxr 11248 (,)cioo 13327 [,]cicc 13330 volcvol 25343

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 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 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 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 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-q 12934 df-rp 12978 df-xadd 13096 df-ioo 13331 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-seq 13970 df-exp 14031 df-hash 14294 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-rlim 15437 df-sum 15637 df-xmet 21229 df-met 21230 df-ovol 25344 df-vol 25345

This theorem is referenced by: volioofmpt 45263 voliooicof 45265 ovolval3 45916 ovolval5lem2 45922

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