volioof - Mathbox for Glauco Siliprandi

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Theorem volioof 45983

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 25487	. 2 ⊢ vol:dom vol⟶(0[,]+∞)
2		ioof 13469	. . . . 5 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
3		ffn 6711	. . . . 5 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
4	2, 3	ax-mp 5	. . . 4 ⊢ (,) Fn (ℝ^* × ℝ^*)
5		df-ov 7413	. . . . . . . 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 8032	. . . . . . . . 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 6885	. . . . . . 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 25523	. . . . . 6 ⊢ ((1^st ‘𝑥)(,)(2^nd ‘𝑥)) ∈ dom vol
12	10, 11	eqeltrdi 2843	. . . . 5 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → ((,)‘𝑥) ∈ dom vol)
13	12	rgen 3054	. . . 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 7114	. . 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 6735	. 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 1540 ∈ wcel 2109 ∀wral 3052 𝒫 cpw 4580 ⟨cop 4612 × cxp 5657 dom cdm 5659 ∘ ccom 5663 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 1^st c1st 7991 2^nd c2nd 7992 ℝcr 11133 0cc0 11134 +∞cpnf 11271 ℝ^*cxr 11273 (,)cioo 13367 [,]cicc 13370 volcvol 25421

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-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

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-op 4613 df-uni 4889 df-int 4928 df-iun 4974 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-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9459 df-inf 9460 df-oi 9529 df-dju 9920 df-card 9958 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-n0 12507 df-z 12594 df-uz 12858 df-q 12970 df-rp 13014 df-xadd 13134 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-xmet 21313 df-met 21314 df-ovol 25422 df-vol 25423

This theorem is referenced by: volioofmpt 45990 voliooicof 45992 ovolval3 46643 ovolval5lem2 46649

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