volioof - Mathbox for Glauco Siliprandi

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Theorem volioof 44690

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 25038	. 2 ⊢ vol:dom vol⟶(0[,]+∞)
2		ioof 13421	. . . . 5 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
3		ffn 6715	. . . . 5 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
4	2, 3	ax-mp 5	. . . 4 ⊢ (,) Fn (ℝ^* × ℝ^*)
5		df-ov 7409	. . . . . . . 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 8011	. . . . . . . . 9 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
8	7	eqcomd 2739	. . . . . . . 8 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ = 𝑥)
9	8	fveq2d 6893	. . . . . . 7 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → ((,)‘⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) = ((,)‘𝑥))
10	6, 9	eqtr2d 2774	. . . . . 6 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → ((,)‘𝑥) = ((1^st ‘𝑥)(,)(2^nd ‘𝑥)))
11		ioombl 25074	. . . . . 6 ⊢ ((1^st ‘𝑥)(,)(2^nd ‘𝑥)) ∈ dom vol
12	10, 11	eqeltrdi 2842	. . . . 5 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → ((,)‘𝑥) ∈ dom vol)
13	12	rgen 3064	. . . 4 ⊢ ∀𝑥 ∈ (ℝ^* × ℝ^*)((,)‘𝑥) ∈ dom vol
14	4, 13	pm3.2i 472	. . 3 ⊢ ((,) Fn (ℝ^* × ℝ^) ∧ ∀𝑥 ∈ (ℝ^ × ℝ^*)((,)‘𝑥) ∈ dom vol)
15		ffnfv 7115	. . 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 6739	. 2 ⊢ ((vol:dom vol⟶(0[,]+∞) ∧ (,):(ℝ^* × ℝ^)⟶dom vol) → (vol ∘ (,)):(ℝ^ × ℝ^*)⟶(0[,]+∞))
18	1, 16, 17	mp2an 691	1 ⊢ (vol ∘ (,)):(ℝ^* × ℝ^*)⟶(0[,]+∞)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 𝒫 cpw 4602 ⟨cop 4634 × cxp 5674 dom cdm 5676 ∘ ccom 5680 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 1^st c1st 7970 2^nd c2nd 7971 ℝcr 11106 0cc0 11107 +∞cpnf 11242 ℝ^*cxr 11244 (,)cioo 13321 [,]cicc 13324 volcvol 24972

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-2o 8464 df-er 8700 df-map 8819 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-oi 9502 df-dju 9893 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-q 12930 df-rp 12972 df-xadd 13090 df-ioo 13325 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-rlim 15430 df-sum 15630 df-xmet 20930 df-met 20931 df-ovol 24973 df-vol 24974

This theorem is referenced by: volioofmpt 44697 voliooicof 44699 ovolval3 45350 ovolval5lem2 45356

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