volioof - Mathbox for Glauco Siliprandi

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Theorem volioof 45943

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 25578	. 2 ⊢ vol:dom vol⟶(0[,]+∞)
2		ioof 13484	. . . . 5 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
3		ffn 6737	. . . . 5 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
4	2, 3	ax-mp 5	. . . 4 ⊢ (,) Fn (ℝ^* × ℝ^*)
5		df-ov 7434	. . . . . . . 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 8052	. . . . . . . . 9 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
8	7	eqcomd 2741	. . . . . . . 8 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ = 𝑥)
9	8	fveq2d 6911	. . . . . . 7 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → ((,)‘⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) = ((,)‘𝑥))
10	6, 9	eqtr2d 2776	. . . . . 6 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → ((,)‘𝑥) = ((1^st ‘𝑥)(,)(2^nd ‘𝑥)))
11		ioombl 25614	. . . . . 6 ⊢ ((1^st ‘𝑥)(,)(2^nd ‘𝑥)) ∈ dom vol
12	10, 11	eqeltrdi 2847	. . . . 5 ⊢ (𝑥 ∈ (ℝ^* × ℝ^*) → ((,)‘𝑥) ∈ dom vol)
13	12	rgen 3061	. . . 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 7139	. . 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 6761	. 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 1537 ∈ wcel 2106 ∀wral 3059 𝒫 cpw 4605 ⟨cop 4637 × cxp 5687 dom cdm 5689 ∘ ccom 5693 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 1^st c1st 8011 2^nd c2nd 8012 ℝcr 11152 0cc0 11153 +∞cpnf 11290 ℝ^*cxr 11292 (,)cioo 13384 [,]cicc 13387 volcvol 25512

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xadd 13153 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-rlim 15522 df-sum 15720 df-xmet 21375 df-met 21376 df-ovol 25513 df-vol 25514

This theorem is referenced by: volioofmpt 45950 voliooicof 45952 ovolval3 46603 ovolval5lem2 46609

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