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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > volioof | Structured version Visualization version GIF version |
Description: The function that assigns the Lebesgue measure to open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
volioof | ⊢ (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | volf 25045 | . 2 ⊢ vol:dom vol⟶(0[,]+∞) | |
2 | ioof 13423 | . . . . 5 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
3 | ffn 6717 | . . . . 5 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ (,) Fn (ℝ* × ℝ*) |
5 | df-ov 7411 | . . . . . . . 8 ⊢ ((1st ‘𝑥)(,)(2nd ‘𝑥)) = ((,)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ (𝑥 ∈ (ℝ* × ℝ*) → ((1st ‘𝑥)(,)(2nd ‘𝑥)) = ((,)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)) |
7 | 1st2nd2 8013 | . . . . . . . . 9 ⊢ (𝑥 ∈ (ℝ* × ℝ*) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) | |
8 | 7 | eqcomd 2738 | . . . . . . . 8 ⊢ (𝑥 ∈ (ℝ* × ℝ*) → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = 𝑥) |
9 | 8 | fveq2d 6895 | . . . . . . 7 ⊢ (𝑥 ∈ (ℝ* × ℝ*) → ((,)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) = ((,)‘𝑥)) |
10 | 6, 9 | eqtr2d 2773 | . . . . . 6 ⊢ (𝑥 ∈ (ℝ* × ℝ*) → ((,)‘𝑥) = ((1st ‘𝑥)(,)(2nd ‘𝑥))) |
11 | ioombl 25081 | . . . . . 6 ⊢ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ∈ dom vol | |
12 | 10, 11 | eqeltrdi 2841 | . . . . 5 ⊢ (𝑥 ∈ (ℝ* × ℝ*) → ((,)‘𝑥) ∈ dom vol) |
13 | 12 | rgen 3063 | . . . 4 ⊢ ∀𝑥 ∈ (ℝ* × ℝ*)((,)‘𝑥) ∈ dom vol |
14 | 4, 13 | pm3.2i 471 | . . 3 ⊢ ((,) Fn (ℝ* × ℝ*) ∧ ∀𝑥 ∈ (ℝ* × ℝ*)((,)‘𝑥) ∈ dom vol) |
15 | ffnfv 7117 | . . 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 6741 | . 2 ⊢ ((vol:dom vol⟶(0[,]+∞) ∧ (,):(ℝ* × ℝ*)⟶dom vol) → (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞)) | |
18 | 1, 16, 17 | mp2an 690 | 1 ⊢ (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 𝒫 cpw 4602 ⟨cop 4634 × cxp 5674 dom cdm 5676 ∘ ccom 5680 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 1st c1st 7972 2nd c2nd 7973 ℝcr 11108 0cc0 11109 +∞cpnf 11244 ℝ*cxr 11246 (,)cioo 13323 [,]cicc 13326 volcvol 24979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 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 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 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 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-q 12932 df-rp 12974 df-xadd 13092 df-ioo 13327 df-ico 13329 df-icc 13330 df-fz 13484 df-fzo 13627 df-fl 13756 df-seq 13966 df-exp 14027 df-hash 14290 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-clim 15431 df-rlim 15432 df-sum 15632 df-xmet 20936 df-met 20937 df-ovol 24980 df-vol 24981 |
This theorem is referenced by: volioofmpt 44700 voliooicof 44702 ovolval3 45353 ovolval5lem2 45359 |
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