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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > voliooicof | Structured version Visualization version GIF version |
Description: The Lebesgue measure of open intervals is the same as the Lebesgue measure of left-closed right-open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
voliooicof.1 | ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ × ℝ)) |
Ref | Expression |
---|---|
voliooicof | ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = ((vol ∘ [,)) ∘ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | volioof 42629 | . . . . 5 ⊢ (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞) | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞)) |
3 | rexpssxrxp 10675 | . . . . 5 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*) | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → (ℝ × ℝ) ⊆ (ℝ* × ℝ*)) |
5 | voliooicof.1 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ × ℝ)) | |
6 | 2, 4, 5 | fcoss 41839 | . . 3 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹):𝐴⟶(0[,]+∞)) |
7 | 6 | ffnd 6488 | . 2 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) Fn 𝐴) |
8 | volf 24133 | . . . . . 6 ⊢ vol:dom vol⟶(0[,]+∞) | |
9 | 8 | a1i 11 | . . . . 5 ⊢ (𝜑 → vol:dom vol⟶(0[,]+∞)) |
10 | icof 41848 | . . . . . . . . . 10 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ* | |
11 | 10 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*) |
12 | 11, 4, 5 | fcoss 41839 | . . . . . . . 8 ⊢ (𝜑 → ([,) ∘ 𝐹):𝐴⟶𝒫 ℝ*) |
13 | 12 | ffnd 6488 | . . . . . . 7 ⊢ (𝜑 → ([,) ∘ 𝐹) Fn 𝐴) |
14 | 5 | adantr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶(ℝ × ℝ)) |
15 | simpr 488 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
16 | 14, 15 | fvovco 41821 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (([,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥)))) |
17 | 5 | ffvelrnda 6828 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (ℝ × ℝ)) |
18 | xp1st 7703 | . . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘𝑥)) ∈ ℝ) | |
19 | 17, 18 | syl 17 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (1st ‘(𝐹‘𝑥)) ∈ ℝ) |
20 | xp2nd 7704 | . . . . . . . . . . . 12 ⊢ ((𝐹‘𝑥) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘𝑥)) ∈ ℝ) | |
21 | 17, 20 | syl 17 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (2nd ‘(𝐹‘𝑥)) ∈ ℝ) |
22 | 21 | rexrd 10680 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (2nd ‘(𝐹‘𝑥)) ∈ ℝ*) |
23 | icombl 24168 | . . . . . . . . . 10 ⊢ (((1st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ*) → ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))) ∈ dom vol) | |
24 | 19, 22, 23 | syl2anc 587 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))) ∈ dom vol) |
25 | 16, 24 | eqeltrd 2890 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (([,) ∘ 𝐹)‘𝑥) ∈ dom vol) |
26 | 25 | ralrimiva 3149 | . . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (([,) ∘ 𝐹)‘𝑥) ∈ dom vol) |
27 | 13, 26 | jca 515 | . . . . . 6 ⊢ (𝜑 → (([,) ∘ 𝐹) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (([,) ∘ 𝐹)‘𝑥) ∈ dom vol)) |
28 | ffnfv 6859 | . . . . . 6 ⊢ (([,) ∘ 𝐹):𝐴⟶dom vol ↔ (([,) ∘ 𝐹) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (([,) ∘ 𝐹)‘𝑥) ∈ dom vol)) | |
29 | 27, 28 | sylibr 237 | . . . . 5 ⊢ (𝜑 → ([,) ∘ 𝐹):𝐴⟶dom vol) |
30 | fco 6505 | . . . . 5 ⊢ ((vol:dom vol⟶(0[,]+∞) ∧ ([,) ∘ 𝐹):𝐴⟶dom vol) → (vol ∘ ([,) ∘ 𝐹)):𝐴⟶(0[,]+∞)) | |
31 | 9, 29, 30 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (vol ∘ ([,) ∘ 𝐹)):𝐴⟶(0[,]+∞)) |
32 | coass 6085 | . . . . . 6 ⊢ ((vol ∘ [,)) ∘ 𝐹) = (vol ∘ ([,) ∘ 𝐹)) | |
33 | 32 | a1i 11 | . . . . 5 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (vol ∘ ([,) ∘ 𝐹))) |
34 | 33 | feq1d 6472 | . . . 4 ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞) ↔ (vol ∘ ([,) ∘ 𝐹)):𝐴⟶(0[,]+∞))) |
35 | 31, 34 | mpbird 260 | . . 3 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞)) |
36 | 35 | ffnd 6488 | . 2 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹) Fn 𝐴) |
37 | 19, 21 | voliooico 42634 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (vol‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥)))) = (vol‘((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))))) |
38 | 5, 4 | fssd 6502 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ* × ℝ*)) |
39 | 38 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶(ℝ* × ℝ*)) |
40 | 39, 15 | fvvolioof 42631 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((vol ∘ (,)) ∘ 𝐹)‘𝑥) = (vol‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))) |
41 | 39, 15 | fvvolicof 42633 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((vol ∘ [,)) ∘ 𝐹)‘𝑥) = (vol‘((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))))) |
42 | 37, 40, 41 | 3eqtr4d 2843 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((vol ∘ (,)) ∘ 𝐹)‘𝑥) = (((vol ∘ [,)) ∘ 𝐹)‘𝑥)) |
43 | 7, 36, 42 | eqfnfvd 6782 | 1 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = ((vol ∘ [,)) ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ⊆ wss 3881 𝒫 cpw 4497 × cxp 5517 dom cdm 5519 ∘ ccom 5523 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 1st c1st 7669 2nd c2nd 7670 ℝcr 10525 0cc0 10526 +∞cpnf 10661 ℝ*cxr 10663 (,)cioo 12726 [,)cico 12728 [,]cicc 12729 volcvol 24067 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-rest 16688 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-top 21499 df-topon 21516 df-bases 21551 df-cmp 21992 df-ovol 24068 df-vol 24069 |
This theorem is referenced by: ovolval5lem3 43293 |
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