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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 45608 | . . . . 5 ⊢ (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞) | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞)) |
3 | rexpssxrxp 11309 | . . . . 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 44817 | . . 3 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹):𝐴⟶(0[,]+∞)) |
7 | 6 | ffnd 6729 | . 2 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) Fn 𝐴) |
8 | volf 25549 | . . . . . 6 ⊢ vol:dom vol⟶(0[,]+∞) | |
9 | 8 | a1i 11 | . . . . 5 ⊢ (𝜑 → vol:dom vol⟶(0[,]+∞)) |
10 | icof 44826 | . . . . . . . . . 10 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ* | |
11 | 10 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*) |
12 | 11, 4, 5 | fcoss 44817 | . . . . . . . 8 ⊢ (𝜑 → ([,) ∘ 𝐹):𝐴⟶𝒫 ℝ*) |
13 | 12 | ffnd 6729 | . . . . . . 7 ⊢ (𝜑 → ([,) ∘ 𝐹) Fn 𝐴) |
14 | 5 | adantr 479 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶(ℝ × ℝ)) |
15 | simpr 483 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
16 | 14, 15 | fvovco 44800 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (([,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥)))) |
17 | 5 | ffvelcdmda 7098 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (ℝ × ℝ)) |
18 | xp1st 8035 | . . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘𝑥)) ∈ ℝ) | |
19 | 17, 18 | syl 17 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (1st ‘(𝐹‘𝑥)) ∈ ℝ) |
20 | xp2nd 8036 | . . . . . . . . . . . 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 11314 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (2nd ‘(𝐹‘𝑥)) ∈ ℝ*) |
23 | icombl 25584 | . . . . . . . . . 10 ⊢ (((1st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ*) → ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))) ∈ dom vol) | |
24 | 19, 22, 23 | syl2anc 582 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))) ∈ dom vol) |
25 | 16, 24 | eqeltrd 2826 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (([,) ∘ 𝐹)‘𝑥) ∈ dom vol) |
26 | 25 | ralrimiva 3136 | . . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (([,) ∘ 𝐹)‘𝑥) ∈ dom vol) |
27 | 13, 26 | jca 510 | . . . . . 6 ⊢ (𝜑 → (([,) ∘ 𝐹) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (([,) ∘ 𝐹)‘𝑥) ∈ dom vol)) |
28 | ffnfv 7133 | . . . . . 6 ⊢ (([,) ∘ 𝐹):𝐴⟶dom vol ↔ (([,) ∘ 𝐹) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (([,) ∘ 𝐹)‘𝑥) ∈ dom vol)) | |
29 | 27, 28 | sylibr 233 | . . . . 5 ⊢ (𝜑 → ([,) ∘ 𝐹):𝐴⟶dom vol) |
30 | fco 6752 | . . . . 5 ⊢ ((vol:dom vol⟶(0[,]+∞) ∧ ([,) ∘ 𝐹):𝐴⟶dom vol) → (vol ∘ ([,) ∘ 𝐹)):𝐴⟶(0[,]+∞)) | |
31 | 9, 29, 30 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (vol ∘ ([,) ∘ 𝐹)):𝐴⟶(0[,]+∞)) |
32 | coass 6276 | . . . . . 6 ⊢ ((vol ∘ [,)) ∘ 𝐹) = (vol ∘ ([,) ∘ 𝐹)) | |
33 | 32 | a1i 11 | . . . . 5 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (vol ∘ ([,) ∘ 𝐹))) |
34 | 33 | feq1d 6713 | . . . 4 ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞) ↔ (vol ∘ ([,) ∘ 𝐹)):𝐴⟶(0[,]+∞))) |
35 | 31, 34 | mpbird 256 | . . 3 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞)) |
36 | 35 | ffnd 6729 | . 2 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹) Fn 𝐴) |
37 | 19, 21 | voliooico 45613 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (vol‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥)))) = (vol‘((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))))) |
38 | 5, 4 | fssd 6745 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ* × ℝ*)) |
39 | 38 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶(ℝ* × ℝ*)) |
40 | 39, 15 | fvvolioof 45610 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((vol ∘ (,)) ∘ 𝐹)‘𝑥) = (vol‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))) |
41 | 39, 15 | fvvolicof 45612 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((vol ∘ [,)) ∘ 𝐹)‘𝑥) = (vol‘((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))))) |
42 | 37, 40, 41 | 3eqtr4d 2776 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((vol ∘ (,)) ∘ 𝐹)‘𝑥) = (((vol ∘ [,)) ∘ 𝐹)‘𝑥)) |
43 | 7, 36, 42 | eqfnfvd 7047 | 1 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = ((vol ∘ [,)) ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ⊆ wss 3947 𝒫 cpw 4607 × cxp 5680 dom cdm 5682 ∘ ccom 5686 Fn wfn 6549 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 1st c1st 8001 2nd c2nd 8002 ℝcr 11157 0cc0 11158 +∞cpnf 11295 ℝ*cxr 11297 (,)cioo 13378 [,)cico 13380 [,]cicc 13381 volcvol 25483 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-pm 8858 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fi 9454 df-sup 9485 df-inf 9486 df-oi 9553 df-dju 9944 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13146 df-xadd 13147 df-xmul 13148 df-ioo 13382 df-ico 13384 df-icc 13385 df-fz 13539 df-fzo 13682 df-fl 13812 df-seq 14022 df-exp 14082 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 df-rlim 15491 df-sum 15691 df-rest 17437 df-topgen 17458 df-psmet 21335 df-xmet 21336 df-met 21337 df-bl 21338 df-mopn 21339 df-top 22887 df-topon 22904 df-bases 22940 df-cmp 23382 df-ovol 25484 df-vol 25485 |
This theorem is referenced by: ovolval5lem3 46275 |
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