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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 40997 | . . . . 5 ⊢ (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞) | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞)) |
3 | rexpssxrxp 10400 | . . . . 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 40207 | . . 3 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹):𝐴⟶(0[,]+∞)) |
7 | 6 | ffnd 6278 | . 2 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) Fn 𝐴) |
8 | volf 23694 | . . . . . 6 ⊢ vol:dom vol⟶(0[,]+∞) | |
9 | 8 | a1i 11 | . . . . 5 ⊢ (𝜑 → vol:dom vol⟶(0[,]+∞)) |
10 | icof 40216 | . . . . . . . . . 10 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ* | |
11 | 10 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*) |
12 | 11, 4, 5 | fcoss 40207 | . . . . . . . 8 ⊢ (𝜑 → ([,) ∘ 𝐹):𝐴⟶𝒫 ℝ*) |
13 | 12 | ffnd 6278 | . . . . . . 7 ⊢ (𝜑 → ([,) ∘ 𝐹) Fn 𝐴) |
14 | 5 | adantr 474 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶(ℝ × ℝ)) |
15 | simpr 479 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
16 | 14, 15 | fvovco 40188 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (([,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥)))) |
17 | 5 | ffvelrnda 6607 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (ℝ × ℝ)) |
18 | xp1st 7459 | . . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘𝑥)) ∈ ℝ) | |
19 | 17, 18 | syl 17 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (1st ‘(𝐹‘𝑥)) ∈ ℝ) |
20 | xp2nd 7460 | . . . . . . . . . . . 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 10405 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (2nd ‘(𝐹‘𝑥)) ∈ ℝ*) |
23 | icombl 23729 | . . . . . . . . . 10 ⊢ (((1st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ*) → ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))) ∈ dom vol) | |
24 | 19, 22, 23 | syl2anc 581 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))) ∈ dom vol) |
25 | 16, 24 | eqeltrd 2905 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (([,) ∘ 𝐹)‘𝑥) ∈ dom vol) |
26 | 25 | ralrimiva 3174 | . . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (([,) ∘ 𝐹)‘𝑥) ∈ dom vol) |
27 | 13, 26 | jca 509 | . . . . . 6 ⊢ (𝜑 → (([,) ∘ 𝐹) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (([,) ∘ 𝐹)‘𝑥) ∈ dom vol)) |
28 | ffnfv 6636 | . . . . . 6 ⊢ (([,) ∘ 𝐹):𝐴⟶dom vol ↔ (([,) ∘ 𝐹) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (([,) ∘ 𝐹)‘𝑥) ∈ dom vol)) | |
29 | 27, 28 | sylibr 226 | . . . . 5 ⊢ (𝜑 → ([,) ∘ 𝐹):𝐴⟶dom vol) |
30 | fco 6294 | . . . . 5 ⊢ ((vol:dom vol⟶(0[,]+∞) ∧ ([,) ∘ 𝐹):𝐴⟶dom vol) → (vol ∘ ([,) ∘ 𝐹)):𝐴⟶(0[,]+∞)) | |
31 | 9, 29, 30 | syl2anc 581 | . . . 4 ⊢ (𝜑 → (vol ∘ ([,) ∘ 𝐹)):𝐴⟶(0[,]+∞)) |
32 | coass 5894 | . . . . . 6 ⊢ ((vol ∘ [,)) ∘ 𝐹) = (vol ∘ ([,) ∘ 𝐹)) | |
33 | 32 | a1i 11 | . . . . 5 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (vol ∘ ([,) ∘ 𝐹))) |
34 | 33 | feq1d 6262 | . . . 4 ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞) ↔ (vol ∘ ([,) ∘ 𝐹)):𝐴⟶(0[,]+∞))) |
35 | 31, 34 | mpbird 249 | . . 3 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞)) |
36 | 35 | ffnd 6278 | . 2 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹) Fn 𝐴) |
37 | 19, 21 | voliooico 41002 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (vol‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥)))) = (vol‘((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))))) |
38 | 5, 4 | fssd 6291 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ* × ℝ*)) |
39 | 38 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶(ℝ* × ℝ*)) |
40 | 39, 15 | fvvolioof 40999 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((vol ∘ (,)) ∘ 𝐹)‘𝑥) = (vol‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))) |
41 | 39, 15 | fvvolicof 41001 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((vol ∘ [,)) ∘ 𝐹)‘𝑥) = (vol‘((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))))) |
42 | 37, 40, 41 | 3eqtr4d 2870 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((vol ∘ (,)) ∘ 𝐹)‘𝑥) = (((vol ∘ [,)) ∘ 𝐹)‘𝑥)) |
43 | 7, 36, 42 | eqfnfvd 6562 | 1 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = ((vol ∘ [,)) ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∀wral 3116 ⊆ wss 3797 𝒫 cpw 4377 × cxp 5339 dom cdm 5341 ∘ ccom 5345 Fn wfn 6117 ⟶wf 6118 ‘cfv 6122 (class class class)co 6904 1st c1st 7425 2nd c2nd 7426 ℝcr 10250 0cc0 10251 +∞cpnf 10387 ℝ*cxr 10389 (,)cioo 12462 [,)cico 12464 [,]cicc 12465 volcvol 23628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-inf2 8814 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-pre-sup 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-se 5301 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-isom 6131 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-of 7156 df-om 7326 df-1st 7427 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-2o 7826 df-oadd 7829 df-er 8008 df-map 8123 df-pm 8124 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-fi 8585 df-sup 8616 df-inf 8617 df-oi 8683 df-card 9077 df-cda 9304 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-3 11414 df-n0 11618 df-z 11704 df-uz 11968 df-q 12071 df-rp 12112 df-xneg 12231 df-xadd 12232 df-xmul 12233 df-ioo 12466 df-ico 12468 df-icc 12469 df-fz 12619 df-fzo 12760 df-fl 12887 df-seq 13095 df-exp 13154 df-hash 13410 df-cj 14215 df-re 14216 df-im 14217 df-sqrt 14351 df-abs 14352 df-clim 14595 df-rlim 14596 df-sum 14793 df-rest 16435 df-topgen 16456 df-psmet 20097 df-xmet 20098 df-met 20099 df-bl 20100 df-mopn 20101 df-top 21068 df-topon 21085 df-bases 21120 df-cmp 21560 df-ovol 23629 df-vol 23630 |
This theorem is referenced by: ovolval5lem3 41661 |
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