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 38880 | . . . . 5 ⊢ (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞) | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞)) |
3 | rexpssxrxp 9963 | . . . . 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 38397 | . . 3 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹):𝐴⟶(0[,]+∞)) |
7 | ffn 5958 | . . 3 ⊢ (((vol ∘ (,)) ∘ 𝐹):𝐴⟶(0[,]+∞) → ((vol ∘ (,)) ∘ 𝐹) Fn 𝐴) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) Fn 𝐴) |
9 | volf 23104 | . . . . . 6 ⊢ vol:dom vol⟶(0[,]+∞) | |
10 | 9 | a1i 11 | . . . . 5 ⊢ (𝜑 → vol:dom vol⟶(0[,]+∞)) |
11 | icof 38406 | . . . . . . . . . 10 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ* | |
12 | 11 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*) |
13 | 12, 4, 5 | fcoss 38397 | . . . . . . . 8 ⊢ (𝜑 → ([,) ∘ 𝐹):𝐴⟶𝒫 ℝ*) |
14 | ffn 5958 | . . . . . . . 8 ⊢ (([,) ∘ 𝐹):𝐴⟶𝒫 ℝ* → ([,) ∘ 𝐹) Fn 𝐴) | |
15 | 13, 14 | syl 17 | . . . . . . 7 ⊢ (𝜑 → ([,) ∘ 𝐹) Fn 𝐴) |
16 | 5 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶(ℝ × ℝ)) |
17 | simpr 476 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
18 | 16, 17 | fvovco 38376 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (([,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥)))) |
19 | 5 | ffvelrnda 6267 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (ℝ × ℝ)) |
20 | xp1st 7089 | . . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘𝑥)) ∈ ℝ) | |
21 | 19, 20 | syl 17 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (1st ‘(𝐹‘𝑥)) ∈ ℝ) |
22 | xp2nd 7090 | . . . . . . . . . . . 12 ⊢ ((𝐹‘𝑥) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘𝑥)) ∈ ℝ) | |
23 | 19, 22 | syl 17 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (2nd ‘(𝐹‘𝑥)) ∈ ℝ) |
24 | 23 | rexrd 9968 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (2nd ‘(𝐹‘𝑥)) ∈ ℝ*) |
25 | icombl 23139 | . . . . . . . . . 10 ⊢ (((1st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ*) → ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))) ∈ dom vol) | |
26 | 21, 24, 25 | syl2anc 691 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))) ∈ dom vol) |
27 | 18, 26 | eqeltrd 2688 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (([,) ∘ 𝐹)‘𝑥) ∈ dom vol) |
28 | 27 | ralrimiva 2949 | . . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (([,) ∘ 𝐹)‘𝑥) ∈ dom vol) |
29 | 15, 28 | jca 553 | . . . . . 6 ⊢ (𝜑 → (([,) ∘ 𝐹) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (([,) ∘ 𝐹)‘𝑥) ∈ dom vol)) |
30 | ffnfv 6295 | . . . . . 6 ⊢ (([,) ∘ 𝐹):𝐴⟶dom vol ↔ (([,) ∘ 𝐹) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (([,) ∘ 𝐹)‘𝑥) ∈ dom vol)) | |
31 | 29, 30 | sylibr 223 | . . . . 5 ⊢ (𝜑 → ([,) ∘ 𝐹):𝐴⟶dom vol) |
32 | fco 5971 | . . . . 5 ⊢ ((vol:dom vol⟶(0[,]+∞) ∧ ([,) ∘ 𝐹):𝐴⟶dom vol) → (vol ∘ ([,) ∘ 𝐹)):𝐴⟶(0[,]+∞)) | |
33 | 10, 31, 32 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (vol ∘ ([,) ∘ 𝐹)):𝐴⟶(0[,]+∞)) |
34 | coass 5571 | . . . . . 6 ⊢ ((vol ∘ [,)) ∘ 𝐹) = (vol ∘ ([,) ∘ 𝐹)) | |
35 | 34 | a1i 11 | . . . . 5 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (vol ∘ ([,) ∘ 𝐹))) |
36 | 35 | feq1d 5943 | . . . 4 ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞) ↔ (vol ∘ ([,) ∘ 𝐹)):𝐴⟶(0[,]+∞))) |
37 | 33, 36 | mpbird 246 | . . 3 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞)) |
38 | ffn 5958 | . . 3 ⊢ (((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞) → ((vol ∘ [,)) ∘ 𝐹) Fn 𝐴) | |
39 | 37, 38 | syl 17 | . 2 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹) Fn 𝐴) |
40 | 21, 23 | voliooico 38885 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (vol‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥)))) = (vol‘((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))))) |
41 | 5, 4 | fssd 5970 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ* × ℝ*)) |
42 | 41 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶(ℝ* × ℝ*)) |
43 | 42, 17 | fvvolioof 38882 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((vol ∘ (,)) ∘ 𝐹)‘𝑥) = (vol‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))) |
44 | 42, 17 | fvvolicof 38884 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((vol ∘ [,)) ∘ 𝐹)‘𝑥) = (vol‘((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))))) |
45 | 40, 43, 44 | 3eqtr4d 2654 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((vol ∘ (,)) ∘ 𝐹)‘𝑥) = (((vol ∘ [,)) ∘ 𝐹)‘𝑥)) |
46 | 8, 39, 45 | eqfnfvd 6222 | 1 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = ((vol ∘ [,)) ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 𝒫 cpw 4108 × cxp 5036 dom cdm 5038 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 ℝcr 9814 0cc0 9815 +∞cpnf 9950 ℝ*cxr 9952 (,)cioo 12046 [,)cico 12048 [,]cicc 12049 volcvol 23039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-rest 15906 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-cmp 21000 df-ovol 23040 df-vol 23041 |
This theorem is referenced by: ovolval5lem3 39544 |
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