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 23104 | . 2 ⊢ vol:dom vol⟶(0[,]+∞) | |
2 | ioof 12142 | . . . . 5 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
3 | ffn 5958 | . . . . 5 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ (,) Fn (ℝ* × ℝ*) |
5 | df-ov 6552 | . . . . . . . 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 7096 | . . . . . . . . 9 ⊢ (𝑥 ∈ (ℝ* × ℝ*) → 𝑥 = 〈(1st ‘𝑥), (2nd ‘𝑥)〉) | |
8 | 7 | eqcomd 2616 | . . . . . . . 8 ⊢ (𝑥 ∈ (ℝ* × ℝ*) → 〈(1st ‘𝑥), (2nd ‘𝑥)〉 = 𝑥) |
9 | 8 | fveq2d 6107 | . . . . . . 7 ⊢ (𝑥 ∈ (ℝ* × ℝ*) → ((,)‘〈(1st ‘𝑥), (2nd ‘𝑥)〉) = ((,)‘𝑥)) |
10 | 6, 9 | eqtr2d 2645 | . . . . . 6 ⊢ (𝑥 ∈ (ℝ* × ℝ*) → ((,)‘𝑥) = ((1st ‘𝑥)(,)(2nd ‘𝑥))) |
11 | ioombl 23140 | . . . . . 6 ⊢ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ∈ dom vol | |
12 | 10, 11 | syl6eqel 2696 | . . . . 5 ⊢ (𝑥 ∈ (ℝ* × ℝ*) → ((,)‘𝑥) ∈ dom vol) |
13 | 12 | rgen 2906 | . . . 4 ⊢ ∀𝑥 ∈ (ℝ* × ℝ*)((,)‘𝑥) ∈ dom vol |
14 | 4, 13 | pm3.2i 470 | . . 3 ⊢ ((,) Fn (ℝ* × ℝ*) ∧ ∀𝑥 ∈ (ℝ* × ℝ*)((,)‘𝑥) ∈ dom vol) |
15 | ffnfv 6295 | . . 3 ⊢ ((,):(ℝ* × ℝ*)⟶dom vol ↔ ((,) Fn (ℝ* × ℝ*) ∧ ∀𝑥 ∈ (ℝ* × ℝ*)((,)‘𝑥) ∈ dom vol)) | |
16 | 14, 15 | mpbir 220 | . 2 ⊢ (,):(ℝ* × ℝ*)⟶dom vol |
17 | fco 5971 | . 2 ⊢ ((vol:dom vol⟶(0[,]+∞) ∧ (,):(ℝ* × ℝ*)⟶dom vol) → (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞)) | |
18 | 1, 16, 17 | mp2an 704 | 1 ⊢ (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 𝒫 cpw 4108 〈cop 4131 × 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 [,]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-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-xadd 11823 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-xmet 19560 df-met 19561 df-ovol 23040 df-vol 23041 |
This theorem is referenced by: volioofmpt 38887 voliooicof 38889 ovolval3 39537 ovolval5lem2 39543 |
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