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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 25476 | . 2 ⊢ vol:dom vol⟶(0[,]+∞) | |
2 | ioof 13462 | . . . . 5 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
3 | ffn 6725 | . . . . 5 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ (,) Fn (ℝ* × ℝ*) |
5 | df-ov 7427 | . . . . . . . 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 8036 | . . . . . . . . 9 ⊢ (𝑥 ∈ (ℝ* × ℝ*) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) | |
8 | 7 | eqcomd 2733 | . . . . . . . 8 ⊢ (𝑥 ∈ (ℝ* × ℝ*) → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = 𝑥) |
9 | 8 | fveq2d 6904 | . . . . . . 7 ⊢ (𝑥 ∈ (ℝ* × ℝ*) → ((,)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) = ((,)‘𝑥)) |
10 | 6, 9 | eqtr2d 2768 | . . . . . 6 ⊢ (𝑥 ∈ (ℝ* × ℝ*) → ((,)‘𝑥) = ((1st ‘𝑥)(,)(2nd ‘𝑥))) |
11 | ioombl 25512 | . . . . . 6 ⊢ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ∈ dom vol | |
12 | 10, 11 | eqeltrdi 2836 | . . . . 5 ⊢ (𝑥 ∈ (ℝ* × ℝ*) → ((,)‘𝑥) ∈ dom vol) |
13 | 12 | rgen 3059 | . . . 4 ⊢ ∀𝑥 ∈ (ℝ* × ℝ*)((,)‘𝑥) ∈ dom vol |
14 | 4, 13 | pm3.2i 469 | . . 3 ⊢ ((,) Fn (ℝ* × ℝ*) ∧ ∀𝑥 ∈ (ℝ* × ℝ*)((,)‘𝑥) ∈ dom vol) |
15 | ffnfv 7132 | . . 3 ⊢ ((,):(ℝ* × ℝ*)⟶dom vol ↔ ((,) Fn (ℝ* × ℝ*) ∧ ∀𝑥 ∈ (ℝ* × ℝ*)((,)‘𝑥) ∈ dom vol)) | |
16 | 14, 15 | mpbir 230 | . 2 ⊢ (,):(ℝ* × ℝ*)⟶dom vol |
17 | fco 6750 | . 2 ⊢ ((vol:dom vol⟶(0[,]+∞) ∧ (,):(ℝ* × ℝ*)⟶dom vol) → (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞)) | |
18 | 1, 16, 17 | mp2an 690 | 1 ⊢ (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3057 𝒫 cpw 4604 ⟨cop 4636 × cxp 5678 dom cdm 5680 ∘ ccom 5684 Fn wfn 6546 ⟶wf 6547 ‘cfv 6551 (class class class)co 7424 1st c1st 7995 2nd c2nd 7996 ℝcr 11143 0cc0 11144 +∞cpnf 11281 ℝ*cxr 11283 (,)cioo 13362 [,]cicc 13365 volcvol 25410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-inf2 9670 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-2o 8492 df-er 8729 df-map 8851 df-pm 8852 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-sup 9471 df-inf 9472 df-oi 9539 df-dju 9930 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-n0 12509 df-z 12595 df-uz 12859 df-q 12969 df-rp 13013 df-xadd 13131 df-ioo 13366 df-ico 13368 df-icc 13369 df-fz 13523 df-fzo 13666 df-fl 13795 df-seq 14005 df-exp 14065 df-hash 14328 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-clim 15470 df-rlim 15471 df-sum 15671 df-xmet 21277 df-met 21278 df-ovol 25411 df-vol 25412 |
This theorem is referenced by: volioofmpt 45384 voliooicof 45386 ovolval3 46037 ovolval5lem2 46043 |
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