| 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 46002 | . . . . 5 ⊢ (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞) | |
| 2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞)) |
| 3 | rexpssxrxp 11306 | . . . . 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 45215 | . . 3 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹):𝐴⟶(0[,]+∞)) |
| 7 | 6 | ffnd 6737 | . 2 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) Fn 𝐴) |
| 8 | volf 25564 | . . . . . 6 ⊢ vol:dom vol⟶(0[,]+∞) | |
| 9 | 8 | a1i 11 | . . . . 5 ⊢ (𝜑 → vol:dom vol⟶(0[,]+∞)) |
| 10 | icof 45224 | . . . . . . . . . 10 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ* | |
| 11 | 10 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*) |
| 12 | 11, 4, 5 | fcoss 45215 | . . . . . . . 8 ⊢ (𝜑 → ([,) ∘ 𝐹):𝐴⟶𝒫 ℝ*) |
| 13 | 12 | ffnd 6737 | . . . . . . 7 ⊢ (𝜑 → ([,) ∘ 𝐹) Fn 𝐴) |
| 14 | 5 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶(ℝ × ℝ)) |
| 15 | simpr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
| 16 | 14, 15 | fvovco 45198 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (([,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥)))) |
| 17 | 5 | ffvelcdmda 7104 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (ℝ × ℝ)) |
| 18 | xp1st 8046 | . . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘𝑥)) ∈ ℝ) | |
| 19 | 17, 18 | syl 17 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (1st ‘(𝐹‘𝑥)) ∈ ℝ) |
| 20 | xp2nd 8047 | . . . . . . . . . . . 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 11311 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (2nd ‘(𝐹‘𝑥)) ∈ ℝ*) |
| 23 | icombl 25599 | . . . . . . . . . 10 ⊢ (((1st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ*) → ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))) ∈ dom vol) | |
| 24 | 19, 22, 23 | syl2anc 584 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))) ∈ dom vol) |
| 25 | 16, 24 | eqeltrd 2841 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (([,) ∘ 𝐹)‘𝑥) ∈ dom vol) |
| 26 | 25 | ralrimiva 3146 | . . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (([,) ∘ 𝐹)‘𝑥) ∈ dom vol) |
| 27 | 13, 26 | jca 511 | . . . . . 6 ⊢ (𝜑 → (([,) ∘ 𝐹) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (([,) ∘ 𝐹)‘𝑥) ∈ dom vol)) |
| 28 | ffnfv 7139 | . . . . . 6 ⊢ (([,) ∘ 𝐹):𝐴⟶dom vol ↔ (([,) ∘ 𝐹) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (([,) ∘ 𝐹)‘𝑥) ∈ dom vol)) | |
| 29 | 27, 28 | sylibr 234 | . . . . 5 ⊢ (𝜑 → ([,) ∘ 𝐹):𝐴⟶dom vol) |
| 30 | fco 6760 | . . . . 5 ⊢ ((vol:dom vol⟶(0[,]+∞) ∧ ([,) ∘ 𝐹):𝐴⟶dom vol) → (vol ∘ ([,) ∘ 𝐹)):𝐴⟶(0[,]+∞)) | |
| 31 | 9, 29, 30 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (vol ∘ ([,) ∘ 𝐹)):𝐴⟶(0[,]+∞)) |
| 32 | coass 6285 | . . . . . 6 ⊢ ((vol ∘ [,)) ∘ 𝐹) = (vol ∘ ([,) ∘ 𝐹)) | |
| 33 | 32 | a1i 11 | . . . . 5 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (vol ∘ ([,) ∘ 𝐹))) |
| 34 | 33 | feq1d 6720 | . . . 4 ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞) ↔ (vol ∘ ([,) ∘ 𝐹)):𝐴⟶(0[,]+∞))) |
| 35 | 31, 34 | mpbird 257 | . . 3 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞)) |
| 36 | 35 | ffnd 6737 | . 2 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹) Fn 𝐴) |
| 37 | 19, 21 | voliooico 46007 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (vol‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥)))) = (vol‘((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))))) |
| 38 | 5, 4 | fssd 6753 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ* × ℝ*)) |
| 39 | 38 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶(ℝ* × ℝ*)) |
| 40 | 39, 15 | fvvolioof 46004 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((vol ∘ (,)) ∘ 𝐹)‘𝑥) = (vol‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))) |
| 41 | 39, 15 | fvvolicof 46006 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((vol ∘ [,)) ∘ 𝐹)‘𝑥) = (vol‘((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))))) |
| 42 | 37, 40, 41 | 3eqtr4d 2787 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((vol ∘ (,)) ∘ 𝐹)‘𝑥) = (((vol ∘ [,)) ∘ 𝐹)‘𝑥)) |
| 43 | 7, 36, 42 | eqfnfvd 7054 | 1 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = ((vol ∘ [,)) ∘ 𝐹)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 𝒫 cpw 4600 × cxp 5683 dom cdm 5685 ∘ ccom 5689 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 2nd c2nd 8013 ℝcr 11154 0cc0 11155 +∞cpnf 11292 ℝ*cxr 11294 (,)cioo 13387 [,)cico 13389 [,]cicc 13390 volcvol 25498 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-rest 17467 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953 df-cmp 23395 df-ovol 25499 df-vol 25500 |
| This theorem is referenced by: ovolval5lem3 46669 |
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