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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > volioofmpt | Structured version Visualization version GIF version |
Description: ((vol ∘ (,)) ∘ 𝐹) expressed in maps-to notation. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
volioofmpt.x | ⊢ Ⅎ𝑥𝐹 |
volioofmpt.f | ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ* × ℝ*)) |
Ref | Expression |
---|---|
volioofmpt | ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (vol‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2969 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2969 | . . . 4 ⊢ Ⅎ𝑥(vol ∘ (,)) | |
3 | volioofmpt.x | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
4 | 2, 3 | nfco 5524 | . . 3 ⊢ Ⅎ𝑥((vol ∘ (,)) ∘ 𝐹) |
5 | volioof 40992 | . . . . 5 ⊢ (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞) | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝜑 → (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞)) |
7 | volioofmpt.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ* × ℝ*)) | |
8 | fco 6299 | . . . 4 ⊢ (((vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞) ∧ 𝐹:𝐴⟶(ℝ* × ℝ*)) → ((vol ∘ (,)) ∘ 𝐹):𝐴⟶(0[,]+∞)) | |
9 | 6, 7, 8 | syl2anc 579 | . . 3 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹):𝐴⟶(0[,]+∞)) |
10 | 1, 4, 9 | feqmptdf 6502 | . 2 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (((vol ∘ (,)) ∘ 𝐹)‘𝑥))) |
11 | 7 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶(ℝ* × ℝ*)) |
12 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
13 | 11, 12 | fvvolioof 40994 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((vol ∘ (,)) ∘ 𝐹)‘𝑥) = (vol‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))) |
14 | 13 | mpteq2dva 4969 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (((vol ∘ (,)) ∘ 𝐹)‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (vol‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥)))))) |
15 | 10, 14 | eqtrd 2861 | 1 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (vol‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 Ⅎwnfc 2956 ↦ cmpt 4954 × cxp 5344 ∘ ccom 5350 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 1st c1st 7431 2nd c2nd 7432 0cc0 10259 +∞cpnf 10395 ℝ*cxr 10397 (,)cioo 12470 [,]cicc 12473 volcvol 23636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-2o 7832 df-oadd 7835 df-er 8014 df-map 8129 df-pm 8130 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-sup 8623 df-inf 8624 df-oi 8691 df-card 9085 df-cda 9312 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-n0 11626 df-z 11712 df-uz 11976 df-q 12079 df-rp 12120 df-xadd 12240 df-ioo 12474 df-ico 12476 df-icc 12477 df-fz 12627 df-fzo 12768 df-fl 12895 df-seq 13103 df-exp 13162 df-hash 13418 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-clim 14603 df-rlim 14604 df-sum 14801 df-xmet 20106 df-met 20107 df-ovol 23637 df-vol 23638 |
This theorem is referenced by: ovolval5lem2 41655 |
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