![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 2908 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2908 | . . . 4 ⊢ Ⅎ𝑥(vol ∘ (,)) | |
3 | volioofmpt.x | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
4 | 2, 3 | nfco 5826 | . . 3 ⊢ Ⅎ𝑥((vol ∘ (,)) ∘ 𝐹) |
5 | volioof 44302 | . . . . 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 6697 | . . . 4 ⊢ (((vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞) ∧ 𝐹:𝐴⟶(ℝ* × ℝ*)) → ((vol ∘ (,)) ∘ 𝐹):𝐴⟶(0[,]+∞)) | |
9 | 6, 7, 8 | syl2anc 585 | . . 3 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹):𝐴⟶(0[,]+∞)) |
10 | 1, 4, 9 | feqmptdf 6917 | . 2 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (((vol ∘ (,)) ∘ 𝐹)‘𝑥))) |
11 | 7 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶(ℝ* × ℝ*)) |
12 | simpr 486 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
13 | 11, 12 | fvvolioof 44304 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((vol ∘ (,)) ∘ 𝐹)‘𝑥) = (vol‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))) |
14 | 13 | mpteq2dva 5210 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (((vol ∘ (,)) ∘ 𝐹)‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (vol‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥)))))) |
15 | 10, 14 | eqtrd 2777 | 1 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (vol‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Ⅎwnfc 2888 ↦ cmpt 5193 × cxp 5636 ∘ ccom 5642 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 1st c1st 7924 2nd c2nd 7925 0cc0 11058 +∞cpnf 11193 ℝ*cxr 11195 (,)cioo 13271 [,]cicc 13274 volcvol 24843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-map 8774 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-oi 9453 df-dju 9844 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-q 12881 df-rp 12923 df-xadd 13041 df-ioo 13275 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-rlim 15378 df-sum 15578 df-xmet 20805 df-met 20806 df-ovol 24844 df-vol 24845 |
This theorem is referenced by: ovolval5lem2 44968 |
Copyright terms: Public domain | W3C validator |