Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > xpeq2d | Structured version Visualization version GIF version |
Description: Equality deduction for Cartesian product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
Ref | Expression |
---|---|
xpeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
xpeq2d | ⊢ (𝜑 → (𝐶 × 𝐴) = (𝐶 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | xpeq2 5053 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐶 × 𝐴) = (𝐶 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 × cxp 5036 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-opab 4644 df-xp 5044 |
This theorem is referenced by: xpriindi 5180 csbres 5320 fconstg 6005 curry2 7159 fparlem4 7167 fvdiagfn 7788 mapsncnv 7790 xpsneng 7930 xpcdaen 8888 axdc4lem 9160 fpwwe2lem13 9343 expval 12724 imasvscafn 16020 comfffval 16181 fuchom 16444 homafval 16502 setcmon 16560 xpccofval 16645 pwsco2mhm 17194 frmdplusg 17214 mulgfval 17365 mulgval 17366 symgplusg 17632 efgval 17953 psrplusg 19202 psrvscafval 19211 psrvsca 19212 opsrle 19296 evlssca 19343 mpfind 19357 coe1fv 19397 coe1tm 19464 pf1ind 19540 pjfval 19869 frlmval 19911 islindf5 19997 mdetunilem4 20240 mdetunilem9 20245 txindislem 21246 txcmplem2 21255 txhaus 21260 txkgen 21265 xkofvcn 21297 xkoinjcn 21300 cnextval 21675 cnextfval 21676 pcorev2 22636 pcophtb 22637 pi1grplem 22657 pi1inv 22660 dvfval 23467 dvnfval 23491 0dgrb 23806 dgrnznn 23807 dgreq0 23825 dgrmulc 23831 plyrem 23864 facth 23865 fta1 23867 aaliou2 23899 taylfval 23917 taylpfval 23923 0ofval 27026 aciunf1 28845 indval2 29404 sxbrsigalem3 29661 sxbrsigalem2 29675 eulerpartlemgu 29766 sseqval 29777 sconpht 30465 sconpht2 30474 sconpi1 30475 cvmlift2lem11 30549 cvmlift2lem12 30550 cvmlift2lem13 30551 cvmlift3lem9 30563 mexval 30653 mexval2 30654 mdvval 30655 mpstval 30686 elima4 30924 bj-xtageq 32169 matunitlindflem1 32575 poimirlem32 32611 ismrer1 32807 lflsc0N 33388 lkrscss 33403 lfl1dim 33426 lfl1dim2N 33427 ldualvs 33442 mzpclval 36306 mzpcl1 36310 mendvsca 36780 dvconstbi 37555 expgrowth 37556 csbresgOLD 38077 |
Copyright terms: Public domain | W3C validator |