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Mirrors > Home > MPE Home > Th. List > sqxpeqd | Structured version Visualization version GIF version |
Description: Equality deduction for a Cartesian square, see Wikipedia "Cartesian product", https://en.wikipedia.org/wiki/Cartesian_product#n-ary_Cartesian_power. (Contributed by AV, 13-Jan-2020.) |
Ref | Expression |
---|---|
xpeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
sqxpeqd | ⊢ (𝜑 → (𝐴 × 𝐴) = (𝐵 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1, 1 | xpeq12d 5064 | 1 ⊢ (𝜑 → (𝐴 × 𝐴) = (𝐵 × 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: xpcoid 5593 hartogslem1 8330 isfin6 9005 fpwwe2cbv 9331 fpwwe2lem2 9333 fpwwe2lem3 9334 fpwwe2lem8 9338 fpwwe2lem12 9342 fpwwe2lem13 9343 fpwwe2 9344 fpwwecbv 9345 fpwwelem 9346 canthwelem 9351 canthwe 9352 pwfseqlem4 9363 prdsval 15938 imasval 15994 imasaddfnlem 16011 comfffval 16181 comfeq 16189 oppcval 16196 sscfn1 16300 sscfn2 16301 isssc 16303 ssceq 16309 reschomf 16314 isfunc 16347 idfuval 16359 funcres 16379 funcpropd 16383 fucval 16441 fucpropd 16460 homafval 16502 setcval 16550 catcval 16569 estrcval 16587 estrchomfeqhom 16599 hofval 16715 hofpropd 16730 islat 16870 istsr 17040 cnvtsr 17045 isdir 17055 tsrdir 17061 intopsn 17076 frmdval 17211 resgrpplusfrn 17259 opsrval 19295 matval 20036 ustval 21816 trust 21843 utop2nei 21864 utop3cls 21865 utopreg 21866 ussval 21873 ressuss 21877 tususs 21884 fmucnd 21906 cfilufg 21907 trcfilu 21908 neipcfilu 21910 ispsmet 21919 prdsdsf 21982 prdsxmet 21984 ressprdsds 21986 xpsdsfn2 21993 xpsxmetlem 21994 xpsmet 21997 isxms 22062 isms 22064 xmspropd 22088 mspropd 22089 setsxms 22094 setsms 22095 imasf1oxms 22104 imasf1oms 22105 ressxms 22140 ressms 22141 prdsxmslem2 22144 metuval 22164 nmpropd2 22209 ngppropd 22251 tngngp2 22266 pi1addf 22655 pi1addval 22656 iscms 22950 cmspropd 22954 cmsss 22955 rrxds 22989 rrxmfval 22997 minveclem3a 23006 dvlip2 23562 dchrval 24759 brcgr 25580 issh 27449 qtophaus 29231 prsssdm 29291 ordtrestNEW 29295 ordtrest2NEW 29297 isrrext 29372 sibfof 29729 mdvval 30655 msrval 30689 mthmpps 30733 funtransport 31308 fvtransport 31309 bj-diagval 32267 prdsbnd2 32764 cnpwstotbnd 32766 isrngo 32866 isrngod 32867 rngosn3 32893 isdivrngo 32919 drngoi 32920 isgrpda 32924 ldualset 33430 aomclem8 36649 intopval 41628 rngcvalALTV 41753 rngcval 41754 rnghmsubcsetclem1 41767 rngccat 41770 rngchomrnghmresALTV 41788 ringcvalALTV 41799 ringcval 41800 rhmsubcsetclem1 41813 ringccat 41816 rhmsubcrngclem1 41819 rhmsubcrngc 41821 srhmsubc 41868 rhmsubc 41882 srhmsubcALTV 41887 elpglem3 42255 |
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