Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prjcp1 Unicode version

Theorem prjcp1 25084
Description: Projection of a cross product. (Contributed by FL, 5-Oct-2009.)
Assertion
Ref Expression
prjcp1  |-  ( B  =/=  (/)  ->  ( 1st " ( A  X.  B
) )  =  A )

Proof of Theorem prjcp1
StepHypRef Expression
1 relxp 4794 . . 3  |-  Rel  ( A  X.  B )
2 prjdmn 25082 . . 3  |-  ( Rel  ( A  X.  B
)  ->  ( 1st " ( A  X.  B
) )  =  dom  ( A  X.  B
) )
31, 2mp1i 11 . 2  |-  ( B  =/=  (/)  ->  ( 1st " ( A  X.  B
) )  =  dom  ( A  X.  B
) )
4 dmxp 4897 . 2  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
53, 4eqtrd 2315 1  |-  ( B  =/=  (/)  ->  ( 1st " ( A  X.  B
) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    =/= wne 2446   (/)c0 3455    X. cxp 4687   dom cdm 4689   "cima 4692   Rel wrel 4694   1stc1st 6120
This theorem is referenced by:  prjpacp1  25127
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123
  Copyright terms: Public domain W3C validator