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Theorem rnxpid 5109
Description: The range of a square cross product. (Contributed by FL, 17-May-2010.)
Assertion
Ref Expression
rnxpid  |-  ran  ( A  X.  A )  =  A

Proof of Theorem rnxpid
StepHypRef Expression
1 rn0 4936 . . 3  |-  ran  (/)  =  (/)
2 xpeq2 4704 . . . . 5  |-  ( A  =  (/)  ->  ( A  X.  A )  =  ( A  X.  (/) ) )
3 xp0 5098 . . . . 5  |-  ( A  X.  (/) )  =  (/)
42, 3syl6eq 2331 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  A )  =  (/) )
54rneqd 4906 . . 3  |-  ( A  =  (/)  ->  ran  ( A  X.  A )  =  ran  (/) )
6 id 19 . . 3  |-  ( A  =  (/)  ->  A  =  (/) )
71, 5, 63eqtr4a 2341 . 2  |-  ( A  =  (/)  ->  ran  ( A  X.  A )  =  A )
8 rnxp 5106 . 2  |-  ( A  =/=  (/)  ->  ran  ( A  X.  A )  =  A )
97, 8pm2.61ine 2522 1  |-  ran  ( A  X.  A )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1623   (/)c0 3455    X. cxp 4687   ran crn 4690
This theorem is referenced by:  sofld  5121  fpwwe2lem13  8264  ovoliunlem1  18861  mgmrddd  25366  relrded  25742  relrcat  25763
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-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-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700
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