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Theorem rnxpid 5305
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 5130 . . 3  |-  ran  (/)  =  (/)
2 xpeq2 4896 . . . . 5  |-  ( A  =  (/)  ->  ( A  X.  A )  =  ( A  X.  (/) ) )
3 xp0 5294 . . . . 5  |-  ( A  X.  (/) )  =  (/)
42, 3syl6eq 2486 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  A )  =  (/) )
54rneqd 5100 . . 3  |-  ( A  =  (/)  ->  ran  ( A  X.  A )  =  ran  (/) )
6 id 21 . . 3  |-  ( A  =  (/)  ->  A  =  (/) )
71, 5, 63eqtr4a 2496 . 2  |-  ( A  =  (/)  ->  ran  ( A  X.  A )  =  A )
8 rnxp 5302 . 2  |-  ( A  =/=  (/)  ->  ran  ( A  X.  A )  =  A )
97, 8pm2.61ine 2682 1  |-  ran  ( A  X.  A )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1653   (/)c0 3630    X. cxp 4879   ran crn 4882
This theorem is referenced by:  sofld  5321  fpwwe2lem13  8522  ustimasn  18263  utopbas  18270  restutop  18272  ovoliunlem1  19403  metideq  24293  mblfinlem1  26255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-dm 4891  df-rn 4892
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