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Theorem rnxpid 5191
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 5018 . . 3  |-  ran  (/)  =  (/)
2 xpeq2 4786 . . . . 5  |-  ( A  =  (/)  ->  ( A  X.  A )  =  ( A  X.  (/) ) )
3 xp0 5180 . . . . 5  |-  ( A  X.  (/) )  =  (/)
42, 3syl6eq 2406 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  A )  =  (/) )
54rneqd 4988 . . 3  |-  ( A  =  (/)  ->  ran  ( A  X.  A )  =  ran  (/) )
6 id 19 . . 3  |-  ( A  =  (/)  ->  A  =  (/) )
71, 5, 63eqtr4a 2416 . 2  |-  ( A  =  (/)  ->  ran  ( A  X.  A )  =  A )
8 rnxp 5188 . 2  |-  ( A  =/=  (/)  ->  ran  ( A  X.  A )  =  A )
97, 8pm2.61ine 2597 1  |-  ran  ( A  X.  A )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1642   (/)c0 3531    X. cxp 4769   ran crn 4772
This theorem is referenced by:  sofld  5203  fpwwe2lem13  8354  ovoliunlem1  18965  ustimasn  23532  utopbas  23539  restutop  23541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-xp 4777  df-rel 4778  df-cnv 4779  df-dm 4781  df-rn 4782
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