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Theorem unixpid 5207
Description: Field of a square cross product. (Contributed by FL, 10-Oct-2009.)
Assertion
Ref Expression
unixpid  |-  U. U. ( A  X.  A
)  =  A

Proof of Theorem unixpid
StepHypRef Expression
1 xpeq1 4703 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  A )  =  ( (/)  X.  A
) )
2 xp0r 4768 . . . 4  |-  ( (/)  X.  A )  =  (/)
31, 2syl6eq 2331 . . 3  |-  ( A  =  (/)  ->  ( A  X.  A )  =  (/) )
4 unieq 3836 . . . . 5  |-  ( ( A  X.  A )  =  (/)  ->  U. ( A  X.  A )  = 
U. (/) )
54unieqd 3838 . . . 4  |-  ( ( A  X.  A )  =  (/)  ->  U. U. ( A  X.  A
)  =  U. U. (/) )
6 uni0 3854 . . . . . 6  |-  U. (/)  =  (/)
76unieqi 3837 . . . . 5  |-  U. U. (/)  =  U. (/)
87, 6eqtri 2303 . . . 4  |-  U. U. (/)  =  (/)
9 eqtr 2300 . . . . 5  |-  ( ( U. U. ( A  X.  A )  = 
U. U. (/)  /\  U. U. (/)  =  (/) )  ->  U. U. ( A  X.  A
)  =  (/) )
10 eqtr 2300 . . . . . . 7  |-  ( ( U. U. ( A  X.  A )  =  (/)  /\  (/)  =  A )  ->  U. U. ( A  X.  A )  =  A )
1110expcom 424 . . . . . 6  |-  ( (/)  =  A  ->  ( U. U. ( A  X.  A
)  =  (/)  ->  U. U. ( A  X.  A
)  =  A ) )
1211eqcoms 2286 . . . . 5  |-  ( A  =  (/)  ->  ( U. U. ( A  X.  A
)  =  (/)  ->  U. U. ( A  X.  A
)  =  A ) )
139, 12syl5com 26 . . . 4  |-  ( ( U. U. ( A  X.  A )  = 
U. U. (/)  /\  U. U. (/)  =  (/) )  ->  ( A  =  (/)  ->  U. U. ( A  X.  A
)  =  A ) )
145, 8, 13sylancl 643 . . 3  |-  ( ( A  X.  A )  =  (/)  ->  ( A  =  (/)  ->  U. U. ( A  X.  A
)  =  A ) )
153, 14mpcom 32 . 2  |-  ( A  =  (/)  ->  U. U. ( A  X.  A
)  =  A )
16 df-ne 2448 . . 3  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
17 xpnz 5099 . . . 4  |-  ( ( A  =/=  (/)  /\  A  =/=  (/) )  <->  ( A  X.  A )  =/=  (/) )
18 unixp 5205 . . . . 5  |-  ( ( A  X.  A )  =/=  (/)  ->  U. U. ( A  X.  A )  =  ( A  u.  A
) )
19 unidm 3318 . . . . 5  |-  ( A  u.  A )  =  A
2018, 19syl6eq 2331 . . . 4  |-  ( ( A  X.  A )  =/=  (/)  ->  U. U. ( A  X.  A )  =  A )
2117, 20sylbi 187 . . 3  |-  ( ( A  =/=  (/)  /\  A  =/=  (/) )  ->  U. U. ( A  X.  A
)  =  A )
2216, 16, 21sylancbr 647 . 2  |-  ( -.  A  =  (/)  ->  U. U. ( A  X.  A
)  =  A )
2315, 22pm2.61i 156 1  |-  U. U. ( A  X.  A
)  =  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    =/= wne 2446    u. cun 3150   (/)c0 3455   U.cuni 3827    X. cxp 4687
This theorem is referenced by:  psss  14323  scprefat  25071  scprefat2  25072
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-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  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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