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Theorem unixpid 5363
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 4851 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  A )  =  ( (/)  X.  A
) )
2 xp0r 4915 . . . 4  |-  ( (/)  X.  A )  =  (/)
31, 2syl6eq 2452 . . 3  |-  ( A  =  (/)  ->  ( A  X.  A )  =  (/) )
4 unieq 3984 . . . . 5  |-  ( ( A  X.  A )  =  (/)  ->  U. ( A  X.  A )  = 
U. (/) )
54unieqd 3986 . . . 4  |-  ( ( A  X.  A )  =  (/)  ->  U. U. ( A  X.  A
)  =  U. U. (/) )
6 uni0 4002 . . . . . 6  |-  U. (/)  =  (/)
76unieqi 3985 . . . . 5  |-  U. U. (/)  =  U. (/)
87, 6eqtri 2424 . . . 4  |-  U. U. (/)  =  (/)
9 eqtr 2421 . . . . 5  |-  ( ( U. U. ( A  X.  A )  = 
U. U. (/)  /\  U. U. (/)  =  (/) )  ->  U. U. ( A  X.  A
)  =  (/) )
10 eqtr 2421 . . . . . . 7  |-  ( ( U. U. ( A  X.  A )  =  (/)  /\  (/)  =  A )  ->  U. U. ( A  X.  A )  =  A )
1110expcom 425 . . . . . 6  |-  ( (/)  =  A  ->  ( U. U. ( A  X.  A
)  =  (/)  ->  U. U. ( A  X.  A
)  =  A ) )
1211eqcoms 2407 . . . . 5  |-  ( A  =  (/)  ->  ( U. U. ( A  X.  A
)  =  (/)  ->  U. U. ( A  X.  A
)  =  A ) )
139, 12syl5com 28 . . . 4  |-  ( ( U. U. ( A  X.  A )  = 
U. U. (/)  /\  U. U. (/)  =  (/) )  ->  ( A  =  (/)  ->  U. U. ( A  X.  A
)  =  A ) )
145, 8, 13sylancl 644 . . 3  |-  ( ( A  X.  A )  =  (/)  ->  ( A  =  (/)  ->  U. U. ( A  X.  A
)  =  A ) )
153, 14mpcom 34 . 2  |-  ( A  =  (/)  ->  U. U. ( A  X.  A
)  =  A )
16 df-ne 2569 . . 3  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
17 xpnz 5251 . . . 4  |-  ( ( A  =/=  (/)  /\  A  =/=  (/) )  <->  ( A  X.  A )  =/=  (/) )
18 unixp 5361 . . . . 5  |-  ( ( A  X.  A )  =/=  (/)  ->  U. U. ( A  X.  A )  =  ( A  u.  A
) )
19 unidm 3450 . . . . 5  |-  ( A  u.  A )  =  A
2018, 19syl6eq 2452 . . . 4  |-  ( ( A  X.  A )  =/=  (/)  ->  U. U. ( A  X.  A )  =  A )
2117, 20sylbi 188 . . 3  |-  ( ( A  =/=  (/)  /\  A  =/=  (/) )  ->  U. U. ( A  X.  A
)  =  A )
2216, 16, 21sylancbr 648 . 2  |-  ( -.  A  =  (/)  ->  U. U. ( A  X.  A
)  =  A )
2315, 22pm2.61i 158 1  |-  U. U. ( A  X.  A
)  =  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    =/= wne 2567    u. cun 3278   (/)c0 3588   U.cuni 3975    X. cxp 4835
This theorem is referenced by:  psss  14601
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848
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