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Theorem unixpid 5433
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 4921 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  A )  =  ( (/)  X.  A
) )
2 xp0r 4985 . . . 4  |-  ( (/)  X.  A )  =  (/)
31, 2syl6eq 2490 . . 3  |-  ( A  =  (/)  ->  ( A  X.  A )  =  (/) )
4 unieq 4048 . . . . 5  |-  ( ( A  X.  A )  =  (/)  ->  U. ( A  X.  A )  = 
U. (/) )
54unieqd 4050 . . . 4  |-  ( ( A  X.  A )  =  (/)  ->  U. U. ( A  X.  A
)  =  U. U. (/) )
6 uni0 4066 . . . . . 6  |-  U. (/)  =  (/)
76unieqi 4049 . . . . 5  |-  U. U. (/)  =  U. (/)
87, 6eqtri 2462 . . . 4  |-  U. U. (/)  =  (/)
9 eqtr 2459 . . . . 5  |-  ( ( U. U. ( A  X.  A )  = 
U. U. (/)  /\  U. U. (/)  =  (/) )  ->  U. U. ( A  X.  A
)  =  (/) )
10 eqtr 2459 . . . . . . 7  |-  ( ( U. U. ( A  X.  A )  =  (/)  /\  (/)  =  A )  ->  U. U. ( A  X.  A )  =  A )
1110expcom 426 . . . . . 6  |-  ( (/)  =  A  ->  ( U. U. ( A  X.  A
)  =  (/)  ->  U. U. ( A  X.  A
)  =  A ) )
1211eqcoms 2445 . . . . 5  |-  ( A  =  (/)  ->  ( U. U. ( A  X.  A
)  =  (/)  ->  U. U. ( A  X.  A
)  =  A ) )
139, 12syl5com 29 . . . 4  |-  ( ( U. U. ( A  X.  A )  = 
U. U. (/)  /\  U. U. (/)  =  (/) )  ->  ( A  =  (/)  ->  U. U. ( A  X.  A
)  =  A ) )
145, 8, 13sylancl 645 . . 3  |-  ( ( A  X.  A )  =  (/)  ->  ( A  =  (/)  ->  U. U. ( A  X.  A
)  =  A ) )
153, 14mpcom 35 . 2  |-  ( A  =  (/)  ->  U. U. ( A  X.  A
)  =  A )
16 df-ne 2607 . . 3  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
17 xpnz 5321 . . . 4  |-  ( ( A  =/=  (/)  /\  A  =/=  (/) )  <->  ( A  X.  A )  =/=  (/) )
18 unixp 5431 . . . . 5  |-  ( ( A  X.  A )  =/=  (/)  ->  U. U. ( A  X.  A )  =  ( A  u.  A
) )
19 unidm 3476 . . . . 5  |-  ( A  u.  A )  =  A
2018, 19syl6eq 2490 . . . 4  |-  ( ( A  X.  A )  =/=  (/)  ->  U. U. ( A  X.  A )  =  A )
2117, 20sylbi 189 . . 3  |-  ( ( A  =/=  (/)  /\  A  =/=  (/) )  ->  U. U. ( A  X.  A
)  =  A )
2216, 16, 21sylancbr 649 . 2  |-  ( -.  A  =  (/)  ->  U. U. ( A  X.  A
)  =  A )
2315, 22pm2.61i 159 1  |-  U. U. ( A  X.  A
)  =  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    =/= wne 2605    u. cun 3304   (/)c0 3613   U.cuni 4039    X. cxp 4905
This theorem is referenced by:  psss  14677
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 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-xp 4913  df-rel 4914  df-cnv 4915  df-dm 4917  df-rn 4918
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