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Theorem unixpid 5310
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 4806 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  A )  =  ( (/)  X.  A
) )
2 xp0r 4871 . . . 4  |-  ( (/)  X.  A )  =  (/)
31, 2syl6eq 2414 . . 3  |-  ( A  =  (/)  ->  ( A  X.  A )  =  (/) )
4 unieq 3938 . . . . 5  |-  ( ( A  X.  A )  =  (/)  ->  U. ( A  X.  A )  = 
U. (/) )
54unieqd 3940 . . . 4  |-  ( ( A  X.  A )  =  (/)  ->  U. U. ( A  X.  A
)  =  U. U. (/) )
6 uni0 3956 . . . . . 6  |-  U. (/)  =  (/)
76unieqi 3939 . . . . 5  |-  U. U. (/)  =  U. (/)
87, 6eqtri 2386 . . . 4  |-  U. U. (/)  =  (/)
9 eqtr 2383 . . . . 5  |-  ( ( U. U. ( A  X.  A )  = 
U. U. (/)  /\  U. U. (/)  =  (/) )  ->  U. U. ( A  X.  A
)  =  (/) )
10 eqtr 2383 . . . . . . 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 2369 . . . . 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 2531 . . 3  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
17 xpnz 5202 . . . 4  |-  ( ( A  =/=  (/)  /\  A  =/=  (/) )  <->  ( A  X.  A )  =/=  (/) )
18 unixp 5308 . . . . 5  |-  ( ( A  X.  A )  =/=  (/)  ->  U. U. ( A  X.  A )  =  ( A  u.  A
) )
19 unidm 3406 . . . . 5  |-  ( A  u.  A )  =  A
2018, 19syl6eq 2414 . . . 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 1647    =/= wne 2529    u. cun 3236   (/)c0 3543   U.cuni 3929    X. cxp 4790
This theorem is referenced by:  psss  14533
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-xp 4798  df-rel 4799  df-cnv 4800  df-dm 4802  df-rn 4803
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