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Theorem xpcan 5441
Description: Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.)
Assertion
Ref Expression
xpcan  |-  ( C  =/=  (/)  ->  ( ( C  X.  A )  =  ( C  X.  B
)  <->  A  =  B
) )

Proof of Theorem xpcan
StepHypRef Expression
1 xp11 5440 . . 3  |-  ( ( C  =/=  (/)  /\  A  =/=  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  <->  ( C  =  C  /\  A  =  B ) ) )
2 eqid 2467 . . . 4  |-  C  =  C
32biantrur 506 . . 3  |-  ( A  =  B  <->  ( C  =  C  /\  A  =  B ) )
41, 3syl6bbr 263 . 2  |-  ( ( C  =/=  (/)  /\  A  =/=  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  <->  A  =  B ) )
5 nne 2668 . . . 4  |-  ( -.  A  =/=  (/)  <->  A  =  (/) )
6 simpr 461 . . . . 5  |-  ( ( C  =/=  (/)  /\  A  =  (/) )  ->  A  =  (/) )
7 xpeq2 5014 . . . . . . . . . 10  |-  ( A  =  (/)  ->  ( C  X.  A )  =  ( C  X.  (/) ) )
8 xp0 5423 . . . . . . . . . 10  |-  ( C  X.  (/) )  =  (/)
97, 8syl6eq 2524 . . . . . . . . 9  |-  ( A  =  (/)  ->  ( C  X.  A )  =  (/) )
109eqeq1d 2469 . . . . . . . 8  |-  ( A  =  (/)  ->  ( ( C  X.  A )  =  ( C  X.  B )  <->  (/)  =  ( C  X.  B ) ) )
11 eqcom 2476 . . . . . . . 8  |-  ( (/)  =  ( C  X.  B )  <->  ( C  X.  B )  =  (/) )
1210, 11syl6bb 261 . . . . . . 7  |-  ( A  =  (/)  ->  ( ( C  X.  A )  =  ( C  X.  B )  <->  ( C  X.  B )  =  (/) ) )
1312adantl 466 . . . . . 6  |-  ( ( C  =/=  (/)  /\  A  =  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  <->  ( C  X.  B )  =  (/) ) )
14 df-ne 2664 . . . . . . . 8  |-  ( C  =/=  (/)  <->  -.  C  =  (/) )
15 xpeq0 5425 . . . . . . . . 9  |-  ( ( C  X.  B )  =  (/)  <->  ( C  =  (/)  \/  B  =  (/) ) )
16 orel1 382 . . . . . . . . 9  |-  ( -.  C  =  (/)  ->  (
( C  =  (/)  \/  B  =  (/) )  ->  B  =  (/) ) )
1715, 16syl5bi 217 . . . . . . . 8  |-  ( -.  C  =  (/)  ->  (
( C  X.  B
)  =  (/)  ->  B  =  (/) ) )
1814, 17sylbi 195 . . . . . . 7  |-  ( C  =/=  (/)  ->  ( ( C  X.  B )  =  (/)  ->  B  =  (/) ) )
1918adantr 465 . . . . . 6  |-  ( ( C  =/=  (/)  /\  A  =  (/) )  ->  (
( C  X.  B
)  =  (/)  ->  B  =  (/) ) )
2013, 19sylbid 215 . . . . 5  |-  ( ( C  =/=  (/)  /\  A  =  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  ->  B  =  (/) ) )
21 eqtr3 2495 . . . . 5  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  A  =  B )
226, 20, 21syl6an 545 . . . 4  |-  ( ( C  =/=  (/)  /\  A  =  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  ->  A  =  B )
)
235, 22sylan2b 475 . . 3  |-  ( ( C  =/=  (/)  /\  -.  A  =/=  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  ->  A  =  B )
)
24 xpeq2 5014 . . 3  |-  ( A  =  B  ->  ( C  X.  A )  =  ( C  X.  B
) )
2523, 24impbid1 203 . 2  |-  ( ( C  =/=  (/)  /\  -.  A  =/=  (/) )  ->  (
( C  X.  A
)  =  ( C  X.  B )  <->  A  =  B ) )
264, 25pm2.61dan 789 1  |-  ( C  =/=  (/)  ->  ( ( C  X.  A )  =  ( C  X.  B
)  <->  A  =  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    =/= wne 2662   (/)c0 3785    X. cxp 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010
This theorem is referenced by: (None)
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