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Theorem xpcan2 5429
Description: Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.)
Assertion
Ref Expression
xpcan2  |-  ( C  =/=  (/)  ->  ( ( A  X.  C )  =  ( B  X.  C
)  <->  A  =  B
) )

Proof of Theorem xpcan2
StepHypRef Expression
1 xp11 5427 . . 3  |-  ( ( A  =/=  (/)  /\  C  =/=  (/) )  ->  (
( A  X.  C
)  =  ( B  X.  C )  <->  ( A  =  B  /\  C  =  C ) ) )
2 eqid 2454 . . . 4  |-  C  =  C
32biantru 503 . . 3  |-  ( A  =  B  <->  ( A  =  B  /\  C  =  C ) )
41, 3syl6bbr 263 . 2  |-  ( ( A  =/=  (/)  /\  C  =/=  (/) )  ->  (
( A  X.  C
)  =  ( B  X.  C )  <->  A  =  B ) )
5 nne 2655 . . 3  |-  ( -.  A  =/=  (/)  <->  A  =  (/) )
6 simpl 455 . . . . 5  |-  ( ( A  =  (/)  /\  C  =/=  (/) )  ->  A  =  (/) )
7 xpeq1 5002 . . . . . . . . . 10  |-  ( A  =  (/)  ->  ( A  X.  C )  =  ( (/)  X.  C
) )
8 0xp 5069 . . . . . . . . . 10  |-  ( (/)  X.  C )  =  (/)
97, 8syl6eq 2511 . . . . . . . . 9  |-  ( A  =  (/)  ->  ( A  X.  C )  =  (/) )
109eqeq1d 2456 . . . . . . . 8  |-  ( A  =  (/)  ->  ( ( A  X.  C )  =  ( B  X.  C )  <->  (/)  =  ( B  X.  C ) ) )
11 eqcom 2463 . . . . . . . 8  |-  ( (/)  =  ( B  X.  C )  <->  ( B  X.  C )  =  (/) )
1210, 11syl6bb 261 . . . . . . 7  |-  ( A  =  (/)  ->  ( ( A  X.  C )  =  ( B  X.  C )  <->  ( B  X.  C )  =  (/) ) )
1312adantr 463 . . . . . 6  |-  ( ( A  =  (/)  /\  C  =/=  (/) )  ->  (
( A  X.  C
)  =  ( B  X.  C )  <->  ( B  X.  C )  =  (/) ) )
14 df-ne 2651 . . . . . . . 8  |-  ( C  =/=  (/)  <->  -.  C  =  (/) )
15 xpeq0 5412 . . . . . . . . 9  |-  ( ( B  X.  C )  =  (/)  <->  ( B  =  (/)  \/  C  =  (/) ) )
16 orel2 381 . . . . . . . . 9  |-  ( -.  C  =  (/)  ->  (
( B  =  (/)  \/  C  =  (/) )  ->  B  =  (/) ) )
1715, 16syl5bi 217 . . . . . . . 8  |-  ( -.  C  =  (/)  ->  (
( B  X.  C
)  =  (/)  ->  B  =  (/) ) )
1814, 17sylbi 195 . . . . . . 7  |-  ( C  =/=  (/)  ->  ( ( B  X.  C )  =  (/)  ->  B  =  (/) ) )
1918adantl 464 . . . . . 6  |-  ( ( A  =  (/)  /\  C  =/=  (/) )  ->  (
( B  X.  C
)  =  (/)  ->  B  =  (/) ) )
2013, 19sylbid 215 . . . . 5  |-  ( ( A  =  (/)  /\  C  =/=  (/) )  ->  (
( A  X.  C
)  =  ( B  X.  C )  ->  B  =  (/) ) )
21 eqtr3 2482 . . . . 5  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  A  =  B )
226, 20, 21syl6an 543 . . . 4  |-  ( ( A  =  (/)  /\  C  =/=  (/) )  ->  (
( A  X.  C
)  =  ( B  X.  C )  ->  A  =  B )
)
23 xpeq1 5002 . . . 4  |-  ( A  =  B  ->  ( A  X.  C )  =  ( B  X.  C
) )
2422, 23impbid1 203 . . 3  |-  ( ( A  =  (/)  /\  C  =/=  (/) )  ->  (
( A  X.  C
)  =  ( B  X.  C )  <->  A  =  B ) )
255, 24sylanb 470 . 2  |-  ( ( -.  A  =/=  (/)  /\  C  =/=  (/) )  ->  (
( A  X.  C
)  =  ( B  X.  C )  <->  A  =  B ) )
264, 25pm2.61ian 788 1  |-  ( C  =/=  (/)  ->  ( ( A  X.  C )  =  ( B  X.  C
)  <->  A  =  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1398    =/= wne 2649   (/)c0 3783    X. cxp 4986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-dm 4998  df-rn 4999
This theorem is referenced by:  vcoprnelem  25669
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