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Theorem xpcoid 5367
Description: Composition of two square Cartesian products. (Contributed by Thierry Arnoux, 14-Jan-2018.)
Assertion
Ref Expression
xpcoid  |-  ( ( A  X.  A )  o.  ( A  X.  A ) )  =  ( A  X.  A
)

Proof of Theorem xpcoid
StepHypRef Expression
1 co01 5340 . . 3  |-  ( (/)  o.  (/) )  =  (/)
2 id 23 . . . . . 6  |-  ( A  =  (/)  ->  A  =  (/) )
32sqxpeqd 4851 . . . . 5  |-  ( A  =  (/)  ->  ( A  X.  A )  =  ( (/)  X.  (/) ) )
4 0xp 4906 . . . . 5  |-  ( (/)  X.  (/) )  =  (/)
53, 4syl6eq 2461 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  A )  =  (/) )
65, 5coeq12d 4990 . . 3  |-  ( A  =  (/)  ->  ( ( A  X.  A )  o.  ( A  X.  A ) )  =  ( (/)  o.  (/) ) )
71, 6, 53eqtr4a 2471 . 2  |-  ( A  =  (/)  ->  ( ( A  X.  A )  o.  ( A  X.  A ) )  =  ( A  X.  A
) )
8 xpco 5366 . 2  |-  ( A  =/=  (/)  ->  ( ( A  X.  A )  o.  ( A  X.  A
) )  =  ( A  X.  A ) )
97, 8pm2.61ine 2718 1  |-  ( ( A  X.  A )  o.  ( A  X.  A ) )  =  ( A  X.  A
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1407   (/)c0 3740    X. cxp 4823    o. ccom 4829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-br 4398  df-opab 4456  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834
This theorem is referenced by:  utop2nei  21047
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