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Theorem xpcoid 5541
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 5515 . . 3  |-  ( (/)  o.  (/) )  =  (/)
2 id 22 . . . . . 6  |-  ( A  =  (/)  ->  A  =  (/) )
32, 2xpeq12d 5019 . . . . 5  |-  ( A  =  (/)  ->  ( A  X.  A )  =  ( (/)  X.  (/) ) )
4 0xp 5073 . . . . 5  |-  ( (/)  X.  (/) )  =  (/)
53, 4syl6eq 2519 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  A )  =  (/) )
65, 5coeq12d 5160 . . 3  |-  ( A  =  (/)  ->  ( ( A  X.  A )  o.  ( A  X.  A ) )  =  ( (/)  o.  (/) ) )
71, 6, 53eqtr4a 2529 . 2  |-  ( A  =  (/)  ->  ( ( A  X.  A )  o.  ( A  X.  A ) )  =  ( A  X.  A
) )
8 xpco 5540 . 2  |-  ( A  =/=  (/)  ->  ( ( A  X.  A )  o.  ( A  X.  A
) )  =  ( A  X.  A ) )
97, 8pm2.61ine 2775 1  |-  ( ( A  X.  A )  o.  ( A  X.  A ) )  =  ( A  X.  A
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374   (/)c0 3780    X. cxp 4992    o. ccom 4998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003
This theorem is referenced by:  utop2nei  20483
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