Theorem xpcoid 5676

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

Step	Hyp	Ref	Expression
1		co01 5650	. . 3 |- ( (/) o. (/) ) = (/)
2		id 22	. . . . . 6 |- ( A = (/) -> A = (/) )
3	2	sqxpeqd 5141	. . . . 5 |- ( A = (/) -> ( A X. A ) = ( (/) X. (/) ) )
4		0xp 5199	. . . . 5 |- ( (/) X. (/) ) = (/)
5	3, 4	syl6eq 2672	. . . 4 |- ( A = (/) -> ( A X. A ) = (/) )
6	5, 5	coeq12d 5286	. . 3 |- ( A = (/) -> ( ( A X. A ) o. ( A X. A ) ) = ( (/) o. (/) ) )
7	1, 6, 5	3eqtr4a 2682	. 2 |- ( A = (/) -> ( ( A X. A ) o. ( A X. A ) ) = ( A X. A ) )
8		xpco 5675	. 2 |- ( A =/= (/) -> ( ( A X. A ) o. ( A X. A ) ) = ( A X. A ) )
9	7, 8	pm2.61ine 2877	1 |- ( ( A X. A ) o. ( A X. A ) ) = ( A X. A )

Syntax hints:   = wceq 1483   (/)c0 3915   X. cxp 5112   o. ccom 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123

This theorem is referenced by:  utop2nei  22054