Theorem xpcan2 5571

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

Step	Hyp	Ref	Expression
1		xp11 5569	. . 3 |- ( ( A =/= (/) /\ C =/= (/) ) -> ( ( A X. C ) = ( B X. C ) <-> ( A = B /\ C = C ) ) )
2		eqid 2622	. . . 4 |- C = C
3	2	biantru 526	. . 3 |- ( A = B <-> ( A = B /\ C = C ) )
4	1, 3	syl6bbr 278	. 2 |- ( ( A =/= (/) /\ C =/= (/) ) -> ( ( A X. C ) = ( B X. C ) <-> A = B ) )
5		nne 2798	. . 3 |- ( -. A =/= (/) <-> A = (/) )
6		simpl 473	. . . . 5 |- ( ( A = (/) /\ C =/= (/) ) -> A = (/) )
7		xpeq1 5128	. . . . . . . . . 10 |- ( A = (/) -> ( A X. C ) = ( (/) X. C ) )
8		0xp 5199	. . . . . . . . . 10 |- ( (/) X. C ) = (/)
9	7, 8	syl6eq 2672	. . . . . . . . 9 |- ( A = (/) -> ( A X. C ) = (/) )
10	9	eqeq1d 2624	. . . . . . . 8 |- ( A = (/) -> ( ( A X. C ) = ( B X. C ) <-> (/) = ( B X. C ) ) )
11		eqcom 2629	. . . . . . . 8 |- ( (/) = ( B X. C ) <-> ( B X. C ) = (/) )
12	10, 11	syl6bb 276	. . . . . . 7 |- ( A = (/) -> ( ( A X. C ) = ( B X. C ) <-> ( B X. C ) = (/) ) )
13	12	adantr 481	. . . . . 6 |- ( ( A = (/) /\ C =/= (/) ) -> ( ( A X. C ) = ( B X. C ) <-> ( B X. C ) = (/) ) )
14		df-ne 2795	. . . . . . . 8 |- ( C =/= (/) <-> -. C = (/) )
15		xpeq0 5554	. . . . . . . . 9 |- ( ( B X. C ) = (/) <-> ( B = (/) \/ C = (/) ) )
16		orel2 398	. . . . . . . . 9 |- ( -. C = (/) -> ( ( B = (/) \/ C = (/) ) -> B = (/) ) )
17	15, 16	syl5bi 232	. . . . . . . 8 |- ( -. C = (/) -> ( ( B X. C ) = (/) -> B = (/) ) )
18	14, 17	sylbi 207	. . . . . . 7 |- ( C =/= (/) -> ( ( B X. C ) = (/) -> B = (/) ) )
19	18	adantl 482	. . . . . 6 |- ( ( A = (/) /\ C =/= (/) ) -> ( ( B X. C ) = (/) -> B = (/) ) )
20	13, 19	sylbid 230	. . . . 5 |- ( ( A = (/) /\ C =/= (/) ) -> ( ( A X. C ) = ( B X. C ) -> B = (/) ) )
21		eqtr3 2643	. . . . 5 |- ( ( A = (/) /\ B = (/) ) -> A = B )
22	6, 20, 21	syl6an 568	. . . 4 |- ( ( A = (/) /\ C =/= (/) ) -> ( ( A X. C ) = ( B X. C ) -> A = B ) )
23		xpeq1 5128	. . . 4 |- ( A = B -> ( A X. C ) = ( B X. C ) )
24	22, 23	impbid1 215	. . 3 |- ( ( A = (/) /\ C =/= (/) ) -> ( ( A X. C ) = ( B X. C ) <-> A = B ) )
25	5, 24	sylanb 489	. 2 |- ( ( -. A =/= (/) /\ C =/= (/) ) -> ( ( A X. C ) = ( B X. C ) <-> A = B ) )
26	4, 25	pm2.61ian 831	1 |- ( C =/= (/) -> ( ( A X. C ) = ( B X. C ) <-> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   =/= wne 2794   (/)c0 3915   X. cxp 5112

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-dm 5124  df-rn 5125

This theorem is referenced by: (None)