Theorem xpcan2 4350

Description: Cancellation law for cross-product.

Assertion

Ref	Expression
xpcan2	|- (C =/= (/) -> ((A X. C) = (B X. C) <-> A = B))

Proof of Theorem xpcan2

Step	Hyp	Ref	Expression
1		xp11 4347	. . 3 |- ((A =/= (/) /\ C =/= (/)) -> ((A X. C) = (B X. C) <-> (A = B /\ C = C)))
2		eqid 1884	. . . 4 |- C = C
3	2	biantru 793	. . 3 |- (A = B <-> (A = B /\ C = C))
4	1, 3	syl6bbr 597	. 2 |- ((A =/= (/) /\ C =/= (/)) -> ((A X. C) = (B X. C) <-> A = B))
5		xpeq1 4016	. . . . . . . . . . 11 |- (A = (/) -> (A X. C) = ((/) X. C))
6		xp0r 4065	. . . . . . . . . . 11 |- ((/) X. C) = (/)
7	5, 6	syl6eq 1944	. . . . . . . . . 10 |- (A = (/) -> (A X. C) = (/))
8	7	eqeq1d 1892	. . . . . . . . 9 |- (A = (/) -> ((A X. C) = (B X. C) <-> (/) = (B X. C)))
9		eqcom 1886	. . . . . . . . 9 |- ((/) = (B X. C) <-> (B X. C) = (/))
10	8, 9	syl6bb 595	. . . . . . . 8 |- (A = (/) -> ((A X. C) = (B X. C) <-> (B X. C) = (/)))
11	10	adantr 425	. . . . . . 7 |- ((A = (/) /\ C =/= (/)) -> ((A X. C) = (B X. C) <-> (B X. C) = (/)))
12		df-ne 2019	. . . . . . . . 9 |- (C =/= (/) <-> -. C = (/))
13		orel2 272	. . . . . . . . . 10 |- (-. C = (/) -> ((B = (/) \/ C = (/)) -> B = (/)))
14		xpeq0 4336	. . . . . . . . . 10 |- ((B X. C) = (/) <-> (B = (/) \/ C = (/)))
15	13, 14	syl5ib 223	. . . . . . . . 9 |- (-. C = (/) -> ((B X. C) = (/) -> B = (/)))
16	12, 15	sylbi 216	. . . . . . . 8 |- (C =/= (/) -> ((B X. C) = (/) -> B = (/)))
17	16	adantl 424	. . . . . . 7 |- ((A = (/) /\ C =/= (/)) -> ((B X. C) = (/) -> B = (/)))
18	11, 17	sylbid 220	. . . . . 6 |- ((A = (/) /\ C =/= (/)) -> ((A X. C) = (B X. C) -> B = (/)))
19		simpl 346	. . . . . 6 |- ((A = (/) /\ C =/= (/)) -> A = (/))
20	18, 19	jctild 662	. . . . 5 |- ((A = (/) /\ C =/= (/)) -> ((A X. C) = (B X. C) -> (A = (/) /\ B = (/))))
21		eqtr3 1907	. . . . 5 |- ((A = (/) /\ B = (/)) -> A = B)
22	20, 21	syl6 25	. . . 4 |- ((A = (/) /\ C =/= (/)) -> ((A X. C) = (B X. C) -> A = B))
23		xpeq1 4016	. . . 4 |- (A = B -> (A X. C) = (B X. C))
24	22, 23	impbid1 575	. . 3 |- ((A = (/) /\ C =/= (/)) -> ((A X. C) = (B X. C) <-> A = B))
25		nne 2021	. . 3 |- (-. A =/= (/) <-> A = (/))
26	24, 25	sylanb 498	. 2 |- ((-. A =/= (/) /\ C =/= (/)) -> ((A X. C) = (B X. C) <-> A = B))
27	4, 26	pm2.61ian 534	1 |- (C =/= (/) -> ((A X. C) = (B X. C) <-> A = B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   =/= wne 2017  (/)c0 2875   X. cxp 3984

This theorem is referenced by:  vcoprnelem 9529

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005

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