Theorem unixpid 5363

Description: Field of a square cross product. (Contributed by FL, 10-Oct-2009.)

Assertion

Ref	Expression
unixpid	|- U. U. ( A X. A ) = A

Proof of Theorem unixpid

Step	Hyp	Ref	Expression
1		xpeq1 4851	. . . 4 |- ( A = (/) -> ( A X. A ) = ( (/) X. A ) )
2		xp0r 4915	. . . 4 |- ( (/) X. A ) = (/)
3	1, 2	syl6eq 2452	. . 3 |- ( A = (/) -> ( A X. A ) = (/) )
4		unieq 3984	. . . . 5 |- ( ( A X. A ) = (/) -> U. ( A X. A ) = U. (/) )
5	4	unieqd 3986	. . . 4 |- ( ( A X. A ) = (/) -> U. U. ( A X. A ) = U. U. (/) )
6		uni0 4002	. . . . . 6 |- U. (/) = (/)
7	6	unieqi 3985	. . . . 5 |- U. U. (/) = U. (/)
8	7, 6	eqtri 2424	. . . 4 |- U. U. (/) = (/)
9		eqtr 2421	. . . . 5 |- ( ( U. U. ( A X. A ) = U. U. (/) /\ U. U. (/) = (/) ) -> U. U. ( A X. A ) = (/) )
10		eqtr 2421	. . . . . . 7 |- ( ( U. U. ( A X. A ) = (/) /\ (/) = A ) -> U. U. ( A X. A ) = A )
11	10	expcom 425	. . . . . 6 |- ( (/) = A -> ( U. U. ( A X. A ) = (/) -> U. U. ( A X. A ) = A ) )
12	11	eqcoms 2407	. . . . 5 |- ( A = (/) -> ( U. U. ( A X. A ) = (/) -> U. U. ( A X. A ) = A ) )
13	9, 12	syl5com 28	. . . 4 |- ( ( U. U. ( A X. A ) = U. U. (/) /\ U. U. (/) = (/) ) -> ( A = (/) -> U. U. ( A X. A ) = A ) )
14	5, 8, 13	sylancl 644	. . 3 |- ( ( A X. A ) = (/) -> ( A = (/) -> U. U. ( A X. A ) = A ) )
15	3, 14	mpcom 34	. 2 |- ( A = (/) -> U. U. ( A X. A ) = A )
16		df-ne 2569	. . 3 |- ( A =/= (/) <-> -. A = (/) )
17		xpnz 5251	. . . 4 |- ( ( A =/= (/) /\ A =/= (/) ) <-> ( A X. A ) =/= (/) )
18		unixp 5361	. . . . 5 |- ( ( A X. A ) =/= (/) -> U. U. ( A X. A ) = ( A u. A ) )
19		unidm 3450	. . . . 5 |- ( A u. A ) = A
20	18, 19	syl6eq 2452	. . . 4 |- ( ( A X. A ) =/= (/) -> U. U. ( A X. A ) = A )
21	17, 20	sylbi 188	. . 3 |- ( ( A =/= (/) /\ A =/= (/) ) -> U. U. ( A X. A ) = A )
22	16, 16, 21	sylancbr 648	. 2 |- ( -. A = (/) -> U. U. ( A X. A ) = A )
23	15, 22	pm2.61i 158	1 |- U. U. ( A X. A ) = A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   = wceq 1649   =/= wne 2567   u. cun 3278   (/)c0 3588   U.cuni 3975   X. cxp 4835

This theorem is referenced by:  psss  14601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848