Theorem xpcom 4884

Description: Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.)

Assertion

Ref	Expression
xpcom	|- ( E. x x e. B -> ( ( B X. C ) o. ( A X. B ) ) = ( A X. C ) )

Distinct variable groups:   x, A   x, B   x, C

Proof of Theorem xpcom

Dummy variables a c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ibar 295	. . . 4 |- ( E. x x e. B -> ( ( a e. A /\ c e. C ) <-> ( E. x x e. B /\ ( a e. A /\ c e. C ) ) ) )
2		ancom 262	. . . . . . . 8 |- ( ( a e. A /\ x e. B ) <-> ( x e. B /\ a e. A ) )
3	2	anbi1i 445	. . . . . . 7 |- ( ( ( a e. A /\ x e. B ) /\ ( x e. B /\ c e. C ) ) <-> ( ( x e. B /\ a e. A ) /\ ( x e. B /\ c e. C ) ) )
4		brxp 4393	. . . . . . . 8 |- ( a ( A X. B ) x <-> ( a e. A /\ x e. B ) )
5		brxp 4393	. . . . . . . 8 |- ( x ( B X. C ) c <-> ( x e. B /\ c e. C ) )
6	4, 5	anbi12i 447	. . . . . . 7 |- ( ( a ( A X. B ) x /\ x ( B X. C ) c ) <-> ( ( a e. A /\ x e. B ) /\ ( x e. B /\ c e. C ) ) )
7		anandi 554	. . . . . . 7 |- ( ( x e. B /\ ( a e. A /\ c e. C ) ) <-> ( ( x e. B /\ a e. A ) /\ ( x e. B /\ c e. C ) ) )
8	3, 6, 7	3bitr4i 210	. . . . . 6 |- ( ( a ( A X. B ) x /\ x ( B X. C ) c ) <-> ( x e. B /\ ( a e. A /\ c e. C ) ) )
9	8	exbii 1536	. . . . 5 |- ( E. x ( a ( A X. B ) x /\ x ( B X. C ) c ) <-> E. x ( x e. B /\ ( a e. A /\ c e. C ) ) )
10		19.41v 1823	. . . . 5 |- ( E. x ( x e. B /\ ( a e. A /\ c e. C ) ) <-> ( E. x x e. B /\ ( a e. A /\ c e. C ) ) )
11	9, 10	bitr2i 183	. . . 4 |- ( ( E. x x e. B /\ ( a e. A /\ c e. C ) ) <-> E. x ( a ( A X. B ) x /\ x ( B X. C ) c ) )
12	1, 11	syl6rbb 195	. . 3 |- ( E. x x e. B -> ( E. x ( a ( A X. B ) x /\ x ( B X. C ) c ) <-> ( a e. A /\ c e. C ) ) )
13	12	opabbidv 3844	. 2 |- ( E. x x e. B -> { <. a , c >. | E. x ( a ( A X. B ) x /\ x ( B X. C ) c ) } = { <. a , c >. | ( a e. A /\ c e. C ) } )
14		df-co 4372	. 2 |- ( ( B X. C ) o. ( A X. B ) ) = { <. a , c >. | E. x ( a ( A X. B ) x /\ x ( B X. C ) c ) }
15		df-xp 4369	. 2 |- ( A X. C ) = { <. a , c >. | ( a e. A /\ c e. C ) }
16	13, 14, 15	3eqtr4g 2138	1 |- ( E. x x e. B -> ( ( B X. C ) o. ( A X. B ) ) = ( A X. C ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   E.wex 1421   e. wcel 1433   class class class wbr 3785   {copab 3838   X. cxp 4361   o. ccom 4367

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-co 4372

This theorem is referenced by: (None)