Theorem xpcom 5216

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 301	. . . 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 266	. . . . . . . 8 |- ( ( a e. A /\ x e. B ) <-> ( x e. B /\ a e. A ) )
3	2	anbi1i 458	. . . . . . 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 4694	. . . . . . . 8 |- ( a ( A X. B ) x <-> ( a e. A /\ x e. B ) )
5		brxp 4694	. . . . . . . 8 |- ( x ( B X. C ) c <-> ( x e. B /\ c e. C ) )
6	4, 5	anbi12i 460	. . . . . . 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 590	. . . . . . 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 212	. . . . . 6 |- ( ( a ( A X. B ) x /\ x ( B X. C ) c ) <-> ( x e. B /\ ( a e. A /\ c e. C ) ) )
9	8	exbii 1619	. . . . 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 1917	. . . . 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 185	. . . 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	bitr2di 197	. . 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 4099	. 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 4672	. 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 4669	. 2 |- ( A X. C ) = { <. a , c >. | ( a e. A /\ c e. C ) }
16	13, 14, 15	3eqtr4g 2254	1 |- ( E. x x e. B -> ( ( B X. C ) o. ( A X. B ) ) = ( A X. C ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   E.wex 1506   e. wcel 2167   class class class wbr 4033   {copab 4093   X. cxp 4661   o. ccom 4667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-co 4672

This theorem is referenced by: (None)