Theorem xpcom 5157

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 299	. . . 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 264	. . . . . . . 8 |- ( ( a e. A /\ x e. B ) <-> ( x e. B /\ a e. A ) )
3	2	anbi1i 455	. . . . . . 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 4642	. . . . . . . 8 |- ( a ( A X. B ) x <-> ( a e. A /\ x e. B ) )
5		brxp 4642	. . . . . . . 8 |- ( x ( B X. C ) c <-> ( x e. B /\ c e. C ) )
6	4, 5	anbi12i 457	. . . . . . 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 585	. . . . . . 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 211	. . . . . 6 |- ( ( a ( A X. B ) x /\ x ( B X. C ) c ) <-> ( x e. B /\ ( a e. A /\ c e. C ) ) )
9	8	exbii 1598	. . . . 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 1895	. . . . 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 184	. . . 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 196	. . 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 4055	. 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 4620	. 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 4617	. 2 |- ( A X. C ) = { <. a , c >. | ( a e. A /\ c e. C ) }
16	13, 14, 15	3eqtr4g 2228	1 |- ( E. x x e. B -> ( ( B X. C ) o. ( A X. B ) ) = ( A X. C ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   class class class wbr 3989   {copab 4049   X. cxp 4609   o. ccom 4615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-co 4620

This theorem is referenced by: (None)