Theorem xpcogend 13713

Description: The most interesting case of the composition of two cross products. (Contributed by RP, 24-Dec-2019.)

Hypothesis

Ref	Expression
xpcogend.1	|- ( ph -> ( B i^i C ) =/= (/) )

Assertion

Ref	Expression
xpcogend	|- ( ph -> ( ( C X. D ) o. ( A X. B ) ) = ( A X. D ) )

Proof of Theorem xpcogend

Dummy variables x z y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpcogend.1	. . . . . 6 |- ( ph -> ( B i^i C ) =/= (/) )
2		n0 3931	. . . . . . 7 |- ( ( B i^i C ) =/= (/) <-> E. y y e. ( B i^i C ) )
3		elin 3796	. . . . . . . 8 |- ( y e. ( B i^i C ) <-> ( y e. B /\ y e. C ) )
4	3	exbii 1774	. . . . . . 7 |- ( E. y y e. ( B i^i C ) <-> E. y ( y e. B /\ y e. C ) )
5	2, 4	bitri 264	. . . . . 6 |- ( ( B i^i C ) =/= (/) <-> E. y ( y e. B /\ y e. C ) )
6	1, 5	sylib 208	. . . . 5 |- ( ph -> E. y ( y e. B /\ y e. C ) )
7	6	biantrud 528	. . . 4 |- ( ph -> ( ( x e. A /\ z e. D ) <-> ( ( x e. A /\ z e. D ) /\ E. y ( y e. B /\ y e. C ) ) ) )
8		brxp 5147	. . . . . . 7 |- ( x ( A X. B ) y <-> ( x e. A /\ y e. B ) )
9		brxp 5147	. . . . . . . 8 |- ( y ( C X. D ) z <-> ( y e. C /\ z e. D ) )
10		ancom 466	. . . . . . . 8 |- ( ( y e. C /\ z e. D ) <-> ( z e. D /\ y e. C ) )
11	9, 10	bitri 264	. . . . . . 7 |- ( y ( C X. D ) z <-> ( z e. D /\ y e. C ) )
12	8, 11	anbi12i 733	. . . . . 6 |- ( ( x ( A X. B ) y /\ y ( C X. D ) z ) <-> ( ( x e. A /\ y e. B ) /\ ( z e. D /\ y e. C ) ) )
13	12	exbii 1774	. . . . 5 |- ( E. y ( x ( A X. B ) y /\ y ( C X. D ) z ) <-> E. y ( ( x e. A /\ y e. B ) /\ ( z e. D /\ y e. C ) ) )
14		an4 865	. . . . . 6 |- ( ( ( x e. A /\ y e. B ) /\ ( z e. D /\ y e. C ) ) <-> ( ( x e. A /\ z e. D ) /\ ( y e. B /\ y e. C ) ) )
15	14	exbii 1774	. . . . 5 |- ( E. y ( ( x e. A /\ y e. B ) /\ ( z e. D /\ y e. C ) ) <-> E. y ( ( x e. A /\ z e. D ) /\ ( y e. B /\ y e. C ) ) )
16		19.42v 1918	. . . . 5 |- ( E. y ( ( x e. A /\ z e. D ) /\ ( y e. B /\ y e. C ) ) <-> ( ( x e. A /\ z e. D ) /\ E. y ( y e. B /\ y e. C ) ) )
17	13, 15, 16	3bitri 286	. . . 4 |- ( E. y ( x ( A X. B ) y /\ y ( C X. D ) z ) <-> ( ( x e. A /\ z e. D ) /\ E. y ( y e. B /\ y e. C ) ) )
18	7, 17	syl6rbbr 279	. . 3 |- ( ph -> ( E. y ( x ( A X. B ) y /\ y ( C X. D ) z ) <-> ( x e. A /\ z e. D ) ) )
19	18	opabbidv 4716	. 2 |- ( ph -> { <. x , z >. | E. y ( x ( A X. B ) y /\ y ( C X. D ) z ) } = { <. x , z >. | ( x e. A /\ z e. D ) } )
20		df-co 5123	. 2 |- ( ( C X. D ) o. ( A X. B ) ) = { <. x , z >. | E. y ( x ( A X. B ) y /\ y ( C X. D ) z ) }
21		df-xp 5120	. 2 |- ( A X. D ) = { <. x , z >. | ( x e. A /\ z e. D ) }
22	19, 20, 21	3eqtr4g 2681	1 |- ( ph -> ( ( C X. D ) o. ( A X. B ) ) = ( A X. D ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   i^i cin 3573   (/)c0 3915   class class class wbr 4653   {copab 4712   X. cxp 5112   o. ccom 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-co 5123

This theorem is referenced by:  xpcoidgend  13714