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Theorem xpcogend 13113
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.
StepHypRef Expression
1 xpcogend.1 . . . . . 6  |-  ( ph  ->  ( B  i^i  C
)  =/=  (/) )
2 n0 3732 . . . . . . 7  |-  ( ( B  i^i  C )  =/=  (/)  <->  E. y  y  e.  ( B  i^i  C
) )
3 elin 3608 . . . . . . . 8  |-  ( y  e.  ( B  i^i  C )  <->  ( y  e.  B  /\  y  e.  C ) )
43exbii 1726 . . . . . . 7  |-  ( E. y  y  e.  ( B  i^i  C )  <->  E. y ( y  e.  B  /\  y  e.  C ) )
52, 4bitri 257 . . . . . 6  |-  ( ( B  i^i  C )  =/=  (/)  <->  E. y ( y  e.  B  /\  y  e.  C ) )
61, 5sylib 201 . . . . 5  |-  ( ph  ->  E. y ( y  e.  B  /\  y  e.  C ) )
76biantrud 515 . . . 4  |-  ( ph  ->  ( ( x  e.  A  /\  z  e.  D )  <->  ( (
x  e.  A  /\  z  e.  D )  /\  E. y ( y  e.  B  /\  y  e.  C ) ) ) )
8 brxp 4870 . . . . . . 7  |-  ( x ( A  X.  B
) y  <->  ( x  e.  A  /\  y  e.  B ) )
9 brxp 4870 . . . . . . . 8  |-  ( y ( C  X.  D
) z  <->  ( y  e.  C  /\  z  e.  D ) )
10 ancom 457 . . . . . . . 8  |-  ( ( y  e.  C  /\  z  e.  D )  <->  ( z  e.  D  /\  y  e.  C )
)
119, 10bitri 257 . . . . . . 7  |-  ( y ( C  X.  D
) z  <->  ( z  e.  D  /\  y  e.  C ) )
128, 11anbi12i 711 . . . . . 6  |-  ( ( x ( A  X.  B ) y  /\  y ( C  X.  D ) z )  <-> 
( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  D  /\  y  e.  C )
) )
1312exbii 1726 . . . . 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 840 . . . . . 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
) ) )
1514exbii 1726 . . . . 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 1842 . . . . 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 ) ) )
1713, 15, 163bitri 279 . . . 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 ) ) )
187, 17syl6rbbr 272 . . 3  |-  ( ph  ->  ( E. y ( x ( A  X.  B ) y  /\  y ( C  X.  D ) z )  <-> 
( x  e.  A  /\  z  e.  D
) ) )
1918opabbidv 4459 . 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 4848 . 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 4845 . 2  |-  ( A  X.  D )  =  { <. x ,  z
>.  |  ( x  e.  A  /\  z  e.  D ) }
2219, 20, 213eqtr4g 2530 1  |-  ( ph  ->  ( ( C  X.  D )  o.  ( A  X.  B ) )  =  ( A  X.  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641    i^i cin 3389   (/)c0 3722   class class class wbr 4395   {copab 4453    X. cxp 4837    o. ccom 4843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-co 4848
This theorem is referenced by:  xpcoidgend  13114
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