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Theorem xpcom 5170
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.
StepHypRef 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 )
)
32anbi1i 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 4653 . . . . . . . 8  |-  ( a ( A  X.  B
) x  <->  ( a  e.  A  /\  x  e.  B ) )
5 brxp 4653 . . . . . . . 8  |-  ( x ( B  X.  C
) c  <->  ( x  e.  B  /\  c  e.  C ) )
64, 5anbi12i 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
) ) )
83, 6, 73bitr4i 212 . . . . . 6  |-  ( ( a ( A  X.  B ) x  /\  x ( B  X.  C ) c )  <-> 
( x  e.  B  /\  ( a  e.  A  /\  c  e.  C
) ) )
98exbii 1605 . . . . 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 1902 . . . . 5  |-  ( E. x ( x  e.  B  /\  ( a  e.  A  /\  c  e.  C ) )  <->  ( E. x  x  e.  B  /\  ( a  e.  A  /\  c  e.  C
) ) )
119, 10bitr2i 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 ) )
121, 11bitr2di 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
) ) )
1312opabbidv 4066 . 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 4631 . 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 4628 . 2  |-  ( A  X.  C )  =  { <. a ,  c
>.  |  ( a  e.  A  /\  c  e.  C ) }
1613, 14, 153eqtr4g 2235 1  |-  ( E. x  x  e.  B  ->  ( ( B  X.  C )  o.  ( A  X.  B ) )  =  ( A  X.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353   E.wex 1492    e. wcel 2148   class class class wbr 4000   {copab 4060    X. cxp 4620    o. ccom 4626
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4628  df-co 4631
This theorem is referenced by: (None)
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