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Theorem xpcom 4964
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 295 . . . 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 262 . . . . . . . 8  |-  ( ( a  e.  A  /\  x  e.  B )  <->  ( x  e.  B  /\  a  e.  A )
)
32anbi1i 446 . . . . . . 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 4458 . . . . . . . 8  |-  ( a ( A  X.  B
) x  <->  ( a  e.  A  /\  x  e.  B ) )
5 brxp 4458 . . . . . . . 8  |-  ( x ( B  X.  C
) c  <->  ( x  e.  B  /\  c  e.  C ) )
64, 5anbi12i 448 . . . . . . 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 557 . . . . . . 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 210 . . . . . 6  |-  ( ( a ( A  X.  B ) x  /\  x ( B  X.  C ) c )  <-> 
( x  e.  B  /\  ( a  e.  A  /\  c  e.  C
) ) )
98exbii 1541 . . . . 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 1830 . . . . 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 183 . . . 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, 11syl6rbb 195 . . 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 3896 . 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 4437 . 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 4434 . 2  |-  ( A  X.  C )  =  { <. a ,  c
>.  |  ( a  e.  A  /\  c  e.  C ) }
1613, 14, 153eqtr4g 2145 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 102    = wceq 1289   E.wex 1426    e. wcel 1438   class class class wbr 3837   {copab 3890    X. cxp 4426    o. ccom 4432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-co 4437
This theorem is referenced by: (None)
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