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Theorem xpcom 4864
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 285 . . . 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 253 . . . . . . . 8  |-  ( ( a  e.  A  /\  x  e.  B )  <->  ( x  e.  B  /\  a  e.  A )
)
32anbi1i 431 . . . . . . 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 4375 . . . . . . . 8  |-  ( a ( A  X.  B
) x  <->  ( a  e.  A  /\  x  e.  B ) )
5 brxp 4375 . . . . . . . 8  |-  ( x ( B  X.  C
) c  <->  ( x  e.  B  /\  c  e.  C ) )
64, 5anbi12i 433 . . . . . . 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 524 . . . . . . 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 201 . . . . . 6  |-  ( ( a ( A  X.  B ) x  /\  x ( B  X.  C ) c )  <-> 
( x  e.  B  /\  ( a  e.  A  /\  c  e.  C
) ) )
98exbii 1496 . . . . 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 1782 . . . . 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 174 . . . 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 186 . . 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 3823 . 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 4354 . 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 4351 . 2  |-  ( A  X.  C )  =  { <. a ,  c
>.  |  ( a  e.  A  /\  c  e.  C ) }
1613, 14, 153eqtr4g 2097 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 97    = wceq 1243   E.wex 1381    e. wcel 1393   class class class wbr 3764   {copab 3817    X. cxp 4343    o. ccom 4349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-co 4354
This theorem is referenced by: (None)
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