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Theorem inxp 4761
Description: The intersection of two cross products. Exercise 9 of [TakeutiZaring] p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
inxp  |-  ( ( A  X.  B )  i^i  ( C  X.  D ) )  =  ( ( A  i^i  C )  X.  ( B  i^i  D ) )

Proof of Theorem inxp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inopab 4759 . . 3  |-  ( {
<. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }  i^i  {
<. x ,  y >.  |  ( x  e.  C  /\  y  e.  D ) } )  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  ( x  e.  C  /\  y  e.  D ) ) }
2 an4 586 . . . . 5  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( x  e.  C  /\  y  e.  D ) )  <->  ( (
x  e.  A  /\  x  e.  C )  /\  ( y  e.  B  /\  y  e.  D
) ) )
3 elin 3318 . . . . . 6  |-  ( x  e.  ( A  i^i  C )  <->  ( x  e.  A  /\  x  e.  C ) )
4 elin 3318 . . . . . 6  |-  ( y  e.  ( B  i^i  D )  <->  ( y  e.  B  /\  y  e.  D ) )
53, 4anbi12i 460 . . . . 5  |-  ( ( x  e.  ( A  i^i  C )  /\  y  e.  ( B  i^i  D ) )  <->  ( (
x  e.  A  /\  x  e.  C )  /\  ( y  e.  B  /\  y  e.  D
) ) )
62, 5bitr4i 187 . . . 4  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( x  e.  C  /\  y  e.  D ) )  <->  ( x  e.  ( A  i^i  C
)  /\  y  e.  ( B  i^i  D ) ) )
76opabbii 4070 . . 3  |-  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  ( x  e.  C  /\  y  e.  D ) ) }  =  { <. x ,  y >.  |  ( x  e.  ( A  i^i  C )  /\  y  e.  ( B  i^i  D ) ) }
81, 7eqtri 2198 . 2  |-  ( {
<. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }  i^i  {
<. x ,  y >.  |  ( x  e.  C  /\  y  e.  D ) } )  =  { <. x ,  y >.  |  ( x  e.  ( A  i^i  C )  /\  y  e.  ( B  i^i  D ) ) }
9 df-xp 4632 . . 3  |-  ( A  X.  B )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  B ) }
10 df-xp 4632 . . 3  |-  ( C  X.  D )  =  { <. x ,  y
>.  |  ( x  e.  C  /\  y  e.  D ) }
119, 10ineq12i 3334 . 2  |-  ( ( A  X.  B )  i^i  ( C  X.  D ) )  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }  i^i  { <. x ,  y >.  |  ( x  e.  C  /\  y  e.  D ) } )
12 df-xp 4632 . 2  |-  ( ( A  i^i  C )  X.  ( B  i^i  D ) )  =  { <. x ,  y >.  |  ( x  e.  ( A  i^i  C
)  /\  y  e.  ( B  i^i  D ) ) }
138, 11, 123eqtr4i 2208 1  |-  ( ( A  X.  B )  i^i  ( C  X.  D ) )  =  ( ( A  i^i  C )  X.  ( B  i^i  D ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1353    e. wcel 2148    i^i cin 3128   {copab 4063    X. cxp 4624
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 4121  ax-pow 4174  ax-pr 4209
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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-opab 4065  df-xp 4632  df-rel 4633
This theorem is referenced by:  xpindi  4762  xpindir  4763  dmxpin  4849  xpssres  4942  xpdisj1  5053  xpdisj2  5054  imainrect  5074  xpima1  5075  xpima2m  5076  hashxp  10801  txbas  13651  txrest  13669  metreslem  13773
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