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Theorem inxp 4834
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 4832 . . 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 797 . . . . 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 3371 . . . . . 6  |-  ( x  e.  ( A  i^i  C )  <->  ( x  e.  A  /\  x  e.  C ) )
4 elin 3371 . . . . . 6  |-  ( y  e.  ( B  i^i  D )  <->  ( y  e.  B  /\  y  e.  D ) )
53, 4anbi12i 678 . . . . 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 243 . . . 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 4099 . . 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 2316 . 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 4711 . . 3  |-  ( A  X.  B )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  B ) }
10 df-xp 4711 . . 3  |-  ( C  X.  D )  =  { <. x ,  y
>.  |  ( x  e.  C  /\  y  e.  D ) }
119, 10ineq12i 3381 . 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 4711 . 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 2326 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 358    = wceq 1632    e. wcel 1696    i^i cin 3164   {copab 4092    X. cxp 4703
This theorem is referenced by:  xpindi  4835  xpindir  4836  dmxpin  4915  xpssres  5005  xpdisj1  5117  xpdisj2  5118  imainrect  5135  curry1  6226  curry2  6229  fpar  6238  marypha1lem  7202  fpwwe2lem13  8280  hashxplem  11401  sscres  13716  gsumxp  15243  pjfval  16622  pjpm  16624  txbas  17278  txcls  17315  txrest  17341  metreslem  17942  ressxms  18087  ressms  18088  xpima  23217  mbfmcst  23579  0rrv  23669  domrancur1b  25303  domrancur1c  25305  selsubf3  26094
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-xp 4711  df-rel 4712
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