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Theorem inxp 4792
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
StepHypRef Expression
1 inopab 4790 . . 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 800 . . . . 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 3319 . . . . . 6  |-  ( x  e.  ( A  i^i  C )  <->  ( x  e.  A  /\  x  e.  C ) )
4 elin 3319 . . . . . 6  |-  ( y  e.  ( B  i^i  D )  <->  ( y  e.  B  /\  y  e.  D ) )
53, 4anbi12i 681 . . . . 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 245 . . . 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 4043 . . 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 2276 . 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 4661 . . 3  |-  ( A  X.  B )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  B ) }
10 df-xp 4661 . . 3  |-  ( C  X.  D )  =  { <. x ,  y
>.  |  ( x  e.  C  /\  y  e.  D ) }
119, 10ineq12i 3329 . 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 4661 . 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 2286 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 360    = wceq 1619    e. wcel 1621    i^i cin 3112   {copab 4036    X. cxp 4645
This theorem is referenced by:  xpindi  4793  xpindir  4794  dmxpin  4873  xpssres  4963  xpdisj1  5075  xpdisj2  5076  imainrect  5093  curry1  6130  curry2  6133  fpar  6142  marypha1lem  7140  fpwwe2lem13  8218  hashxplem  11336  sscres  13648  gsumxp  15175  pjfval  16554  pjpm  16556  txbas  17210  txcls  17247  txrest  17273  metreslem  17874  ressxms  18019  ressms  18020  domrancur1b  24553  domrancur1c  24555  selsubf3  25344
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-opab 4038  df-xp 4661  df-rel 4662
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