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Theorem inxp 4999
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 4997 . . 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 798 . . . . 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 3522 . . . . . 6  |-  ( x  e.  ( A  i^i  C )  <->  ( x  e.  A  /\  x  e.  C ) )
4 elin 3522 . . . . . 6  |-  ( y  e.  ( B  i^i  D )  <->  ( y  e.  B  /\  y  e.  D ) )
53, 4anbi12i 679 . . . . 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 244 . . . 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 4264 . . 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 2455 . 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 4876 . . 3  |-  ( A  X.  B )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  B ) }
10 df-xp 4876 . . 3  |-  ( C  X.  D )  =  { <. x ,  y
>.  |  ( x  e.  C  /\  y  e.  D ) }
119, 10ineq12i 3532 . 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 4876 . 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 2465 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 359    = wceq 1652    e. wcel 1725    i^i cin 3311   {copab 4257    X. cxp 4868
This theorem is referenced by:  xpindi  5000  xpindir  5001  dmxpin  5082  xpssres  5172  xpdisj1  5286  xpdisj2  5287  imainrect  5304  xpima  5305  curry1  6430  curry2  6433  fpar  6442  marypha1lem  7430  fpwwe2lem13  8509  hashxplem  11688  sscres  14015  gsumxp  15542  pjfval  16925  pjpm  16927  txbas  17591  txcls  17628  txrest  17655  trust  18251  ressuss  18285  trcfilu  18316  metreslem  18384  ressxms  18547  ressms  18548  mbfmcst  24601  0rrv  24701
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259  df-xp 4876  df-rel 4877
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