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Theorem xpriindi 5128
Description: Distributive law for Cartesian product over relativized indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
xpriindi  |-  ( C  X.  ( D  i^i  |^|_
x  e.  A  B
) )  =  ( ( C  X.  D
)  i^i  |^|_ x  e.  A  ( C  X.  B ) )
Distinct variable groups:    x, A    x, C
Allowed substitution hints:    B( x)    D( x)

Proof of Theorem xpriindi
StepHypRef Expression
1 iineq1 4330 . . . . . . 7  |-  ( A  =  (/)  ->  |^|_ x  e.  A  B  =  |^|_
x  e.  (/)  B )
2 0iin 4373 . . . . . . 7  |-  |^|_ x  e.  (/)  B  =  _V
31, 2syl6eq 2511 . . . . . 6  |-  ( A  =  (/)  ->  |^|_ x  e.  A  B  =  _V )
43ineq2d 3686 . . . . 5  |-  ( A  =  (/)  ->  ( D  i^i  |^|_ x  e.  A  B )  =  ( D  i^i  _V )
)
5 inv1 3811 . . . . 5  |-  ( D  i^i  _V )  =  D
64, 5syl6eq 2511 . . . 4  |-  ( A  =  (/)  ->  ( D  i^i  |^|_ x  e.  A  B )  =  D )
76xpeq2d 5012 . . 3  |-  ( A  =  (/)  ->  ( C  X.  ( D  i^i  |^|_
x  e.  A  B
) )  =  ( C  X.  D ) )
8 iineq1 4330 . . . . . 6  |-  ( A  =  (/)  ->  |^|_ x  e.  A  ( C  X.  B )  =  |^|_ x  e.  (/)  ( C  X.  B ) )
9 0iin 4373 . . . . . 6  |-  |^|_ x  e.  (/)  ( C  X.  B )  =  _V
108, 9syl6eq 2511 . . . . 5  |-  ( A  =  (/)  ->  |^|_ x  e.  A  ( C  X.  B )  =  _V )
1110ineq2d 3686 . . . 4  |-  ( A  =  (/)  ->  ( ( C  X.  D )  i^i  |^|_ x  e.  A  ( C  X.  B
) )  =  ( ( C  X.  D
)  i^i  _V )
)
12 inv1 3811 . . . 4  |-  ( ( C  X.  D )  i^i  _V )  =  ( C  X.  D
)
1311, 12syl6eq 2511 . . 3  |-  ( A  =  (/)  ->  ( ( C  X.  D )  i^i  |^|_ x  e.  A  ( C  X.  B
) )  =  ( C  X.  D ) )
147, 13eqtr4d 2498 . 2  |-  ( A  =  (/)  ->  ( C  X.  ( D  i^i  |^|_
x  e.  A  B
) )  =  ( ( C  X.  D
)  i^i  |^|_ x  e.  A  ( C  X.  B ) ) )
15 xpindi 5125 . . 3  |-  ( C  X.  ( D  i^i  |^|_
x  e.  A  B
) )  =  ( ( C  X.  D
)  i^i  ( C  X.  |^|_ x  e.  A  B ) )
16 xpiindi 5127 . . . 4  |-  ( A  =/=  (/)  ->  ( C  X.  |^|_ x  e.  A  B )  =  |^|_ x  e.  A  ( C  X.  B ) )
1716ineq2d 3686 . . 3  |-  ( A  =/=  (/)  ->  ( ( C  X.  D )  i^i  ( C  X.  |^|_ x  e.  A  B ) )  =  ( ( C  X.  D )  i^i  |^|_ x  e.  A  ( C  X.  B
) ) )
1815, 17syl5eq 2507 . 2  |-  ( A  =/=  (/)  ->  ( C  X.  ( D  i^i  |^|_ x  e.  A  B ) )  =  ( ( C  X.  D )  i^i  |^|_ x  e.  A  ( C  X.  B
) ) )
1914, 18pm2.61ine 2767 1  |-  ( C  X.  ( D  i^i  |^|_
x  e.  A  B
) )  =  ( ( C  X.  D
)  i^i  |^|_ x  e.  A  ( C  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    =/= wne 2649   _Vcvv 3106    i^i cin 3460   (/)c0 3783   |^|_ciin 4316    X. cxp 4986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-iin 4318  df-opab 4498  df-xp 4994  df-rel 4995
This theorem is referenced by: (None)
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