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Theorem xpriindi 5138
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 4340 . . . . . . 7  |-  ( A  =  (/)  ->  |^|_ x  e.  A  B  =  |^|_
x  e.  (/)  B )
2 0iin 4383 . . . . . . 7  |-  |^|_ x  e.  (/)  B  =  _V
31, 2syl6eq 2524 . . . . . 6  |-  ( A  =  (/)  ->  |^|_ x  e.  A  B  =  _V )
43ineq2d 3700 . . . . 5  |-  ( A  =  (/)  ->  ( D  i^i  |^|_ x  e.  A  B )  =  ( D  i^i  _V )
)
5 inv1 3812 . . . . 5  |-  ( D  i^i  _V )  =  D
64, 5syl6eq 2524 . . . 4  |-  ( A  =  (/)  ->  ( D  i^i  |^|_ x  e.  A  B )  =  D )
76xpeq2d 5023 . . 3  |-  ( A  =  (/)  ->  ( C  X.  ( D  i^i  |^|_
x  e.  A  B
) )  =  ( C  X.  D ) )
8 iineq1 4340 . . . . . 6  |-  ( A  =  (/)  ->  |^|_ x  e.  A  ( C  X.  B )  =  |^|_ x  e.  (/)  ( C  X.  B ) )
9 0iin 4383 . . . . . 6  |-  |^|_ x  e.  (/)  ( C  X.  B )  =  _V
108, 9syl6eq 2524 . . . . 5  |-  ( A  =  (/)  ->  |^|_ x  e.  A  ( C  X.  B )  =  _V )
1110ineq2d 3700 . . . 4  |-  ( A  =  (/)  ->  ( ( C  X.  D )  i^i  |^|_ x  e.  A  ( C  X.  B
) )  =  ( ( C  X.  D
)  i^i  _V )
)
12 inv1 3812 . . . 4  |-  ( ( C  X.  D )  i^i  _V )  =  ( C  X.  D
)
1311, 12syl6eq 2524 . . 3  |-  ( A  =  (/)  ->  ( ( C  X.  D )  i^i  |^|_ x  e.  A  ( C  X.  B
) )  =  ( C  X.  D ) )
147, 13eqtr4d 2511 . 2  |-  ( A  =  (/)  ->  ( C  X.  ( D  i^i  |^|_
x  e.  A  B
) )  =  ( ( C  X.  D
)  i^i  |^|_ x  e.  A  ( C  X.  B ) ) )
15 xpindi 5135 . . 3  |-  ( C  X.  ( D  i^i  |^|_
x  e.  A  B
) )  =  ( ( C  X.  D
)  i^i  ( C  X.  |^|_ x  e.  A  B ) )
16 xpiindi 5137 . . . 4  |-  ( A  =/=  (/)  ->  ( C  X.  |^|_ x  e.  A  B )  =  |^|_ x  e.  A  ( C  X.  B ) )
1716ineq2d 3700 . . 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 2520 . 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 2780 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 1379    =/= wne 2662   _Vcvv 3113    i^i cin 3475   (/)c0 3785   |^|_ciin 4326    X. cxp 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-iin 4328  df-opab 4506  df-xp 5005  df-rel 5006
This theorem is referenced by: (None)
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