MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpriindi Structured version   Unicode version

Theorem xpriindi 4987
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 4311 . . . . . . 7  |-  ( A  =  (/)  ->  |^|_ x  e.  A  B  =  |^|_
x  e.  (/)  B )
2 0iin 4354 . . . . . . 7  |-  |^|_ x  e.  (/)  B  =  _V
31, 2syl6eq 2479 . . . . . 6  |-  ( A  =  (/)  ->  |^|_ x  e.  A  B  =  _V )
43ineq2d 3664 . . . . 5  |-  ( A  =  (/)  ->  ( D  i^i  |^|_ x  e.  A  B )  =  ( D  i^i  _V )
)
5 inv1 3789 . . . . 5  |-  ( D  i^i  _V )  =  D
64, 5syl6eq 2479 . . . 4  |-  ( A  =  (/)  ->  ( D  i^i  |^|_ x  e.  A  B )  =  D )
76xpeq2d 4874 . . 3  |-  ( A  =  (/)  ->  ( C  X.  ( D  i^i  |^|_
x  e.  A  B
) )  =  ( C  X.  D ) )
8 iineq1 4311 . . . . . 6  |-  ( A  =  (/)  ->  |^|_ x  e.  A  ( C  X.  B )  =  |^|_ x  e.  (/)  ( C  X.  B ) )
9 0iin 4354 . . . . . 6  |-  |^|_ x  e.  (/)  ( C  X.  B )  =  _V
108, 9syl6eq 2479 . . . . 5  |-  ( A  =  (/)  ->  |^|_ x  e.  A  ( C  X.  B )  =  _V )
1110ineq2d 3664 . . . 4  |-  ( A  =  (/)  ->  ( ( C  X.  D )  i^i  |^|_ x  e.  A  ( C  X.  B
) )  =  ( ( C  X.  D
)  i^i  _V )
)
12 inv1 3789 . . . 4  |-  ( ( C  X.  D )  i^i  _V )  =  ( C  X.  D
)
1311, 12syl6eq 2479 . . 3  |-  ( A  =  (/)  ->  ( ( C  X.  D )  i^i  |^|_ x  e.  A  ( C  X.  B
) )  =  ( C  X.  D ) )
147, 13eqtr4d 2466 . 2  |-  ( A  =  (/)  ->  ( C  X.  ( D  i^i  |^|_
x  e.  A  B
) )  =  ( ( C  X.  D
)  i^i  |^|_ x  e.  A  ( C  X.  B ) ) )
15 xpindi 4984 . . 3  |-  ( C  X.  ( D  i^i  |^|_
x  e.  A  B
) )  =  ( ( C  X.  D
)  i^i  ( C  X.  |^|_ x  e.  A  B ) )
16 xpiindi 4986 . . . 4  |-  ( A  =/=  (/)  ->  ( C  X.  |^|_ x  e.  A  B )  =  |^|_ x  e.  A  ( C  X.  B ) )
1716ineq2d 3664 . . 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 2475 . 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 2737 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 1437    =/= wne 2618   _Vcvv 3081    i^i cin 3435   (/)c0 3761   |^|_ciin 4297    X. cxp 4848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-iin 4299  df-opab 4480  df-xp 4856  df-rel 4857
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator