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

Theorem xpindir 5135
Description: Distributive law for Cartesian product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
Assertion
Ref Expression
xpindir  |-  ( ( A  i^i  B )  X.  C )  =  ( ( A  X.  C )  i^i  ( B  X.  C ) )

Proof of Theorem xpindir
StepHypRef Expression
1 inxp 5133 . 2  |-  ( ( A  X.  C )  i^i  ( B  X.  C ) )  =  ( ( A  i^i  B )  X.  ( C  i^i  C ) )
2 inidm 3707 . . 3  |-  ( C  i^i  C )  =  C
32xpeq2i 5020 . 2  |-  ( ( A  i^i  B )  X.  ( C  i^i  C ) )  =  ( ( A  i^i  B
)  X.  C )
41, 3eqtr2i 2497 1  |-  ( ( A  i^i  B )  X.  C )  =  ( ( A  X.  C )  i^i  ( B  X.  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    i^i cin 3475    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-opab 4506  df-xp 5005  df-rel 5006
This theorem is referenced by:  resres  5284  resindi  5287  imainrect  5446  resdmres  5496  cdaassen  8558  txhaus  19880  ustund  20456
  Copyright terms: Public domain W3C validator