ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  muladdi Unicode version

Theorem muladdi 8321
Description: Product of two sums. (Contributed by NM, 17-May-1999.)
Hypotheses
Ref Expression
mulm1.1  |-  A  e.  CC
mulneg.2  |-  B  e.  CC
subdi.3  |-  C  e.  CC
muladdi.4  |-  D  e.  CC
Assertion
Ref Expression
muladdi  |-  ( ( A  +  B )  x.  ( C  +  D ) )  =  ( ( ( A  x.  C )  +  ( D  x.  B
) )  +  ( ( A  x.  D
)  +  ( C  x.  B ) ) )

Proof of Theorem muladdi
StepHypRef Expression
1 mulm1.1 . 2  |-  A  e.  CC
2 mulneg.2 . 2  |-  B  e.  CC
3 subdi.3 . 2  |-  C  e.  CC
4 muladdi.4 . 2  |-  D  e.  CC
5 muladd 8296 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B )  x.  ( C  +  D )
)  =  ( ( ( A  x.  C
)  +  ( D  x.  B ) )  +  ( ( A  x.  D )  +  ( C  x.  B
) ) ) )
61, 2, 3, 4, 5mp4an 425 1  |-  ( ( A  +  B )  x.  ( C  +  D ) )  =  ( ( ( A  x.  C )  +  ( D  x.  B
) )  +  ( ( A  x.  D
)  +  ( C  x.  B ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1348    e. wcel 2141  (class class class)co 5851   CCcc 7765    + caddc 7770    x. cmul 7772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-addcl 7863  ax-mulcl 7865  ax-addcom 7867  ax-mulcom 7868  ax-addass 7869  ax-distr 7871
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-iota 5158  df-fv 5204  df-ov 5854
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator