MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mul32i Structured version   Unicode version

Theorem mul32i 9764
Description: Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.)
Hypotheses
Ref Expression
mul.1  |-  A  e.  CC
mul.2  |-  B  e.  CC
mul.3  |-  C  e.  CC
Assertion
Ref Expression
mul32i  |-  ( ( A  x.  B )  x.  C )  =  ( ( A  x.  C )  x.  B
)

Proof of Theorem mul32i
StepHypRef Expression
1 mul.1 . 2  |-  A  e.  CC
2 mul.2 . 2  |-  B  e.  CC
3 mul.3 . 2  |-  C  e.  CC
4 mul32 9735 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( ( A  x.  C )  x.  B ) )
51, 2, 3, 4mp3an 1319 1  |-  ( ( A  x.  B )  x.  C )  =  ( ( A  x.  C )  x.  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    e. wcel 1762  (class class class)co 6275   CCcc 9479    x. cmul 9486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-mulcom 9545  ax-mulass 9547
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-iota 5542  df-fv 5587  df-ov 6278
This theorem is referenced by:  8th4div3  10748  faclbnd4lem1  12326  dec5nprm  14400  dec2nprm  14401  karatsuba  14418  quart1lem  22907  log2ublem2  22999  log2ub  23001  normlem3  25691  bcseqi  25699  bpoly4  29384
  Copyright terms: Public domain W3C validator