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

Theorem mul32i 9264
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 9235 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( ( A  x.  C )  x.  B ) )
51, 2, 3, 4mp3an 1280 1  |-  ( ( A  x.  B )  x.  C )  =  ( ( A  x.  C )  x.  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726  (class class class)co 6083   CCcc 8990    x. cmul 8997
This theorem is referenced by:  8th4div3  10193  faclbnd4lem1  11586  dec5nprm  13404  dec2nprm  13405  karatsuba  13422  quart1lem  20697  log2ublem2  20789  log2ub  20791  normlem3  22616  bcseqi  22624  bpoly4  26107
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-mulcom 9056  ax-mulass 9058
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086
  Copyright terms: Public domain W3C validator