Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mul12i Structured version   Unicode version

Theorem mul12i 9266
 Description: Commutative/associative law that swaps the first two factors in a triple product. (Contributed by NM, 11-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
Hypotheses
Ref Expression
mul.1
mul.2
mul.3
Assertion
Ref Expression
mul12i

Proof of Theorem mul12i
StepHypRef Expression
1 mul.1 . 2
2 mul.2 . 2
3 mul.3 . 2
4 mul12 9237 . 2
51, 2, 3, 4mp3an 1280 1
 Colors of variables: wff set class Syntax hints:   wceq 1653   wcel 1726  (class class class)co 6084  cc 8993   cmul 9000 This theorem is referenced by:  faclbnd4lem1  11589  decsplit  13424  root1eq1  20644  cxpeq  20646  1cubrlem  20686  efiatan2  20762  2efiatan  20763  tanatan  20764  log2ublem2  20792  log2ublem3  20793  bposlem8  21080  ip1ilem  22332  ipasslem10  22345  polid2i  22664  ax5seglem7  25879  bpoly3  26109 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 9059  ax-mulass 9061 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 4216  df-iota 5421  df-fv 5465  df-ov 6087
 Copyright terms: Public domain W3C validator