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

Theorem mul12i 9796
 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 9767 . 2
51, 2, 3, 4mp3an 1324 1
 Colors of variables: wff setvar class Syntax hints:  =wceq 1395  e.wcel 1818  (class class class)co 6296   cc 9511   cmul 9518 This theorem is referenced by:  faclbnd4lem1  12371  decsplit  14569  root1eq1  23129  cxpeq  23131  1cubrlem  23172  efiatan2  23248  2efiatan  23249  tanatan  23250  log2ublem2  23278  log2ublem3  23279  bposlem8  23566  ax5seglem7  24238  ip1ilem  25741  ipasslem10  25754  polid2i  26074  bpoly3  29820 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-mulcom 9577  ax-mulass 9579 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-iota 5556  df-fv 5601  df-ov 6299
 Copyright terms: Public domain W3C validator