MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caov32 Structured version   Visualization version   GIF version

Theorem caov32 7369
Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
Hypotheses
Ref Expression
caov.1 𝐴 ∈ V
caov.2 𝐵 ∈ V
caov.3 𝐶 ∈ V
caov.com (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
caov.ass ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
Assertion
Ref Expression
caov32 ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧

Proof of Theorem caov32
StepHypRef Expression
1 caov.2 . . . 4 𝐵 ∈ V
2 caov.3 . . . 4 𝐶 ∈ V
3 caov.com . . . 4 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
41, 2, 3caovcom 7339 . . 3 (𝐵𝐹𝐶) = (𝐶𝐹𝐵)
54oveq2i 7160 . 2 (𝐴𝐹(𝐵𝐹𝐶)) = (𝐴𝐹(𝐶𝐹𝐵))
6 caov.1 . . 3 𝐴 ∈ V
7 caov.ass . . 3 ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
86, 1, 2, 7caovass 7342 . 2 ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))
96, 2, 1, 7caovass 7342 . 2 ((𝐴𝐹𝐶)𝐹𝐵) = (𝐴𝐹(𝐶𝐹𝐵))
105, 8, 93eqtr4i 2857 1 ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2115  Vcvv 3480  (class class class)co 7149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ral 3138  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7152
This theorem is referenced by:  caov31  7371  addassnq  10378  ltexprlem7  10462  mulcmpblnrlem  10490  recexsrlem  10523  mulgt0sr  10525
  Copyright terms: Public domain W3C validator