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

Theorem caovassd 7336
Description: Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovassg.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
caovassd.2 (𝜑𝐴𝑆)
caovassd.3 (𝜑𝐵𝑆)
caovassd.4 (𝜑𝐶𝑆)
Assertion
Ref Expression
caovassd (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem caovassd
StepHypRef Expression
1 id 22 . 2 (𝜑𝜑)
2 caovassd.2 . 2 (𝜑𝐴𝑆)
3 caovassd.3 . 2 (𝜑𝐵𝑆)
4 caovassd.4 . 2 (𝜑𝐶𝑆)
5 caovassg.1 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
65caovassg 7335 . 2 ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
71, 2, 3, 4, 6syl13anc 1364 1 (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  (class class class)co 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148
This theorem is referenced by:  caov32d  7357  caov12d  7358  caov13d  7360  caov4d  7361  seqf1olem2a  13396  grprinvlem  17871  grprinvd  17872  grpridd  17873  grprcan  18075
  Copyright terms: Public domain W3C validator