Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isasslaw Structured version   Visualization version   GIF version

Theorem isasslaw 45386
Description: The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (Revised by AV, 13-Jan-2020.)
Assertion
Ref Expression
isasslaw (( 𝑉𝑀𝑊) → ( assLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
Distinct variable groups:   𝑥,𝑀,𝑦,𝑧   𝑥, ,𝑦,𝑧
Allowed substitution hints:   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)

Proof of Theorem isasslaw
Dummy variables 𝑚 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 485 . . 3 ((𝑜 = 𝑚 = 𝑀) → 𝑚 = 𝑀)
2 id 22 . . . . . . . 8 (𝑜 = 𝑜 = )
3 oveq 7281 . . . . . . . 8 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
4 eqidd 2739 . . . . . . . 8 (𝑜 = 𝑧 = 𝑧)
52, 3, 4oveq123d 7296 . . . . . . 7 (𝑜 = → ((𝑥𝑜𝑦)𝑜𝑧) = ((𝑥 𝑦) 𝑧))
6 eqidd 2739 . . . . . . . 8 (𝑜 = 𝑥 = 𝑥)
7 oveq 7281 . . . . . . . 8 (𝑜 = → (𝑦𝑜𝑧) = (𝑦 𝑧))
82, 6, 7oveq123d 7296 . . . . . . 7 (𝑜 = → (𝑥𝑜(𝑦𝑜𝑧)) = (𝑥 (𝑦 𝑧)))
95, 8eqeq12d 2754 . . . . . 6 (𝑜 = → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
109adantr 481 . . . . 5 ((𝑜 = 𝑚 = 𝑀) → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
111, 10raleqbidv 3336 . . . 4 ((𝑜 = 𝑚 = 𝑀) → (∀𝑧𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
121, 11raleqbidv 3336 . . 3 ((𝑜 = 𝑚 = 𝑀) → (∀𝑦𝑚𝑧𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
131, 12raleqbidv 3336 . 2 ((𝑜 = 𝑚 = 𝑀) → (∀𝑥𝑚𝑦𝑚𝑧𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
14 df-asslaw 45382 . 2 assLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚𝑧𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
1513, 14brabga 5447 1 (( 𝑉𝑀𝑊) → ( assLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wral 3064   class class class wbr 5074  (class class class)co 7275   assLaw casslaw 45378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-iota 6391  df-fv 6441  df-ov 7278  df-asslaw 45382
This theorem is referenced by:  asslawass  45387  sgrpplusgaopALT  45389  isassintop  45404  assintopass  45408  sgrp2sgrp  45422
  Copyright terms: Public domain W3C validator