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Theorem isasslaw 47332
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 483 . . 3 ((𝑜 = 𝑚 = 𝑀) → 𝑚 = 𝑀)
2 id 22 . . . . . . . 8 (𝑜 = 𝑜 = )
3 oveq 7432 . . . . . . . 8 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
4 eqidd 2729 . . . . . . . 8 (𝑜 = 𝑧 = 𝑧)
52, 3, 4oveq123d 7447 . . . . . . 7 (𝑜 = → ((𝑥𝑜𝑦)𝑜𝑧) = ((𝑥 𝑦) 𝑧))
6 eqidd 2729 . . . . . . . 8 (𝑜 = 𝑥 = 𝑥)
7 oveq 7432 . . . . . . . 8 (𝑜 = → (𝑦𝑜𝑧) = (𝑦 𝑧))
82, 6, 7oveq123d 7447 . . . . . . 7 (𝑜 = → (𝑥𝑜(𝑦𝑜𝑧)) = (𝑥 (𝑦 𝑧)))
95, 8eqeq12d 2744 . . . . . 6 (𝑜 = → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
109adantr 479 . . . . 5 ((𝑜 = 𝑚 = 𝑀) → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
111, 10raleqbidv 3340 . . . 4 ((𝑜 = 𝑚 = 𝑀) → (∀𝑧𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
121, 11raleqbidv 3340 . . 3 ((𝑜 = 𝑚 = 𝑀) → (∀𝑦𝑚𝑧𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
131, 12raleqbidv 3340 . 2 ((𝑜 = 𝑚 = 𝑀) → (∀𝑥𝑚𝑦𝑚𝑧𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
14 df-asslaw 47328 . 2 assLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚𝑧𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
1513, 14brabga 5540 1 (( 𝑉𝑀𝑊) → ( assLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3058   class class class wbr 5152  (class class class)co 7426   assLaw casslaw 47324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-iota 6505  df-fv 6561  df-ov 7429  df-asslaw 47328
This theorem is referenced by:  asslawass  47333  sgrpplusgaopALT  47335  isassintop  47350  assintopass  47354  sgrp2sgrp  47368
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