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Theorem isasslaw 46212
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 486 . . 3 ((𝑜 = 𝑚 = 𝑀) → 𝑚 = 𝑀)
2 id 22 . . . . . . . 8 (𝑜 = 𝑜 = )
3 oveq 7364 . . . . . . . 8 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
4 eqidd 2734 . . . . . . . 8 (𝑜 = 𝑧 = 𝑧)
52, 3, 4oveq123d 7379 . . . . . . 7 (𝑜 = → ((𝑥𝑜𝑦)𝑜𝑧) = ((𝑥 𝑦) 𝑧))
6 eqidd 2734 . . . . . . . 8 (𝑜 = 𝑥 = 𝑥)
7 oveq 7364 . . . . . . . 8 (𝑜 = → (𝑦𝑜𝑧) = (𝑦 𝑧))
82, 6, 7oveq123d 7379 . . . . . . 7 (𝑜 = → (𝑥𝑜(𝑦𝑜𝑧)) = (𝑥 (𝑦 𝑧)))
95, 8eqeq12d 2749 . . . . . 6 (𝑜 = → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
109adantr 482 . . . . 5 ((𝑜 = 𝑚 = 𝑀) → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
111, 10raleqbidv 3318 . . . 4 ((𝑜 = 𝑚 = 𝑀) → (∀𝑧𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
121, 11raleqbidv 3318 . . 3 ((𝑜 = 𝑚 = 𝑀) → (∀𝑦𝑚𝑧𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
131, 12raleqbidv 3318 . 2 ((𝑜 = 𝑚 = 𝑀) → (∀𝑥𝑚𝑦𝑚𝑧𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
14 df-asslaw 46208 . 2 assLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚𝑧𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
1513, 14brabga 5492 1 (( 𝑉𝑀𝑊) → ( assLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3061   class class class wbr 5106  (class class class)co 7358   assLaw casslaw 46204
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-iota 6449  df-fv 6505  df-ov 7361  df-asslaw 46208
This theorem is referenced by:  asslawass  46213  sgrpplusgaopALT  46215  isassintop  46230  assintopass  46234  sgrp2sgrp  46248
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