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Theorem sgrpplusgaopALT 41621
 Description: Slot 2 (group operation) of a semigroup as extensible structure is an associative operation on the base set. (Contributed by AV, 13-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sgrpplusgaopALT (𝐺 ∈ SGrp → (+g𝐺) assLaw (Base‘𝐺))

Proof of Theorem sgrpplusgaopALT
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 476 . 2 ((𝐺 ∈ Mgm ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧))) → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧)))
2 eqid 2610 . . 3 (Base‘𝐺) = (Base‘𝐺)
3 eqid 2610 . . 3 (+g𝐺) = (+g𝐺)
42, 3issgrp 17108 . 2 (𝐺 ∈ SGrp ↔ (𝐺 ∈ Mgm ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧))))
5 fvex 6113 . . 3 (+g𝐺) ∈ V
6 fvex 6113 . . 3 (Base‘𝐺) ∈ V
7 isasslaw 41618 . . 3 (((+g𝐺) ∈ V ∧ (Base‘𝐺) ∈ V) → ((+g𝐺) assLaw (Base‘𝐺) ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧))))
85, 6, 7mp2an 704 . 2 ((+g𝐺) assLaw (Base‘𝐺) ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧)))
91, 4, 83imtr4i 280 1 (𝐺 ∈ SGrp → (+g𝐺) assLaw (Base‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  Mgmcmgm 17063  SGrpcsgrp 17106   assLaw casslaw 41610 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-iota 5768  df-fv 5812  df-ov 6552  df-sgrp 17107  df-asslaw 41614 This theorem is referenced by: (None)
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