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Theorem sgrpplusgaopALT 42341
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 479 . 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 2760 . . 3 (Base‘𝐺) = (Base‘𝐺)
3 eqid 2760 . . 3 (+g𝐺) = (+g𝐺)
42, 3issgrp 17486 . 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 6362 . . 3 (+g𝐺) ∈ V
6 fvex 6362 . . 3 (Base‘𝐺) ∈ V
7 isasslaw 42338 . . 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 710 . 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 281 1 (𝐺 ∈ SGrp → (+g𝐺) assLaw (Base‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340   class class class wbr 4804  cfv 6049  (class class class)co 6813  Basecbs 16059  +gcplusg 16143  Mgmcmgm 17441  SGrpcsgrp 17484   assLaw casslaw 42330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-iota 6012  df-fv 6057  df-ov 6816  df-sgrp 17485  df-asslaw 42334
This theorem is referenced by: (None)
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