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Theorem sgrp2sgrp 44503
 Description: Equivalence of the two definitions of a semigroup. (Contributed by AV, 16-Jan-2020.)
Assertion
Ref Expression
sgrp2sgrp (𝑀 ∈ SGrpALT ↔ 𝑀 ∈ Smgrp)

Proof of Theorem sgrp2sgrp
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgm2mgm 44502 . . . 4 (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm)
21anbi1i 626 . . 3 ((𝑀 ∈ MgmALT ∧ (+g𝑀) assLaw (Base‘𝑀)) ↔ (𝑀 ∈ Mgm ∧ (+g𝑀) assLaw (Base‘𝑀)))
3 fvex 6658 . . . . . 6 (+g𝑀) ∈ V
4 fvex 6658 . . . . . 6 (Base‘𝑀) ∈ V
53, 4pm3.2i 474 . . . . 5 ((+g𝑀) ∈ V ∧ (Base‘𝑀) ∈ V)
6 isasslaw 44467 . . . . 5 (((+g𝑀) ∈ V ∧ (Base‘𝑀) ∈ V) → ((+g𝑀) assLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧))))
75, 6mp1i 13 . . . 4 (𝑀 ∈ Mgm → ((+g𝑀) assLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧))))
87pm5.32i 578 . . 3 ((𝑀 ∈ Mgm ∧ (+g𝑀) assLaw (Base‘𝑀)) ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧))))
92, 8bitri 278 . 2 ((𝑀 ∈ MgmALT ∧ (+g𝑀) assLaw (Base‘𝑀)) ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧))))
10 eqid 2798 . . 3 (Base‘𝑀) = (Base‘𝑀)
11 eqid 2798 . . 3 (+g𝑀) = (+g𝑀)
1210, 11issgrpALT 44500 . 2 (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ (+g𝑀) assLaw (Base‘𝑀)))
1310, 11issgrp 17896 . 2 (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧))))
149, 12, 133bitr4i 306 1 (𝑀 ∈ SGrpALT ↔ 𝑀 ∈ Smgrp)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  Basecbs 16477  +gcplusg 16559  Mgmcmgm 17844  Smgrpcsgrp 17894   assLaw casslaw 44459  MgmALTcmgm2 44490  SGrpALTcsgrp2 44492 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-iota 6283  df-fv 6332  df-ov 7138  df-mgm 17846  df-sgrp 17895  df-cllaw 44461  df-asslaw 44463  df-mgm2 44494  df-sgrp2 44496 This theorem is referenced by: (None)
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