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Theorem sgrp2sgrp 48149
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 48148 . . . 4 (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm)
21anbi1i 624 . . 3 ((𝑀 ∈ MgmALT ∧ (+g𝑀) assLaw (Base‘𝑀)) ↔ (𝑀 ∈ Mgm ∧ (+g𝑀) assLaw (Base‘𝑀)))
3 fvex 6918 . . . . . 6 (+g𝑀) ∈ V
4 fvex 6918 . . . . . 6 (Base‘𝑀) ∈ V
53, 4pm3.2i 470 . . . . 5 ((+g𝑀) ∈ V ∧ (Base‘𝑀) ∈ V)
6 isasslaw 48113 . . . . 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 574 . . 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 275 . 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 2736 . . 3 (Base‘𝑀) = (Base‘𝑀)
11 eqid 2736 . . 3 (+g𝑀) = (+g𝑀)
1210, 11issgrpALT 48146 . 2 (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ (+g𝑀) assLaw (Base‘𝑀)))
1310, 11issgrp 18734 . 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 303 1 (𝑀 ∈ SGrpALT ↔ 𝑀 ∈ Smgrp)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1539  wcel 2107  wral 3060  Vcvv 3479   class class class wbr 5142  cfv 6560  (class class class)co 7432  Basecbs 17248  +gcplusg 17298  Mgmcmgm 18652  Smgrpcsgrp 18732   assLaw casslaw 48105  MgmALTcmgm2 48136  SGrpALTcsgrp2 48138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-iota 6513  df-fv 6568  df-ov 7435  df-mgm 18654  df-sgrp 18733  df-cllaw 48107  df-asslaw 48109  df-mgm2 48140  df-sgrp2 48142
This theorem is referenced by: (None)
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