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Theorem sgrp2sgrp 46248
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 46247 . . . 4 (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm)
21anbi1i 625 . . 3 ((𝑀 ∈ MgmALT ∧ (+g𝑀) assLaw (Base‘𝑀)) ↔ (𝑀 ∈ Mgm ∧ (+g𝑀) assLaw (Base‘𝑀)))
3 fvex 6856 . . . . . 6 (+g𝑀) ∈ V
4 fvex 6856 . . . . . 6 (Base‘𝑀) ∈ V
53, 4pm3.2i 472 . . . . 5 ((+g𝑀) ∈ V ∧ (Base‘𝑀) ∈ V)
6 isasslaw 46212 . . . . 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 576 . . 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 2733 . . 3 (Base‘𝑀) = (Base‘𝑀)
11 eqid 2733 . . 3 (+g𝑀) = (+g𝑀)
1210, 11issgrpALT 46245 . 2 (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ (+g𝑀) assLaw (Base‘𝑀)))
1310, 11issgrp 18552 . 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 205  wa 397   = wceq 1542  wcel 2107  wral 3061  Vcvv 3444   class class class wbr 5106  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  Mgmcmgm 18500  Smgrpcsgrp 18550   assLaw casslaw 46204  MgmALTcmgm2 46235  SGrpALTcsgrp2 46237
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-ne 2941  df-ral 3062  df-rab 3407  df-v 3446  df-sbc 3741  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-mgm 18502  df-sgrp 18551  df-cllaw 46206  df-asslaw 46208  df-mgm2 46239  df-sgrp2 46241
This theorem is referenced by: (None)
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