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Theorem sgrp2sgrp 46917
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 46916 . . . 4 (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm)
21anbi1i 623 . . 3 ((𝑀 ∈ MgmALT ∧ (+g𝑀) assLaw (Base‘𝑀)) ↔ (𝑀 ∈ Mgm ∧ (+g𝑀) assLaw (Base‘𝑀)))
3 fvex 6904 . . . . . 6 (+g𝑀) ∈ V
4 fvex 6904 . . . . . 6 (Base‘𝑀) ∈ V
53, 4pm3.2i 470 . . . . 5 ((+g𝑀) ∈ V ∧ (Base‘𝑀) ∈ V)
6 isasslaw 46881 . . . . 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 2731 . . 3 (Base‘𝑀) = (Base‘𝑀)
11 eqid 2731 . . 3 (+g𝑀) = (+g𝑀)
1210, 11issgrpALT 46914 . 2 (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ (+g𝑀) assLaw (Base‘𝑀)))
1310, 11issgrp 18648 . 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 395   = wceq 1540  wcel 2105  wral 3060  Vcvv 3473   class class class wbr 5148  cfv 6543  (class class class)co 7412  Basecbs 17151  +gcplusg 17204  Mgmcmgm 18566  Smgrpcsgrp 18646   assLaw casslaw 46873  MgmALTcmgm2 46904  SGrpALTcsgrp2 46906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-iota 6495  df-fv 6551  df-ov 7415  df-mgm 18568  df-sgrp 18647  df-cllaw 46875  df-asslaw 46877  df-mgm2 46908  df-sgrp2 46910
This theorem is referenced by: (None)
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