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Theorem sgrp2sgrp 42465
Description: Equivalence of the two definitions of a semigroup. (Contributed by AV, 16-Jan-2020.)
Assertion
Ref Expression
sgrp2sgrp (𝑀 ∈ SGrpALT ↔ 𝑀 ∈ SGrp)

Proof of Theorem sgrp2sgrp
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgm2mgm 42464 . . . 4 (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm)
21anbi1i 617 . . 3 ((𝑀 ∈ MgmALT ∧ (+g𝑀) assLaw (Base‘𝑀)) ↔ (𝑀 ∈ Mgm ∧ (+g𝑀) assLaw (Base‘𝑀)))
3 fvex 6388 . . . . . 6 (+g𝑀) ∈ V
4 fvex 6388 . . . . . 6 (Base‘𝑀) ∈ V
53, 4pm3.2i 462 . . . . 5 ((+g𝑀) ∈ V ∧ (Base‘𝑀) ∈ V)
6 isasslaw 42429 . . . . 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 570 . . 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 266 . 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 2765 . . 3 (Base‘𝑀) = (Base‘𝑀)
11 eqid 2765 . . 3 (+g𝑀) = (+g𝑀)
1210, 11issgrpALT 42462 . 2 (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ (+g𝑀) assLaw (Base‘𝑀)))
1310, 11issgrp 17553 . 2 (𝑀 ∈ SGrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧))))
149, 12, 133bitr4i 294 1 (𝑀 ∈ SGrpALT ↔ 𝑀 ∈ SGrp)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350   class class class wbr 4809  cfv 6068  (class class class)co 6842  Basecbs 16132  +gcplusg 16216  Mgmcmgm 17508  SGrpcsgrp 17551   assLaw casslaw 42421  MgmALTcmgm2 42452  SGrpALTcsgrp2 42454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-iota 6031  df-fv 6076  df-ov 6845  df-mgm 17510  df-sgrp 17552  df-cllaw 42423  df-asslaw 42425  df-mgm2 42456  df-sgrp2 42458
This theorem is referenced by: (None)
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