Theorem sgrp2sgrp 42392

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.

Step	Hyp	Ref	Expression
1		mgm2mgm 42391	. . . 4 ⊢ (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm)
2	1	anbi1i 733	. . 3 ⊢ ((𝑀 ∈ MgmALT ∧ (+g‘𝑀) assLaw (Base‘𝑀)) ↔ (𝑀 ∈ Mgm ∧ (+g‘𝑀) assLaw (Base‘𝑀)))
3		fvex 6363	. . . . . 6 ⊢ (+g‘𝑀) ∈ V
4		fvex 6363	. . . . . 6 ⊢ (Base‘𝑀) ∈ V
5	3, 4	pm3.2i 470	. . . . 5 ⊢ ((+g‘𝑀) ∈ V ∧ (Base‘𝑀) ∈ V)
6		isasslaw 42356	. . . . 5 ⊢ (((+g‘𝑀) ∈ V ∧ (Base‘𝑀) ∈ V) → ((+g‘𝑀) assLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧))))
7	5, 6	mp1i 13	. . . 4 ⊢ (𝑀 ∈ Mgm → ((+g‘𝑀) assLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧))))
8	7	pm5.32i 672	. . 3 ⊢ ((𝑀 ∈ Mgm ∧ (+g‘𝑀) assLaw (Base‘𝑀)) ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧))))
9	2, 8	bitri 264	. 2 ⊢ ((𝑀 ∈ MgmALT ∧ (+g‘𝑀) assLaw (Base‘𝑀)) ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧))))
10		eqid 2760	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
11		eqid 2760	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
12	10, 11	issgrpALT 42389	. 2 ⊢ (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ (+g‘𝑀) assLaw (Base‘𝑀)))
13	10, 11	issgrp 17506	. 2 ⊢ (𝑀 ∈ SGrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧))))
14	9, 12, 13	3bitr4i 292	1 ⊢ (𝑀 ∈ SGrpALT ↔ 𝑀 ∈ SGrp)

Syntax hints:   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050  Vcvv 3340   class class class wbr 4804  ‘cfv 6049  (class class class)co 6814  Basecbs 16079  +_gcplusg 16163  Mgmcmgm 17461  SGrpcsgrp 17504   assLaw casslaw 42348  MgmALTcmgm2 42379  SGrpALTcsgrp2 42381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-iota 6012  df-fv 6057  df-ov 6817  df-mgm 17463  df-sgrp 17505  df-cllaw 42350  df-asslaw 42352  df-mgm2 42383  df-sgrp2 42385

This theorem is referenced by: (None)