Theorem sgrp2sgrp 48814

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.

Step	Hyp	Ref	Expression
1		mgm2mgm 48813	. . . 4 ⊢ (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm)
2	1	anbi1i 633	. . 3 ⊢ ((𝑀 ∈ MgmALT ∧ (+g‘𝑀) assLaw (Base‘𝑀)) ↔ (𝑀 ∈ Mgm ∧ (+g‘𝑀) assLaw (Base‘𝑀)))
3		fvex 6876	. . . . . 6 ⊢ (+g‘𝑀) ∈ V
4		fvex 6876	. . . . . 6 ⊢ (Base‘𝑀) ∈ V
5	3, 4	pm3.2i 474	. . . . 5 ⊢ ((+g‘𝑀) ∈ V ∧ (Base‘𝑀) ∈ V)
6		isasslaw 48778	. . . . 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 582	. . 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 277	. 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 2761	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
11		eqid 2761	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
12	10, 11	issgrpALT 48811	. 2 ⊢ (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ (+g‘𝑀) assLaw (Base‘𝑀)))
13	10, 11	issgrp 18737	. 2 ⊢ (𝑀 ∈ Smgrp ↔ (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 305	1 ⊢ (𝑀 ∈ SGrpALT ↔ 𝑀 ∈ Smgrp)

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   class class class wbr 5099  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  +_gcplusg 17269  Mgmcmgm 18655  Smgrpcsgrp 18735   assLaw casslaw 48770  MgmALTcmgm2 48801  SGrpALTcsgrp2 48803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-iota 6473  df-fv 6525  df-ov 7395  df-mgm 18657  df-sgrp 18736  df-cllaw 48772  df-asslaw 48774  df-mgm2 48805  df-sgrp2 48807

This theorem is referenced by: (None)