Theorem sgrp2sgrp 48206

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 48205	. . . 4 ⊢ (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm)
2	1	anbi1i 624	. . 3 ⊢ ((𝑀 ∈ MgmALT ∧ (+g‘𝑀) assLaw (Base‘𝑀)) ↔ (𝑀 ∈ Mgm ∧ (+g‘𝑀) assLaw (Base‘𝑀)))
3		fvex 6873	. . . . . 6 ⊢ (+g‘𝑀) ∈ V
4		fvex 6873	. . . . . 6 ⊢ (Base‘𝑀) ∈ V
5	3, 4	pm3.2i 470	. . . . 5 ⊢ ((+g‘𝑀) ∈ V ∧ (Base‘𝑀) ∈ V)
6		isasslaw 48170	. . . . 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 574	. . 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 275	. 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 2730	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
11		eqid 2730	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
12	10, 11	issgrpALT 48203	. 2 ⊢ (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ (+g‘𝑀) assLaw (Base‘𝑀)))
13	10, 11	issgrp 18653	. 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 303	1 ⊢ (𝑀 ∈ SGrpALT ↔ 𝑀 ∈ Smgrp)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  Vcvv 3450   class class class wbr 5109  ‘cfv 6513  (class class class)co 7389  Basecbs 17185  +_gcplusg 17226  Mgmcmgm 18571  Smgrpcsgrp 18651   assLaw casslaw 48162  MgmALTcmgm2 48193  SGrpALTcsgrp2 48195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rab 3409  df-v 3452  df-sbc 3756  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-iota 6466  df-fv 6521  df-ov 7392  df-mgm 18573  df-sgrp 18652  df-cllaw 48164  df-asslaw 48166  df-mgm2 48197  df-sgrp2 48199

This theorem is referenced by: (None)