Theorem sgrpmgm 18922

Description: A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)

Assertion

Ref	Expression
sgrpmgm	⊢ (𝑀 ∈ Smgrp → 𝑀 ∈ Mgm)

Proof of Theorem sgrpmgm

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2760	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2760	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3	1, 2	issgrp 18918	. 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))))
4	3	simplbi 502	1 ⊢ (𝑀 ∈ Smgrp → 𝑀 ∈ Mgm)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ‘cfv 6543  (class class class)co 7424  Basecbs 17396  +_gcplusg 17437  Mgmcmgm 18823  Smgrpcsgrp 18916

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6497  df-fv 6551  df-ov 7427  df-sgrp 18917

This theorem is used by:  sgrpcl  18924  mndmgm  18939  gsumsgrpccat  19045  sgrpssmgm  19141  dfgrp2  19182  dfgrp3e  19259  mulgnndir  19322  mulgnnass  19328  rngcl  20395  rng1zr  20413  isrnghmmul  20681  idrnghm  20697  c0rnghm  20796