Theorem sgrpmgm 18683

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 2739	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2739	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3	1, 2	issgrp 18679	. 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 497	1 ⊢ (𝑀 ∈ Smgrp → 𝑀 ∈ Mgm)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  +_gcplusg 17211  Mgmcmgm 18597  Smgrpcsgrp 18677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-nul 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-ov 7359  df-sgrp 18678

This theorem is referenced by:  sgrpcl  18685  mndmgm  18700  gsumsgrpccat  18799  sgrpssmgm  18895  dfgrp2  18929  dfgrp3e  19007  mulgnndir  19070  mulgnnass  19076  rngcl  20136  isrnghmmul  20413  idrnghm  20429  c0rnghm  20507