Theorem sgrpmgm 18649

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 2724	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2724	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3	1, 2	issgrp 18645	. 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 1533   ∈ wcel 2098  ∀wral 3053  ‘cfv 6534  (class class class)co 7402  Basecbs 17145  +_gcplusg 17198  Mgmcmgm 18563  Smgrpcsgrp 18643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-nul 5297

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-iota 6486  df-fv 6542  df-ov 7405  df-sgrp 18644

This theorem is referenced by:  sgrpcl  18651  mndmgm  18666  gsumsgrpccat  18757  sgrpssmgm  18850  dfgrp2  18884  dfgrp3e  18960  mulgnndir  19022  mulgnnass  19028  rngcl  20061  isrnghmmul  20336  idrnghm  20352  c0rnghm  20427