Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sgrpmgm Structured version   Visualization version   GIF version

Theorem sgrpmgm 17906
 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.
StepHypRef Expression
1 eqid 2824 . . 3 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2824 . . 3 (+g𝑀) = (+g𝑀)
31, 2issgrp 17902 . 2 (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧))))
43simplbi 501 1 (𝑀 ∈ Smgrp → 𝑀 ∈ Mgm)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  ∀wral 3133  ‘cfv 6343  (class class class)co 7149  Basecbs 16483  +gcplusg 16565  Mgmcmgm 17850  Smgrpcsgrp 17900 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5196 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7152  df-sgrp 17901 This theorem is referenced by:  mndmgm  17918  gsumsgrpccat  18004  sgrpssmgm  18098  dfgrp2  18128  dfgrp3e  18199  mulgnndir  18256  mulgnnass  18262  rngcl  44433  isrnghmmul  44443  idrnghm  44458  c0rnghm  44463
 Copyright terms: Public domain W3C validator