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Theorem sgrpmgm 17336
Description: A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
Assertion
Ref Expression
sgrpmgm (𝑀 ∈ SGrp → 𝑀 ∈ Mgm)

Proof of Theorem sgrpmgm
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2651 . . 3 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2651 . . 3 (+g𝑀) = (+g𝑀)
31, 2issgrp 17332 . 2 (𝑀 ∈ SGrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧))))
43simplbi 475 1 (𝑀 ∈ SGrp → 𝑀 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  wral 2941  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  Mgmcmgm 17287  SGrpcsgrp 17330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-sgrp 17331
This theorem is referenced by:  mndmgm  17347  sgrpssmgm  17467  dfgrp2  17494  dfgrp3e  17562  mulgnndir  17616  mulgnnass  17623  rngcl  42208  isrnghmmul  42218  idrnghm  42233  c0rnghm  42238
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