Theorem sgrpmgm 17112

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.

Step	Hyp	Ref	Expression
1		eqid 2610	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2610	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3	1, 2	issgrp 17108	. 2 ⊢ (𝑀 ∈ SGrp ↔ (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 475	1 ⊢ (𝑀 ∈ SGrp → 𝑀 ∈ Mgm)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +_gcplusg 15768  Mgmcmgm 17063  SGrpcsgrp 17106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-sgrp 17107

This theorem is referenced by:  mndmgm  17123  sgrpssmgm  17243  dfgrp2  17270  dfgrp3e  17338  mulgnndir  17392  mulgnnass  17399  rngcl  41673  isrnghmmul  41683  idrnghm  41698  c0rnghm  41703