Theorem sgrpmgm 18632

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 2731	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2731	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3	1, 2	issgrp 18628	. 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 1541   ∈ wcel 2111  ∀wral 3047  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  +_gcplusg 17161  Mgmcmgm 18546  Smgrpcsgrp 18626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ov 7349  df-sgrp 18627

This theorem is referenced by:  sgrpcl  18634  mndmgm  18649  gsumsgrpccat  18748  sgrpssmgm  18841  dfgrp2  18875  dfgrp3e  18953  mulgnndir  19016  mulgnnass  19022  rngcl  20082  isrnghmmul  20360  idrnghm  20376  c0rnghm  20450