Theorem sgrpmgm 17675

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 2777	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2777	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3	1, 2	issgrp 17671	. 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 493	1 ⊢ (𝑀 ∈ SGrp → 𝑀 ∈ Mgm)

Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2106  ∀wral 3089  ‘cfv 6135  (class class class)co 6922  Basecbs 16255  +_gcplusg 16338  Mgmcmgm 17626  SGrpcsgrp 17669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-nul 5025

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-ov 6925  df-sgrp 17670

This theorem is referenced by:  mndmgm  17686  sgrpssmgm  17807  dfgrp2  17834  dfgrp3e  17902  mulgnndir  17955  mulgnnass  17961  rngcl  42891  isrnghmmul  42901  idrnghm  42916  c0rnghm  42921