Theorem bj-smgrpssmgm 37281

Description: Semigroups are magmas. (Contributed by BJ, 12-Apr-2024.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-smgrpssmgm	⊢ Smgrp ⊆ Mgm

Proof of Theorem bj-smgrpssmgm

Dummy variables 𝑔 𝑏 𝑝 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sgrp 18619	. 2 ⊢ Smgrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑝𝑦)𝑝𝑧) = (𝑥𝑝(𝑦𝑝𝑧))}
2	1	ssrab3 4030	1 ⊢ Smgrp ⊆ Mgm

Syntax hints:   = wceq 1541  ∀wral 3045  [wsbc 3739   ⊆ wss 3900  ‘cfv 6477  (class class class)co 7341  Basecbs 17112  +_gcplusg 17153  Mgmcmgm 18538  Smgrpcsgrp 18618

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 2112  ax-9 2120  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3394  df-ss 3917  df-sgrp 18619

This theorem is referenced by:  bj-smgrpssmgmel  37282