Theorem bj-smgrpssmgm 37232

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 18695	. 2 ⊢ Smgrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑝𝑦)𝑝𝑧) = (𝑥𝑝(𝑦𝑝𝑧))}
2	1	ssrab3 4057	1 ⊢ Smgrp ⊆ Mgm

Syntax hints:   = wceq 1540  ∀wral 3051  [wsbc 3765   ⊆ wss 3926  ‘cfv 6530  (class class class)co 7403  Basecbs 17226  +_gcplusg 17269  Mgmcmgm 18614  Smgrpcsgrp 18694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-ss 3943  df-sgrp 18695

This theorem is referenced by:  bj-smgrpssmgmel  37233