Theorem bj-smgrpssmgm 37944

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 18787	. 2 ⊢ Smgrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑝𝑦)𝑝𝑧) = (𝑥𝑝(𝑦𝑝𝑧))}
2	1	ssrab3 4039	1 ⊢ Smgrp ⊆ Mgm

This proof depends on syntax axioms:   = wceq 1570  ∀wral 3082  [wsbc 3747   ⊆ wss 3908  ‘cfv 6540  (class class class)co 7416  Basecbs 17279  +_gcplusg 17320  Mgmcmgm 18706  Smgrpcsgrp 18786

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-ss 3925  df-sgrp 18787

This theorem is used by:  bj-smgrpssmgmel  37945