Theorem bj-smgrpssmgm 37713

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 18734	. 2 ⊢ Smgrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑝𝑦)𝑝𝑧) = (𝑥𝑝(𝑦𝑝𝑧))}
2	1	ssrab3 4035	1 ⊢ Smgrp ⊆ Mgm

Syntax hints:   = wceq 1559  ∀wral 3075  [wsbc 3744   ⊆ wss 3904  ‘cfv 6515  (class class class)co 7390  Basecbs 17226  +_gcplusg 17267  Mgmcmgm 18653  Smgrpcsgrp 18733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-ss 3921  df-sgrp 18734

This theorem is referenced by:  bj-smgrpssmgmel  37714