Theorem bj-smgrpssmgm 37772

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 18767	. 2 ⊢ Smgrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑝𝑦)𝑝𝑧) = (𝑥𝑝(𝑦𝑝𝑧))}
2	1	ssrab3 4038	1 ⊢ Smgrp ⊆ Mgm

Syntax hints:   = wceq 1563  ∀wral 3079  [wsbc 3747   ⊆ wss 3907  ‘cfv 6525  (class class class)co 7400  Basecbs 17259  +_gcplusg 17300  Mgmcmgm 18686  Smgrpcsgrp 18766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-ss 3924  df-sgrp 18767

This theorem is referenced by:  bj-smgrpssmgmel  37773