Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-smgrpssmgm Structured version   Visualization version   GIF version

Theorem bj-smgrpssmgm 37211
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.
StepHypRef Expression
1 df-sgrp 18733 . 2 Smgrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑝𝑦)𝑝𝑧) = (𝑥𝑝(𝑦𝑝𝑧))}
21ssrab3 4092 1 Smgrp ⊆ Mgm
Colors of variables: wff setvar class
Syntax hints:   = wceq 1535  wral 3057  [wsbc 3791  wss 3963  cfv 6558  (class class class)co 7425  Basecbs 17234  +gcplusg 17287  Mgmcmgm 18652  Smgrpcsgrp 18732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-rab 3433  df-ss 3980  df-sgrp 18733
This theorem is referenced by:  bj-smgrpssmgmel  37212
  Copyright terms: Public domain W3C validator