MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mndsgrp Structured version   Visualization version   GIF version

Theorem mndsgrp 17905
Description: A monoid is a semigroup. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) (Proof shortened by AV, 6-Feb-2020.)
Assertion
Ref Expression
mndsgrp (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp)

Proof of Theorem mndsgrp
Dummy variables 𝑒 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2818 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2818 . . 3 (+g𝐺) = (+g𝐺)
31, 2ismnddef 17901 . 2 (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
43simplbi 498 1 (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  Smgrpcsgrp 17888  Mndcmnd 17899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-mnd 17900
This theorem is referenced by:  mndmgm  17906  mndass  17908  gsumccat  17994  mndsssgrp  18037  grpsgrp  18065  mulgnn0dir  18195  mulgnn0ass  18201  ringrng  44078
  Copyright terms: Public domain W3C validator