Theorem mndsgrp 18708

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.

Step	Hyp	Ref	Expression
1		eqid 2736	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2736	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
3	1, 2	ismnddef 18704	. 2 ⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))
4	3	simplbi 496	1 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  Smgrpcsgrp 18686  Mndcmnd 18702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-mnd 18703

This theorem is referenced by:  mndmgm  18709  mndass  18711  gsumccat  18809  mndsssgrp  18905  grpsgrp  18936  mulgnn0dir  19080  mulgnn0ass  19086  ringrng  20266  fidomncyc  42980