Theorem mndsgrp 18718

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 2735	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2735	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
3	1, 2	ismnddef 18714	. 2 ⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))
4	3	simplbi 497	1 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  +_gcplusg 17271  Smgrpcsgrp 18696  Mndcmnd 18712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-nul 5276

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6484  df-fv 6539  df-ov 7408  df-mnd 18713

This theorem is referenced by:  mndmgm  18719  mndass  18721  gsumccat  18819  mndsssgrp  18912  grpsgrp  18943  mulgnn0dir  19087  mulgnn0ass  19093  ringrng  20245  fidomncyc  42558