Theorem mndsgrp 18799

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 2763	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2763	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
3	1, 2	ismnddef 18795	. 2 ⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))
4	3	simplbi 501	1 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  +_gcplusg 17311  Smgrpcsgrp 18777  Mndcmnd 18793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5270

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fv 6546  df-ov 7415  df-mnd 18794

This theorem is referenced by:  mndmgm  18800  mndass  18802  gsumccat  18901  mndsssgrp  18997  grpsgrp  19028  mulgnn0dir  19171  mulgnn0ass  19177  ringrng  20369  fidomncyc  43286