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Theorem mndsgrp 18570
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 2733 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2733 . . 3 (+g𝐺) = (+g𝐺)
31, 2ismnddef 18566 . 2 (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
43simplbi 499 1 (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3061  wrex 3070  cfv 6500  (class class class)co 7361  Basecbs 17091  +gcplusg 17141  Smgrpcsgrp 18553  Mndcmnd 18564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5267
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-iota 6452  df-fv 6508  df-ov 7364  df-mnd 18565
This theorem is referenced by:  mndmgm  18571  mndass  18573  gsumccat  18659  mndsssgrp  18752  grpsgrp  18782  mulgnn0dir  18914  mulgnn0ass  18920  ringrng  46267
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