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Theorem mndsgrp 17220
 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 ∈ SGrp)

Proof of Theorem mndsgrp
Dummy variables 𝑒 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2621 . . 3 (+g𝐺) = (+g𝐺)
31, 2ismnddef 17217 . 2 (𝐺 ∈ Mnd ↔ (𝐺 ∈ SGrp ∧ ∃𝑒 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
43simplbi 476 1 (𝐺 ∈ Mnd → 𝐺 ∈ SGrp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908  ‘cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  SGrpcsgrp 17204  Mndcmnd 17215 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607  df-mnd 17216 This theorem is referenced by:  mndmgm  17221  mndass  17223  mndsssgrp  17342  grpsgrp  17367  mulgnn0dir  17492  mulgnn0ass  17499  ringrng  41164
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