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Theorem ismnd 16538
 Description: The predicate "is a monoid". This is the definig theorem of a monoid by showing that a set is a monoid if and only if it is a set equipped with a closed, everywhere defined internal operation (so, a magma, see mndcl 16544), whose operation is associative (so, a semigroup, see also mndass 16545) and has a two-sided neutral element (see mndid 16548). (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.)
Hypotheses
Ref Expression
ismnd.b
ismnd.p
Assertion
Ref Expression
ismnd
Distinct variable groups:   ,,,   ,,   ,,,   ,,   ,,
Allowed substitution hint:   ()

Proof of Theorem ismnd
StepHypRef Expression
1 ismnd.b . . 3
2 ismnd.p . . 3
31, 2ismnddef 16537 . 2 SGrp
4 rexn0 3902 . . . 4
5 fvprc 5875 . . . . . 6
61, 5syl5eq 2475 . . . . 5
76necon1ai 2651 . . . 4
81, 2issgrpv 16528 . . . 4 SGrp
94, 7, 83syl 18 . . 3 SGrp
109pm5.32ri 642 . 2 SGrp
113, 10bitri 252 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wb 187   wa 370   wceq 1437   wcel 1872   wne 2614  wral 2771  wrex 2772  cvv 3080  c0 3761  cfv 5601  (class class class)co 6305  cbs 15120   cplusg 15189  SGrpcsgrp 16525  cmnd 16534 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-nul 4555  ax-pow 4602 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-iota 5565  df-fv 5609  df-ov 6308  df-mgm 16487  df-sgrp 16526  df-mnd 16536 This theorem is referenced by:  mndclOLD  16546  mndassOLD  16547  mndid  16548  ismndd  16558  mndpropd  16561  mnd1OLD  16577  mhmmnd  16807  signswmnd  29454
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