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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  |-  B  =  ( Base `  G
)
ismnd.p  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
ismnd  |-  ( G  e.  Mnd  <->  ( A. a  e.  B  A. b  e.  B  (
( a  .+  b
)  e.  B  /\  A. c  e.  B  ( ( a  .+  b
)  .+  c )  =  ( a  .+  ( b  .+  c
) ) )  /\  E. e  e.  B  A. a  e.  B  (
( e  .+  a
)  =  a  /\  ( a  .+  e
)  =  a ) ) )
Distinct variable groups:    B, a,
b, c    B, e,
a    G, a, b, c    .+ , a, e    .+ , b,
c
Allowed substitution hint:    G( e)

Proof of Theorem ismnd
StepHypRef Expression
1 ismnd.b . . 3  |-  B  =  ( Base `  G
)
2 ismnd.p . . 3  |-  .+  =  ( +g  `  G )
31, 2ismnddef 16537 . 2  |-  ( G  e.  Mnd  <->  ( G  e. SGrp  /\  E. e  e.  B  A. a  e.  B  ( ( e 
.+  a )  =  a  /\  ( a 
.+  e )  =  a ) ) )
4 rexn0 3902 . . . 4  |-  ( E. e  e.  B  A. a  e.  B  (
( e  .+  a
)  =  a  /\  ( a  .+  e
)  =  a )  ->  B  =/=  (/) )
5 fvprc 5875 . . . . . 6  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
61, 5syl5eq 2475 . . . . 5  |-  ( -.  G  e.  _V  ->  B  =  (/) )
76necon1ai 2651 . . . 4  |-  ( B  =/=  (/)  ->  G  e.  _V )
81, 2issgrpv 16528 . . . 4  |-  ( G  e.  _V  ->  ( G  e. SGrp  <->  A. a  e.  B  A. b  e.  B  ( ( a  .+  b )  e.  B  /\  A. c  e.  B  ( ( a  .+  b )  .+  c
)  =  ( a 
.+  ( b  .+  c ) ) ) ) )
94, 7, 83syl 18 . . 3  |-  ( E. e  e.  B  A. a  e.  B  (
( e  .+  a
)  =  a  /\  ( a  .+  e
)  =  a )  ->  ( G  e. SGrp  <->  A. a  e.  B  A. b  e.  B  (
( a  .+  b
)  e.  B  /\  A. c  e.  B  ( ( a  .+  b
)  .+  c )  =  ( a  .+  ( b  .+  c
) ) ) ) )
109pm5.32ri 642 . 2  |-  ( ( G  e. SGrp  /\  E. e  e.  B  A. a  e.  B  (
( e  .+  a
)  =  a  /\  ( a  .+  e
)  =  a ) )  <->  ( A. a  e.  B  A. b  e.  B  ( (
a  .+  b )  e.  B  /\  A. c  e.  B  ( (
a  .+  b )  .+  c )  =  ( a  .+  ( b 
.+  c ) ) )  /\  E. e  e.  B  A. a  e.  B  ( (
e  .+  a )  =  a  /\  (
a  .+  e )  =  a ) ) )
113, 10bitri 252 1  |-  ( G  e.  Mnd  <->  ( A. a  e.  B  A. b  e.  B  (
( a  .+  b
)  e.  B  /\  A. c  e.  B  ( ( a  .+  b
)  .+  c )  =  ( a  .+  ( b  .+  c
) ) )  /\  E. e  e.  B  A. a  e.  B  (
( e  .+  a
)  =  a  /\  ( a  .+  e
)  =  a ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   E.wrex 2772   _Vcvv 3080   (/)c0 3761   ` cfv 5601  (class class class)co 6305   Basecbs 15120   +g cplusg 15189  SGrpcsgrp 16525   Mndcmnd 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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