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Theorem ismnd 15797
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 15802), whose operation is associative (so, a semigroup, see also mndass 15803) and has a two-sided neutral element (see mndid 15806). (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 15796 . 2  |-  ( G  e.  Mnd  <->  ( G  e. SGrp  /\  E. e  e.  B  A. a  e.  B  ( ( e 
.+  a )  =  a  /\  ( a 
.+  e )  =  a ) ) )
4 rexn0 3936 . . . 4  |-  ( E. e  e.  B  A. a  e.  B  (
( e  .+  a
)  =  a  /\  ( a  .+  e
)  =  a )  ->  B  =/=  (/) )
5 fvprc 5866 . . . . . 6  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
61, 5syl5eq 2520 . . . . 5  |-  ( -.  G  e.  _V  ->  B  =  (/) )
76necon1ai 2698 . . . 4  |-  ( B  =/=  (/)  ->  G  e.  _V )
81, 2issgrpv 15787 . . . 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 20 . . 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 638 . 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 249 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 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818   _Vcvv 3118   (/)c0 3790   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   SGrp csgrp 15784   Mndcmnd 15793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4582  ax-pow 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-ov 6298  df-mgm 15746  df-sgrp 15785  df-mnd 15795
This theorem is referenced by:  mndclOLD  15804  mndassOLD  15805  mndid  15806  ismndd  15816  mndpropd  15819  mnd1OLD  15835  sgrp2nmndlem5  15919  mhmmnd  16064  signswmnd  28339
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