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Theorem ismnd 12712
Description: The predicate "is a monoid". This is the defining 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 12716), whose operation is associative (so, a semigroup, see also mndass 12717) and has a two-sided neutral element (see mndid 12718). (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
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ismnd.b . . 3  |-  B  =  ( Base `  G
)
2 ismnd.p . . 3  |-  .+  =  ( +g  `  G )
31, 2ismnddef 12711 . 2  |-  ( G  e.  Mnd  <->  ( G  e. Smgrp  /\  E. e  e.  B  A. a  e.  B  ( ( e 
.+  a )  =  a  /\  ( a 
.+  e )  =  a ) ) )
4 rexm 3522 . . . . 5  |-  ( E. e  e.  B  A. a  e.  B  (
( e  .+  a
)  =  a  /\  ( a  .+  e
)  =  a )  ->  E. e  e  e.  B )
5 eleq1w 2238 . . . . . 6  |-  ( e  =  w  ->  (
e  e.  B  <->  w  e.  B ) )
65cbvexv 1918 . . . . 5  |-  ( E. e  e  e.  B  <->  E. w  w  e.  B
)
74, 6sylib 122 . . . 4  |-  ( E. e  e.  B  A. a  e.  B  (
( e  .+  a
)  =  a  /\  ( a  .+  e
)  =  a )  ->  E. w  w  e.  B )
81basmex 12503 . . . . 5  |-  ( w  e.  B  ->  G  e.  _V )
98exlimiv 1598 . . . 4  |-  ( E. w  w  e.  B  ->  G  e.  _V )
101, 2issgrpv 12702 . . . 4  |-  ( G  e.  _V  ->  ( G  e. Smgrp  <->  A. a  e.  B  A. b  e.  B  ( ( a  .+  b )  e.  B  /\  A. c  e.  B  ( ( a  .+  b )  .+  c
)  =  ( a 
.+  ( b  .+  c ) ) ) ) )
117, 9, 103syl 17 . . 3  |-  ( E. e  e.  B  A. a  e.  B  (
( e  .+  a
)  =  a  /\  ( a  .+  e
)  =  a )  ->  ( G  e. Smgrp  <->  A. a  e.  B  A. b  e.  B  (
( a  .+  b
)  e.  B  /\  A. c  e.  B  ( ( a  .+  b
)  .+  c )  =  ( a  .+  ( b  .+  c
) ) ) ) )
1211pm5.32ri 455 . 2  |-  ( ( G  e. Smgrp  /\  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 ) ) )
133, 12bitri 184 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 set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1353   E.wex 1492    e. wcel 2148   A.wral 2455   E.wrex 2456   _Vcvv 2737   ` cfv 5212  (class class class)co 5869   Basecbs 12445   +g cplusg 12518  Smgrpcsgrp 12699   Mndcmnd 12709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-cnex 7893  ax-resscn 7894  ax-1re 7896  ax-addrcl 7899
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220  df-ov 5872  df-inn 8909  df-2 8967  df-ndx 12448  df-slot 12449  df-base 12451  df-plusg 12531  df-mgm 12667  df-sgrp 12700  df-mnd 12710
This theorem is referenced by:  mndid  12718  ismndd  12730  mndpropd  12733  mhmmnd  12869
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