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Theorem sgrpidmndm 12700
Description: A semigroup with an identity element which is inhabited is a monoid. Of course there could be monoids with the empty set as identity element, but these cannot be proven to be monoids with this theorem. (Contributed by AV, 29-Jan-2024.)
Hypotheses
Ref Expression
sgrpidmnd.b  |-  B  =  ( Base `  G
)
sgrpidmnd.0  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
sgrpidmndm  |-  ( ( G  e. Smgrp  /\  E. e  e.  B  ( E. w  w  e.  e  /\  e  =  .0.  ) )  ->  G  e.  Mnd )
Distinct variable groups:    B, e, w   
e, G, w    w,  .0.    w, e
Allowed substitution hint:    .0. ( e)

Proof of Theorem sgrpidmndm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp-4r 542 . . . . . . . . . 10  |-  ( ( ( ( ( G  e. Smgrp  /\  e  e.  B )  /\  w  e.  e )  /\  e  =  .0.  )  /\  x  e.  B )  ->  e  e.  B )
2 simpllr 534 . . . . . . . . . . 11  |-  ( ( ( ( ( G  e. Smgrp  /\  e  e.  B )  /\  w  e.  e )  /\  e  =  .0.  )  /\  x  e.  B )  ->  w  e.  e )
3219.8ad 1591 . . . . . . . . . 10  |-  ( ( ( ( ( G  e. Smgrp  /\  e  e.  B )  /\  w  e.  e )  /\  e  =  .0.  )  /\  x  e.  B )  ->  E. w  w  e.  e )
4 simplr 528 . . . . . . . . . . 11  |-  ( ( ( ( ( G  e. Smgrp  /\  e  e.  B )  /\  w  e.  e )  /\  e  =  .0.  )  /\  x  e.  B )  ->  e  =  .0.  )
5 sgrpidmnd.b . . . . . . . . . . . . . 14  |-  B  =  ( Base `  G
)
6 eqid 2177 . . . . . . . . . . . . . 14  |-  ( +g  `  G )  =  ( +g  `  G )
7 sgrpidmnd.0 . . . . . . . . . . . . . 14  |-  .0.  =  ( 0g `  G )
85, 6, 7grpidvalg 12671 . . . . . . . . . . . . 13  |-  ( G  e. Smgrp  ->  .0.  =  ( iota y ( y  e.  B  /\  A. x  e.  B  ( (
y ( +g  `  G
) x )  =  x  /\  ( x ( +g  `  G
) y )  =  x ) ) ) )
98eqeq2d 2189 . . . . . . . . . . . 12  |-  ( G  e. Smgrp  ->  ( e  =  .0.  <->  e  =  ( iota y ( y  e.  B  /\  A. x  e.  B  (
( y ( +g  `  G ) x )  =  x  /\  (
x ( +g  `  G
) y )  =  x ) ) ) ) )
109ad4antr 494 . . . . . . . . . . 11  |-  ( ( ( ( ( G  e. Smgrp  /\  e  e.  B )  /\  w  e.  e )  /\  e  =  .0.  )  /\  x  e.  B )  ->  (
e  =  .0.  <->  e  =  ( iota y ( y  e.  B  /\  A. x  e.  B  (
( y ( +g  `  G ) x )  =  x  /\  (
x ( +g  `  G
) y )  =  x ) ) ) ) )
114, 10mpbid 147 . . . . . . . . . 10  |-  ( ( ( ( ( G  e. Smgrp  /\  e  e.  B )  /\  w  e.  e )  /\  e  =  .0.  )  /\  x  e.  B )  ->  e  =  ( iota y
( y  e.  B  /\  A. x  e.  B  ( ( y ( +g  `  G ) x )  =  x  /\  ( x ( +g  `  G ) y )  =  x ) ) ) )
121, 3, 113jca 1177 . . . . . . . . 9  |-  ( ( ( ( ( G  e. Smgrp  /\  e  e.  B )  /\  w  e.  e )  /\  e  =  .0.  )  /\  x  e.  B )  ->  (
e  e.  B  /\  E. w  w  e.  e  /\  e  =  ( iota y ( y  e.  B  /\  A. x  e.  B  (
( y ( +g  `  G ) x )  =  x  /\  (
x ( +g  `  G
) y )  =  x ) ) ) ) )
13 simpr 110 . . . . . . . . 9  |-  ( ( ( ( ( G  e. Smgrp  /\  e  e.  B )  /\  w  e.  e )  /\  e  =  .0.  )  /\  x  e.  B )  ->  x  e.  B )
14 eleq1w 2238 . . . . . . . . . . . 12  |-  ( y  =  e  ->  (
y  e.  B  <->  e  e.  B ) )
15 oveq1 5875 . . . . . . . . . . . . . 14  |-  ( y  =  e  ->  (
y ( +g  `  G
) x )  =  ( e ( +g  `  G ) x ) )
1615eqeq1d 2186 . . . . . . . . . . . . 13  |-  ( y  =  e  ->  (
( y ( +g  `  G ) x )  =  x  <->  ( e
( +g  `  G ) x )  =  x ) )
1716ovanraleqv 5892 . . . . . . . . . . . 12  |-  ( y  =  e  ->  ( A. x  e.  B  ( ( y ( +g  `  G ) x )  =  x  /\  ( x ( +g  `  G ) y )  =  x )  <->  A. x  e.  B  ( ( e ( +g  `  G ) x )  =  x  /\  ( x ( +g  `  G ) e )  =  x ) ) )
1814, 17anbi12d 473 . . . . . . . . . . 11  |-  ( y  =  e  ->  (
( y  e.  B  /\  A. x  e.  B  ( ( y ( +g  `  G ) x )  =  x  /\  ( x ( +g  `  G ) y )  =  x ) )  <->  ( e  e.  B  /\  A. x  e.  B  ( (
e ( +g  `  G
) x )  =  x  /\  ( x ( +g  `  G
) e )  =  x ) ) ) )
1918iotam 5203 . . . . . . . . . 10  |-  ( ( e  e.  B  /\  E. w  w  e.  e  /\  e  =  ( iota y ( y  e.  B  /\  A. x  e.  B  (
( y ( +g  `  G ) x )  =  x  /\  (
x ( +g  `  G
) y )  =  x ) ) ) )  ->  ( e  e.  B  /\  A. x  e.  B  ( (
e ( +g  `  G
) x )  =  x  /\  ( x ( +g  `  G
) e )  =  x ) ) )
20 rsp 2524 . . . . . . . . . 10  |-  ( A. x  e.  B  (
( e ( +g  `  G ) x )  =  x  /\  (
x ( +g  `  G
) e )  =  x )  ->  (
x  e.  B  -> 
( ( e ( +g  `  G ) x )  =  x  /\  ( x ( +g  `  G ) e )  =  x ) ) )
2119, 20simpl2im 386 . . . . . . . . 9  |-  ( ( e  e.  B  /\  E. w  w  e.  e  /\  e  =  ( iota y ( y  e.  B  /\  A. x  e.  B  (
( y ( +g  `  G ) x )  =  x  /\  (
x ( +g  `  G
) y )  =  x ) ) ) )  ->  ( x  e.  B  ->  ( ( e ( +g  `  G
) x )  =  x  /\  ( x ( +g  `  G
) e )  =  x ) ) )
2212, 13, 21sylc 62 . . . . . . . 8  |-  ( ( ( ( ( G  e. Smgrp  /\  e  e.  B )  /\  w  e.  e )  /\  e  =  .0.  )  /\  x  e.  B )  ->  (
( e ( +g  `  G ) x )  =  x  /\  (
x ( +g  `  G
) e )  =  x ) )
2322ralrimiva 2550 . . . . . . 7  |-  ( ( ( ( G  e. Smgrp  /\  e  e.  B
)  /\  w  e.  e )  /\  e  =  .0.  )  ->  A. x  e.  B  ( (
e ( +g  `  G
) x )  =  x  /\  ( x ( +g  `  G
) e )  =  x ) )
2423exp31 364 . . . . . 6  |-  ( ( G  e. Smgrp  /\  e  e.  B )  ->  (
w  e.  e  -> 
( e  =  .0. 
->  A. x  e.  B  ( ( e ( +g  `  G ) x )  =  x  /\  ( x ( +g  `  G ) e )  =  x ) ) ) )
2524exlimdv 1819 . . . . 5  |-  ( ( G  e. Smgrp  /\  e  e.  B )  ->  ( E. w  w  e.  e  ->  ( e  =  .0.  ->  A. x  e.  B  ( (
e ( +g  `  G
) x )  =  x  /\  ( x ( +g  `  G
) e )  =  x ) ) ) )
2625impd 254 . . . 4  |-  ( ( G  e. Smgrp  /\  e  e.  B )  ->  (
( E. w  w  e.  e  /\  e  =  .0.  )  ->  A. x  e.  B  ( (
e ( +g  `  G
) x )  =  x  /\  ( x ( +g  `  G
) e )  =  x ) ) )
2726reximdva 2579 . . 3  |-  ( G  e. Smgrp  ->  ( E. e  e.  B  ( E. w  w  e.  e  /\  e  =  .0.  )  ->  E. e  e.  B  A. x  e.  B  ( ( e ( +g  `  G ) x )  =  x  /\  ( x ( +g  `  G ) e )  =  x ) ) )
2827imdistani 445 . 2  |-  ( ( G  e. Smgrp  /\  E. e  e.  B  ( E. w  w  e.  e  /\  e  =  .0.  ) )  ->  ( G  e. Smgrp  /\  E. e  e.  B  A. x  e.  B  ( (
e ( +g  `  G
) x )  =  x  /\  ( x ( +g  `  G
) e )  =  x ) ) )
295, 6ismnddef 12698 . 2  |-  ( G  e.  Mnd  <->  ( G  e. Smgrp  /\  E. e  e.  B  A. x  e.  B  ( ( e ( +g  `  G
) x )  =  x  /\  ( x ( +g  `  G
) e )  =  x ) ) )
3028, 29sylibr 134 1  |-  ( ( G  e. Smgrp  /\  E. e  e.  B  ( E. w  w  e.  e  /\  e  =  .0.  ) )  ->  G  e.  Mnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353   E.wex 1492    e. wcel 2148   A.wral 2455   E.wrex 2456   iotacio 5171   ` cfv 5211  (class class class)co 5868   Basecbs 12432   +g cplusg 12505   0gc0g 12640  Smgrpcsgrp 12686   Mndcmnd 12696
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 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
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-csb 3058  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 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-riota 5824  df-ov 5871  df-inn 8896  df-2 8954  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-0g 12642  df-mnd 12697
This theorem is referenced by: (None)
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