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Theorem mndideu 14391
Description: The two-sided identity element of a monoid is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by Mario Carneiro, 8-Dec-2014.)
Hypotheses
Ref Expression
mndlem1.b  |-  B  =  ( Base `  G
)
mndlem1.p  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
mndideu  |-  ( G  e.  Mnd  ->  E! u  e.  B  A. x  e.  B  (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x ) )
Distinct variable groups:    x, u, B    u, G, x    u,  .+ , x

Proof of Theorem mndideu
StepHypRef Expression
1 mndlem1.b . . 3  |-  B  =  ( Base `  G
)
2 mndlem1.p . . 3  |-  .+  =  ( +g  `  G )
31, 2mndid 14390 . 2  |-  ( G  e.  Mnd  ->  E. u  e.  B  A. x  e.  B  ( (
u  .+  x )  =  x  /\  (
x  .+  u )  =  x ) )
4 mgmidmo 14386 . . 3  |-  E* u  e.  B A. x  e.  B  ( ( u 
.+  x )  =  x  /\  ( x 
.+  u )  =  x )
54a1i 10 . 2  |-  ( G  e.  Mnd  ->  E* u  e.  B A. x  e.  B  (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x ) )
6 reu5 2766 . 2  |-  ( E! u  e.  B  A. x  e.  B  (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x )  <->  ( E. u  e.  B  A. x  e.  B  (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x )  /\  E* u  e.  B A. x  e.  B  ( ( u  .+  x )  =  x  /\  ( x  .+  u )  =  x ) ) )
73, 5, 6sylanbrc 645 1  |-  ( G  e.  Mnd  ->  E! u  e.  B  A. x  e.  B  (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   E!wreu 2558   E*wrmo 2559   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   Mndcmnd 14377
This theorem is referenced by:  grpideu  14514  rngideu  15374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165  ax-pow 4204
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-mnd 14383
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