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Theorem mndideu 16502
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
mndcl.b  |-  B  =  ( Base `  G
)
mndcl.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, B    x, G    x,  .+ , u    u, B    u, G    u,  .+

Proof of Theorem mndideu
StepHypRef Expression
1 mndcl.b . . 3  |-  B  =  ( Base `  G
)
2 mndcl.p . . 3  |-  .+  =  ( +g  `  G )
31, 2mndid 16501 . 2  |-  ( G  e.  Mnd  ->  E. u  e.  B  A. x  e.  B  ( (
u  .+  x )  =  x  /\  (
x  .+  u )  =  x ) )
4 mgmidmo 16454 . . 3  |-  E* u  e.  B  A. x  e.  B  ( (
u  .+  x )  =  x  /\  (
x  .+  u )  =  x )
54a1i 11 . 2  |-  ( G  e.  Mnd  ->  E* u  e.  B  A. x  e.  B  (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x ) )
6 reu5 3042 . 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 668 1  |-  ( G  e.  Mnd  ->  E! u  e.  B  A. x  e.  B  (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   A.wral 2773   E.wrex 2774   E!wreu 2775   E*wrmo 2776   ` cfv 5592  (class class class)co 6296   Basecbs 15081   +g cplusg 15150   Mndcmnd 16487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-nul 4547  ax-pow 4594
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-iota 5556  df-fv 5600  df-ov 6299  df-mgm 16440  df-sgrp 16479  df-mnd 16489
This theorem is referenced by:  grpideu  16634  srgideu  17689  ringideu  17739
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