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Theorem mndid 16255
Description: A monoid has a two-sided identity element. (Contributed by NM, 16-Aug-2011.)
Hypotheses
Ref Expression
mndcl.b  |-  B  =  ( Base `  G
)
mndcl.p  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
mndid  |-  ( 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 mndid
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mndcl.b . . 3  |-  B  =  ( Base `  G
)
2 mndcl.p . . 3  |-  .+  =  ( +g  `  G )
31, 2ismnd 16245 . 2  |-  ( G  e.  Mnd  <->  ( A. x  e.  B  A. y  e.  B  (
( x  .+  y
)  e.  B  /\  A. z  e.  B  ( ( x  .+  y
)  .+  z )  =  ( x  .+  ( y  .+  z
) ) )  /\  E. u  e.  B  A. x  e.  B  (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x ) ) )
43simprbi 462 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 367    = wceq 1405    e. wcel 1842   A.wral 2753   E.wrex 2754   ` cfv 5568  (class class class)co 6277   Basecbs 14839   +g cplusg 14907   Mndcmnd 16241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4524  ax-pow 4571
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5532  df-fv 5576  df-ov 6280  df-mgm 16194  df-sgrp 16233  df-mnd 16243
This theorem is referenced by:  mndideu  16256  mndidcl  16260  mndlrid  16262  prds0g  16276
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