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Theorem mndid 14690
Description: A monoid has a two-sided identity element. (Contributed by NM, 16-Aug-2011.)
Hypotheses
Ref Expression
mndlem1.b  |-  B  =  ( Base `  G
)
mndlem1.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, u, B    u, G, x    u,  .+ , x

Proof of Theorem mndid
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mndlem1.b . . 3  |-  B  =  ( Base `  G
)
2 mndlem1.p . . 3  |-  .+  =  ( +g  `  G )
31, 2ismnd 14685 . 2  |-  ( G  e.  Mnd  <->  ( A. x  e.  B  A. y  e.  B  A. z  e.  B  (
( x  .+  y
)  e.  B  /\  ( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )  /\  E. u  e.  B  A. x  e.  B  ( ( u 
.+  x )  =  x  /\  ( x 
.+  u )  =  x ) ) )
43simprbi 451 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 359    = wceq 1652    e. wcel 1725   A.wral 2698   E.wrex 2699   ` cfv 5447  (class class class)co 6074   Basecbs 13462   +g cplusg 13522   Mndcmnd 14677
This theorem is referenced by:  mndideu  14691  mndidcl  14707  mndlrid  14708  prds0g  14722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4331  ax-pow 4370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-iota 5411  df-fv 5455  df-ov 6077  df-mnd 14683
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