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Theorem mndideu 14210
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 14209 . 2  |-  ( G  e.  Mnd  ->  E. u  e.  B  A. x  e.  B  ( (
u  .+  x )  =  x  /\  (
x  .+  u )  =  x ) )
4 mgmidmo 14205 . . 3  |-  E* u
( u  e.  B  /\  A. x  e.  B  ( ( u  .+  x )  =  x  /\  ( x  .+  u )  =  x ) )
54a1i 12 . 2  |-  ( G  e.  Mnd  ->  E* u ( u  e.  B  /\  A. x  e.  B  ( (
u  .+  x )  =  x  /\  (
x  .+  u )  =  x ) ) )
6 reu5 2891 . 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 ( u  e.  B  /\  A. x  e.  B  ( (
u  .+  x )  =  x  /\  (
x  .+  u )  =  x ) ) ) )
73, 5, 6sylanbrc 648 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 6    /\ wa 360    = wceq 1619    e. wcel 1621   E*wmo 2115   A.wral 2509   E.wrex 2510   E!wreu 2511   ` cfv 4592  (class class class)co 5710   Basecbs 13022   +g cplusg 13082   Mndcmnd 14196
This theorem is referenced by:  grpideu  14333  rngideu  15193
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fv 4608  df-ov 5713  df-mnd 14202
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