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Theorem mgmidcl 14712
Description: The identity element of a magma, if it exists, belongs to the base set. (Contributed by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
mndidcl.b  |-  B  =  ( Base `  G
)
mndidcl.o  |-  .0.  =  ( 0g `  G )
mgmidcl.p  |-  .+  =  ( +g  `  G )
mgmidcl.e  |-  ( ph  ->  E. e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )
Assertion
Ref Expression
mgmidcl  |-  ( ph  ->  .0.  e.  B )
Distinct variable groups:    x, e,  .+    .0. , e, x    B, e, x    e, G, x
Allowed substitution hints:    ph( x, e)

Proof of Theorem mgmidcl
StepHypRef Expression
1 eqid 2437 . . 3  |-  .0.  =  .0.
2 mndidcl.b . . . 4  |-  B  =  ( Base `  G
)
3 mndidcl.o . . . 4  |-  .0.  =  ( 0g `  G )
4 mgmidcl.p . . . 4  |-  .+  =  ( +g  `  G )
5 mgmidcl.e . . . 4  |-  ( ph  ->  E. e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )
62, 3, 4, 5ismgmid 14711 . . 3  |-  ( ph  ->  ( (  .0.  e.  B  /\  A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) )  <->  .0.  =  .0.  ) )
71, 6mpbiri 226 . 2  |-  ( ph  ->  (  .0.  e.  B  /\  A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) ) )
87simpld 447 1  |-  ( ph  ->  .0.  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2706   E.wrex 2707   ` cfv 5455  (class class class)co 6082   Basecbs 13470   +g cplusg 13530   0gc0g 13724
This theorem is referenced by:  mndidcl  14715
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-ov 6085  df-riota 6550  df-0g 13728
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