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Theorem mgmidcl 12820
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
ismgmid.b  |-  B  =  ( Base `  G
)
ismgmid.o  |-  .0.  =  ( 0g `  G )
ismgmid.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 2189 . . 3  |-  .0.  =  .0.
2 ismgmid.b . . . 4  |-  B  =  ( Base `  G
)
3 ismgmid.o . . . 4  |-  .0.  =  ( 0g `  G )
4 ismgmid.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 12819 . . 3  |-  ( ph  ->  ( (  .0.  e.  B  /\  A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) )  <->  .0.  =  .0.  ) )
71, 6mpbiri 168 . 2  |-  ( ph  ->  (  .0.  e.  B  /\  A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) ) )
87simpld 112 1  |-  ( ph  ->  .0.  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   A.wral 2468   E.wrex 2469   ` cfv 5231  (class class class)co 5891   Basecbs 12480   +g cplusg 12555   0gc0g 12727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-cnex 7920  ax-resscn 7921  ax-1re 7923  ax-addrcl 7926
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-iota 5193  df-fun 5233  df-fn 5234  df-fv 5239  df-riota 5847  df-ov 5894  df-inn 8938  df-ndx 12483  df-slot 12484  df-base 12486  df-0g 12729
This theorem is referenced by:  mndidcl  12857
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