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Theorem mgmlrid 12690
Description: The identity element of a magma, if it exists, is a left and right identity. (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
mgmlrid  |-  ( (
ph  /\  X  e.  B )  ->  (
(  .0.  .+  X
)  =  X  /\  ( X  .+  .0.  )  =  X ) )
Distinct variable groups:    x, e,  .+    .0. , e, x    B, e, x    e, G, x   
x, X
Allowed substitution hints:    ph( x, e)    X( e)

Proof of Theorem mgmlrid
StepHypRef Expression
1 eqid 2177 . . . 4  |-  .0.  =  .0.
2 ismgmid.b . . . . 5  |-  B  =  ( Base `  G
)
3 ismgmid.o . . . . 5  |-  .0.  =  ( 0g `  G )
4 ismgmid.p . . . . 5  |-  .+  =  ( +g  `  G )
5 mgmidcl.e . . . . 5  |-  ( ph  ->  E. e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )
62, 3, 4, 5ismgmid 12688 . . . 4  |-  ( ph  ->  ( (  .0.  e.  B  /\  A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) )  <->  .0.  =  .0.  ) )
71, 6mpbiri 168 . . 3  |-  ( ph  ->  (  .0.  e.  B  /\  A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) ) )
87simprd 114 . 2  |-  ( ph  ->  A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) )
9 oveq2 5877 . . . . 5  |-  ( x  =  X  ->  (  .0.  .+  x )  =  (  .0.  .+  X
) )
10 id 19 . . . . 5  |-  ( x  =  X  ->  x  =  X )
119, 10eqeq12d 2192 . . . 4  |-  ( x  =  X  ->  (
(  .0.  .+  x
)  =  x  <->  (  .0.  .+  X )  =  X ) )
12 oveq1 5876 . . . . 5  |-  ( x  =  X  ->  (
x  .+  .0.  )  =  ( X  .+  .0.  ) )
1312, 10eqeq12d 2192 . . . 4  |-  ( x  =  X  ->  (
( x  .+  .0.  )  =  x  <->  ( X  .+  .0.  )  =  X ) )
1411, 13anbi12d 473 . . 3  |-  ( x  =  X  ->  (
( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
)  <->  ( (  .0.  .+  X )  =  X  /\  ( X  .+  .0.  )  =  X
) ) )
1514rspccva 2840 . 2  |-  ( ( A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
)  /\  X  e.  B )  ->  (
(  .0.  .+  X
)  =  X  /\  ( X  .+  .0.  )  =  X ) )
168, 15sylan 283 1  |-  ( (
ph  /\  X  e.  B )  ->  (
(  .0.  .+  X
)  =  X  /\  ( X  .+  .0.  )  =  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   ` cfv 5212  (class class class)co 5869   Basecbs 12445   +g cplusg 12518   0gc0g 12653
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-cnex 7893  ax-resscn 7894  ax-1re 7896  ax-addrcl 7899
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220  df-riota 5825  df-ov 5872  df-inn 8909  df-ndx 12448  df-slot 12449  df-base 12451  df-0g 12655
This theorem is referenced by:  mndlrid  12727
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