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Theorem mgmlrid 14389
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
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
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 2283 . . . 4  |-  .0.  =  .0.
2 mndidcl.b . . . . 5  |-  B  =  ( Base `  G
)
3 mndidcl.o . . . . 5  |-  .0.  =  ( 0g `  G )
4 mgmidcl.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 14387 . . . 4  |-  ( ph  ->  ( (  .0.  e.  B  /\  A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) )  <->  .0.  =  .0.  ) )
71, 6mpbiri 224 . . 3  |-  ( ph  ->  (  .0.  e.  B  /\  A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) ) )
87simprd 449 . 2  |-  ( ph  ->  A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) )
9 oveq2 5866 . . . . 5  |-  ( x  =  X  ->  (  .0.  .+  x )  =  (  .0.  .+  X
) )
10 id 19 . . . . 5  |-  ( x  =  X  ->  x  =  X )
119, 10eqeq12d 2297 . . . 4  |-  ( x  =  X  ->  (
(  .0.  .+  x
)  =  x  <->  (  .0.  .+  X )  =  X ) )
12 oveq1 5865 . . . . 5  |-  ( x  =  X  ->  (
x  .+  .0.  )  =  ( X  .+  .0.  ) )
1312, 10eqeq12d 2297 . . . 4  |-  ( x  =  X  ->  (
( x  .+  .0.  )  =  x  <->  ( X  .+  .0.  )  =  X ) )
1411, 13anbi12d 691 . . 3  |-  ( x  =  X  ->  (
( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
)  <->  ( (  .0.  .+  X )  =  X  /\  ( X  .+  .0.  )  =  X
) ) )
1514rspccva 2883 . 2  |-  ( ( A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
)  /\  X  e.  B )  ->  (
(  .0.  .+  X
)  =  X  /\  ( X  .+  .0.  )  =  X ) )
168, 15sylan 457 1  |-  ( (
ph  /\  X  e.  B )  ->  (
(  .0.  .+  X
)  =  X  /\  ( X  .+  .0.  )  =  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400
This theorem is referenced by:  mndlrid  14392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-riota 6304  df-0g 13404
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