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Theorem mgmlrid 14704
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 2435 . . . 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 14702 . . . 4  |-  ( ph  ->  ( (  .0.  e.  B  /\  A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) )  <->  .0.  =  .0.  ) )
71, 6mpbiri 225 . . 3  |-  ( ph  ->  (  .0.  e.  B  /\  A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) ) )
87simprd 450 . 2  |-  ( ph  ->  A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) )
9 oveq2 6081 . . . . 5  |-  ( x  =  X  ->  (  .0.  .+  x )  =  (  .0.  .+  X
) )
10 id 20 . . . . 5  |-  ( x  =  X  ->  x  =  X )
119, 10eqeq12d 2449 . . . 4  |-  ( x  =  X  ->  (
(  .0.  .+  x
)  =  x  <->  (  .0.  .+  X )  =  X ) )
12 oveq1 6080 . . . . 5  |-  ( x  =  X  ->  (
x  .+  .0.  )  =  ( X  .+  .0.  ) )
1312, 10eqeq12d 2449 . . . 4  |-  ( x  =  X  ->  (
( x  .+  .0.  )  =  x  <->  ( X  .+  .0.  )  =  X ) )
1411, 13anbi12d 692 . . 3  |-  ( x  =  X  ->  (
( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
)  <->  ( (  .0.  .+  X )  =  X  /\  ( X  .+  .0.  )  =  X
) ) )
1514rspccva 3043 . 2  |-  ( ( A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
)  /\  X  e.  B )  ->  (
(  .0.  .+  X
)  =  X  /\  ( X  .+  .0.  )  =  X ) )
168, 15sylan 458 1  |-  ( (
ph  /\  X  e.  B )  ->  (
(  .0.  .+  X
)  =  X  /\  ( X  .+  .0.  )  =  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   0gc0g 13715
This theorem is referenced by:  mndlrid  14707
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-riota 6541  df-0g 13719
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