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Theorem grpidd 15435
Description: Deduce the identity element of a monoid from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
grpidd.b  |-  ( ph  ->  B  =  ( Base `  G ) )
grpidd.p  |-  ( ph  ->  .+  =  ( +g  `  G ) )
grpidd.z  |-  ( ph  ->  .0.  e.  B )
grpidd.i  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )
grpidd.j  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .+  .0.  )  =  x )
Assertion
Ref Expression
grpidd  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
Distinct variable groups:    x, G    ph, x    x,  .0.
Allowed substitution hints:    B( x)    .+ ( x)

Proof of Theorem grpidd
StepHypRef Expression
1 eqid 2438 . 2  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2438 . 2  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2438 . 2  |-  ( +g  `  G )  =  ( +g  `  G )
4 grpidd.z . . 3  |-  ( ph  ->  .0.  e.  B )
5 grpidd.b . . 3  |-  ( ph  ->  B  =  ( Base `  G ) )
64, 5eleqtrd 2514 . 2  |-  ( ph  ->  .0.  e.  ( Base `  G ) )
75eleq2d 2505 . . . 4  |-  ( ph  ->  ( x  e.  B  <->  x  e.  ( Base `  G
) ) )
87biimpar 485 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  G )
)  ->  x  e.  B )
9 grpidd.p . . . . . 6  |-  ( ph  ->  .+  =  ( +g  `  G ) )
109adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  .+  =  ( +g  `  G ) )
1110oveqd 6103 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  (  .0.  ( +g  `  G ) x ) )
12 grpidd.i . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )
1311, 12eqtr3d 2472 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  ( +g  `  G
) x )  =  x )
148, 13syldan 470 . 2  |-  ( (
ph  /\  x  e.  ( Base `  G )
)  ->  (  .0.  ( +g  `  G ) x )  =  x )
1510oveqd 6103 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .+  .0.  )  =  ( x ( +g  `  G )  .0.  ) )
16 grpidd.j . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .+  .0.  )  =  x )
1715, 16eqtr3d 2472 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (
x ( +g  `  G
)  .0.  )  =  x )
188, 17syldan 470 . 2  |-  ( (
ph  /\  x  e.  ( Base `  G )
)  ->  ( x
( +g  `  G )  .0.  )  =  x )
191, 2, 3, 6, 14, 18ismgmid2 15430 1  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   ` cfv 5413  (class class class)co 6086   Basecbs 14166   +g cplusg 14230   0gc0g 14370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-iota 5376  df-fun 5415  df-fv 5421  df-riota 6047  df-ov 6089  df-0g 14372
This theorem is referenced by:  ress0g  15442  imasmnd2  15450  isgrpde  15553  xrs0  26087
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