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Theorem grpidd 15760
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 2467 . 2  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2467 . 2  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2467 . 2  |-  ( +g  `  G )  =  ( +g  `  G )
4 grpidd.z . . 3  |-  ( ph  ->  .0.  e.  B )
5 grpidd.b . . 3  |-  ( ph  ->  B  =  ( Base `  G ) )
64, 5eleqtrd 2557 . 2  |-  ( ph  ->  .0.  e.  ( Base `  G ) )
75eleq2d 2537 . . . 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 6301 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  (  .0.  ( +g  `  G ) x ) )
12 grpidd.i . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )
1311, 12eqtr3d 2510 . . 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 6301 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .+  .0.  )  =  ( x ( +g  `  G )  .0.  ) )
16 grpidd.j . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .+  .0.  )  =  x )
1715, 16eqtr3d 2510 . . 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 15755 1  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   ` cfv 5588  (class class class)co 6284   Basecbs 14490   +g cplusg 14555   0gc0g 14695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-riota 6245  df-ov 6287  df-0g 14697
This theorem is referenced by:  ress0g  15768  imasmnd2  15776  isgrpde  15884  xrs0  27353
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