MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpidd Unicode version

Theorem grpidd 14646
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 2388 . 2  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2388 . 2  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2388 . 2  |-  ( +g  `  G )  =  ( +g  `  G )
4 grpidd.z . . 3  |-  ( ph  ->  .0.  e.  B )
5 grpidd.b . . 3  |-  ( ph  ->  B  =  ( Base `  G ) )
64, 5eleqtrd 2464 . 2  |-  ( ph  ->  .0.  e.  ( Base `  G ) )
75eleq2d 2455 . . . 4  |-  ( ph  ->  ( x  e.  B  <->  x  e.  ( Base `  G
) ) )
87biimpar 472 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  G )
)  ->  x  e.  B )
9 grpidd.p . . . . . 6  |-  ( ph  ->  .+  =  ( +g  `  G ) )
109adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  .+  =  ( +g  `  G ) )
1110oveqd 6038 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  (  .0.  ( +g  `  G ) x ) )
12 grpidd.i . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )
1311, 12eqtr3d 2422 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  ( +g  `  G
) x )  =  x )
148, 13syldan 457 . 2  |-  ( (
ph  /\  x  e.  ( Base `  G )
)  ->  (  .0.  ( +g  `  G ) x )  =  x )
1510oveqd 6038 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .+  .0.  )  =  ( x ( +g  `  G )  .0.  ) )
16 grpidd.j . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .+  .0.  )  =  x )
1715, 16eqtr3d 2422 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (
x ( +g  `  G
)  .0.  )  =  x )
188, 17syldan 457 . 2  |-  ( (
ph  /\  x  e.  ( Base `  G )
)  ->  ( x
( +g  `  G )  .0.  )  =  x )
191, 2, 3, 6, 14, 18ismgmid2 14641 1  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   ` cfv 5395  (class class class)co 6021   Basecbs 13397   +g cplusg 13457   0gc0g 13651
This theorem is referenced by:  imasmnd2  14660  isgrpde  14757  ress0g  24022  xrs0  24032
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-iota 5359  df-fun 5397  df-fv 5403  df-ov 6024  df-riota 6486  df-0g 13655
  Copyright terms: Public domain W3C validator