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Theorem grpidval 16014
Description: The value of the identity element of a group. (Contributed by NM, 20-Aug-2011.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
grpidval.b  |-  B  =  ( Base `  G
)
grpidval.p  |-  .+  =  ( +g  `  G )
grpidval.o  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
grpidval  |-  .0.  =  ( iota e ( e  e.  B  /\  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) )
Distinct variable groups:    x, e, B    e, G, x
Allowed substitution hints:    .+ ( x, e)    .0. ( x, e)

Proof of Theorem grpidval
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 grpidval.o . 2  |-  .0.  =  ( 0g `  G )
2 fveq2 5872 . . . . . . . 8  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 grpidval.b . . . . . . . 8  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2516 . . . . . . 7  |-  ( g  =  G  ->  ( Base `  g )  =  B )
54eleq2d 2527 . . . . . 6  |-  ( g  =  G  ->  (
e  e.  ( Base `  g )  <->  e  e.  B ) )
6 fveq2 5872 . . . . . . . . . . 11  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
7 grpidval.p . . . . . . . . . . 11  |-  .+  =  ( +g  `  G )
86, 7syl6eqr 2516 . . . . . . . . . 10  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
98oveqd 6313 . . . . . . . . 9  |-  ( g  =  G  ->  (
e ( +g  `  g
) x )  =  ( e  .+  x
) )
109eqeq1d 2459 . . . . . . . 8  |-  ( g  =  G  ->  (
( e ( +g  `  g ) x )  =  x  <->  ( e  .+  x )  =  x ) )
118oveqd 6313 . . . . . . . . 9  |-  ( g  =  G  ->  (
x ( +g  `  g
) e )  =  ( x  .+  e
) )
1211eqeq1d 2459 . . . . . . . 8  |-  ( g  =  G  ->  (
( x ( +g  `  g ) e )  =  x  <->  ( x  .+  e )  =  x ) )
1310, 12anbi12d 710 . . . . . . 7  |-  ( g  =  G  ->  (
( ( e ( +g  `  g ) x )  =  x  /\  ( x ( +g  `  g ) e )  =  x )  <->  ( ( e 
.+  x )  =  x  /\  ( x 
.+  e )  =  x ) ) )
144, 13raleqbidv 3068 . . . . . 6  |-  ( g  =  G  ->  ( A. x  e.  ( Base `  g ) ( ( e ( +g  `  g ) x )  =  x  /\  (
x ( +g  `  g
) e )  =  x )  <->  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) ) )
155, 14anbi12d 710 . . . . 5  |-  ( g  =  G  ->  (
( e  e.  (
Base `  g )  /\  A. x  e.  (
Base `  g )
( ( e ( +g  `  g ) x )  =  x  /\  ( x ( +g  `  g ) e )  =  x ) )  <->  ( e  e.  B  /\  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) ) ) )
1615iotabidv 5578 . . . 4  |-  ( g  =  G  ->  ( iota e ( e  e.  ( Base `  g
)  /\  A. x  e.  ( Base `  g
) ( ( e ( +g  `  g
) x )  =  x  /\  ( x ( +g  `  g
) e )  =  x ) ) )  =  ( iota e
( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) ) ) )
17 df-0g 14859 . . . 4  |-  0g  =  ( g  e.  _V  |->  ( iota e ( e  e.  ( Base `  g
)  /\  A. x  e.  ( Base `  g
) ( ( e ( +g  `  g
) x )  =  x  /\  ( x ( +g  `  g
) e )  =  x ) ) ) )
18 iotaex 5574 . . . 4  |-  ( iota e ( e  e.  B  /\  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) ) )  e.  _V
1916, 17, 18fvmpt 5956 . . 3  |-  ( G  e.  _V  ->  ( 0g `  G )  =  ( iota e ( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) ) )
20 fvprc 5866 . . . 4  |-  ( -.  G  e.  _V  ->  ( 0g `  G )  =  (/) )
21 euex 2309 . . . . . . 7  |-  ( E! e ( e  e.  B  /\  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) )  ->  E. e ( e  e.  B  /\  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) ) )
22 n0i 3798 . . . . . . . . . 10  |-  ( e  e.  B  ->  -.  B  =  (/) )
23 fvprc 5866 . . . . . . . . . . 11  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
243, 23syl5eq 2510 . . . . . . . . . 10  |-  ( -.  G  e.  _V  ->  B  =  (/) )
2522, 24nsyl2 127 . . . . . . . . 9  |-  ( e  e.  B  ->  G  e.  _V )
2625adantr 465 . . . . . . . 8  |-  ( ( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) )  ->  G  e.  _V )
2726exlimiv 1723 . . . . . . 7  |-  ( E. e ( e  e.  B  /\  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) )  ->  G  e.  _V )
2821, 27syl 16 . . . . . 6  |-  ( E! e ( e  e.  B  /\  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) )  ->  G  e.  _V )
2928con3i 135 . . . . 5  |-  ( -.  G  e.  _V  ->  -.  E! e ( e  e.  B  /\  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) )
30 iotanul 5572 . . . . 5  |-  ( -.  E! e ( e  e.  B  /\  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) )  ->  ( iota e
( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) ) )  =  (/) )
3129, 30syl 16 . . . 4  |-  ( -.  G  e.  _V  ->  ( iota e ( e  e.  B  /\  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) )  =  (/) )
3220, 31eqtr4d 2501 . . 3  |-  ( -.  G  e.  _V  ->  ( 0g `  G )  =  ( iota e
( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) ) ) )
3319, 32pm2.61i 164 . 2  |-  ( 0g
`  G )  =  ( iota e ( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) )
341, 33eqtri 2486 1  |-  .0.  =  ( iota e ( e  e.  B  /\  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1395   E.wex 1613    e. wcel 1819   E!weu 2283   A.wral 2807   _Vcvv 3109   (/)c0 3793   iotacio 5555   ` cfv 5594  (class class class)co 6296   Basecbs 14644   +g cplusg 14712   0gc0g 14857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-0g 14859
This theorem is referenced by:  grpidpropd  16015  0g0  16017  ismgmid  16018  oppgid  16518  dfur2  17283  oppr0  17409  oppr1  17410
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