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Theorem grpidval 14400
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 5541 . . . . . . . 8  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 grpidval.b . . . . . . . 8  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2346 . . . . . . 7  |-  ( g  =  G  ->  ( Base `  g )  =  B )
54eleq2d 2363 . . . . . 6  |-  ( g  =  G  ->  (
e  e.  ( Base `  g )  <->  e  e.  B ) )
6 fveq2 5541 . . . . . . . . . . 11  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
7 grpidval.p . . . . . . . . . . 11  |-  .+  =  ( +g  `  G )
86, 7syl6eqr 2346 . . . . . . . . . 10  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
98oveqd 5891 . . . . . . . . 9  |-  ( g  =  G  ->  (
e ( +g  `  g
) x )  =  ( e  .+  x
) )
109eqeq1d 2304 . . . . . . . 8  |-  ( g  =  G  ->  (
( e ( +g  `  g ) x )  =  x  <->  ( e  .+  x )  =  x ) )
118oveqd 5891 . . . . . . . . 9  |-  ( g  =  G  ->  (
x ( +g  `  g
) e )  =  ( x  .+  e
) )
1211eqeq1d 2304 . . . . . . . 8  |-  ( g  =  G  ->  (
( x ( +g  `  g ) e )  =  x  <->  ( x  .+  e )  =  x ) )
1310, 12anbi12d 691 . . . . . . 7  |-  ( g  =  G  ->  (
( ( e ( +g  `  g ) x )  =  x  /\  ( x ( +g  `  g ) e )  =  x )  <->  ( ( e 
.+  x )  =  x  /\  ( x 
.+  e )  =  x ) ) )
144, 13raleqbidv 2761 . . . . . 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 691 . . . . 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 5256 . . . 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 13420 . . . 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 5252 . . . 4  |-  ( iota e ( e  e.  B  /\  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) ) )  e.  _V
1916, 17, 18fvmpt 5618 . . 3  |-  ( G  e.  _V  ->  ( 0g `  G )  =  ( iota e ( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) ) )
20 fvprc 5535 . . . 4  |-  ( -.  G  e.  _V  ->  ( 0g `  G )  =  (/) )
21 euex 2179 . . . . . . 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 3473 . . . . . . . . . 10  |-  ( e  e.  B  ->  -.  B  =  (/) )
23 fvprc 5535 . . . . . . . . . . 11  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
243, 23syl5eq 2340 . . . . . . . . . 10  |-  ( -.  G  e.  _V  ->  B  =  (/) )
2522, 24nsyl2 119 . . . . . . . . 9  |-  ( e  e.  B  ->  G  e.  _V )
2625adantr 451 . . . . . . . 8  |-  ( ( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) )  ->  G  e.  _V )
2726exlimiv 1624 . . . . . . 7  |-  ( E. e ( e  e.  B  /\  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) )  ->  G  e.  _V )
2821, 27syl 15 . . . . . 6  |-  ( E! e ( e  e.  B  /\  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) )  ->  G  e.  _V )
2928con3i 127 . . . . 5  |-  ( -.  G  e.  _V  ->  -.  E! e ( e  e.  B  /\  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) )
30 iotanul 5250 . . . . 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 15 . . . 4  |-  ( -.  G  e.  _V  ->  ( iota e ( e  e.  B  /\  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) )  =  (/) )
3220, 31eqtr4d 2331 . . 3  |-  ( -.  G  e.  _V  ->  ( 0g `  G )  =  ( iota e
( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) ) ) )
3319, 32pm2.61i 156 . 2  |-  ( 0g
`  G )  =  ( iota e ( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) )
341, 33eqtri 2316 1  |-  .0.  =  ( iota e ( e  e.  B  /\  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   E!weu 2156   A.wral 2556   _Vcvv 2801   (/)c0 3468   iotacio 5233   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416
This theorem is referenced by:  0g0  14402  ismgmid  14403  grpidpropd  14415  oppgid  14845  dfur2  15360  oppr0  15431  oppr1  15432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-0g 13420
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