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Theorem grpidval 14707
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 5728 . . . . . . . 8  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 grpidval.b . . . . . . . 8  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2486 . . . . . . 7  |-  ( g  =  G  ->  ( Base `  g )  =  B )
54eleq2d 2503 . . . . . 6  |-  ( g  =  G  ->  (
e  e.  ( Base `  g )  <->  e  e.  B ) )
6 fveq2 5728 . . . . . . . . . . 11  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
7 grpidval.p . . . . . . . . . . 11  |-  .+  =  ( +g  `  G )
86, 7syl6eqr 2486 . . . . . . . . . 10  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
98oveqd 6098 . . . . . . . . 9  |-  ( g  =  G  ->  (
e ( +g  `  g
) x )  =  ( e  .+  x
) )
109eqeq1d 2444 . . . . . . . 8  |-  ( g  =  G  ->  (
( e ( +g  `  g ) x )  =  x  <->  ( e  .+  x )  =  x ) )
118oveqd 6098 . . . . . . . . 9  |-  ( g  =  G  ->  (
x ( +g  `  g
) e )  =  ( x  .+  e
) )
1211eqeq1d 2444 . . . . . . . 8  |-  ( g  =  G  ->  (
( x ( +g  `  g ) e )  =  x  <->  ( x  .+  e )  =  x ) )
1310, 12anbi12d 692 . . . . . . 7  |-  ( g  =  G  ->  (
( ( e ( +g  `  g ) x )  =  x  /\  ( x ( +g  `  g ) e )  =  x )  <->  ( ( e 
.+  x )  =  x  /\  ( x 
.+  e )  =  x ) ) )
144, 13raleqbidv 2916 . . . . . 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 692 . . . . 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 5439 . . . 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 13727 . . . 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 5435 . . . 4  |-  ( iota e ( e  e.  B  /\  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) ) )  e.  _V
1916, 17, 18fvmpt 5806 . . 3  |-  ( G  e.  _V  ->  ( 0g `  G )  =  ( iota e ( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) ) )
20 fvprc 5722 . . . 4  |-  ( -.  G  e.  _V  ->  ( 0g `  G )  =  (/) )
21 euex 2304 . . . . . . 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 3633 . . . . . . . . . 10  |-  ( e  e.  B  ->  -.  B  =  (/) )
23 fvprc 5722 . . . . . . . . . . 11  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
243, 23syl5eq 2480 . . . . . . . . . 10  |-  ( -.  G  e.  _V  ->  B  =  (/) )
2522, 24nsyl2 121 . . . . . . . . 9  |-  ( e  e.  B  ->  G  e.  _V )
2625adantr 452 . . . . . . . 8  |-  ( ( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) )  ->  G  e.  _V )
2726exlimiv 1644 . . . . . . 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 129 . . . . 5  |-  ( -.  G  e.  _V  ->  -.  E! e ( e  e.  B  /\  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) )
30 iotanul 5433 . . . . 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 2471 . . 3  |-  ( -.  G  e.  _V  ->  ( 0g `  G )  =  ( iota e
( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) ) ) )
3319, 32pm2.61i 158 . 2  |-  ( 0g
`  G )  =  ( iota e ( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) )
341, 33eqtri 2456 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 359   E.wex 1550    = wceq 1652    e. wcel 1725   E!weu 2281   A.wral 2705   _Vcvv 2956   (/)c0 3628   iotacio 5416   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   0gc0g 13723
This theorem is referenced by:  0g0  14709  ismgmid  14710  grpidpropd  14722  oppgid  15152  dfur2  15667  oppr0  15738  oppr1  15739
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-0g 13727
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