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Theorem grpidval 16552
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 5888 . . . . . . . 8  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 grpidval.b . . . . . . . 8  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2514 . . . . . . 7  |-  ( g  =  G  ->  ( Base `  g )  =  B )
54eleq2d 2525 . . . . . 6  |-  ( g  =  G  ->  (
e  e.  ( Base `  g )  <->  e  e.  B ) )
6 fveq2 5888 . . . . . . . . . . 11  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
7 grpidval.p . . . . . . . . . . 11  |-  .+  =  ( +g  `  G )
86, 7syl6eqr 2514 . . . . . . . . . 10  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
98oveqd 6332 . . . . . . . . 9  |-  ( g  =  G  ->  (
e ( +g  `  g
) x )  =  ( e  .+  x
) )
109eqeq1d 2464 . . . . . . . 8  |-  ( g  =  G  ->  (
( e ( +g  `  g ) x )  =  x  <->  ( e  .+  x )  =  x ) )
118oveqd 6332 . . . . . . . . 9  |-  ( g  =  G  ->  (
x ( +g  `  g
) e )  =  ( x  .+  e
) )
1211eqeq1d 2464 . . . . . . . 8  |-  ( g  =  G  ->  (
( x ( +g  `  g ) e )  =  x  <->  ( x  .+  e )  =  x ) )
1310, 12anbi12d 722 . . . . . . 7  |-  ( g  =  G  ->  (
( ( e ( +g  `  g ) x )  =  x  /\  ( x ( +g  `  g ) e )  =  x )  <->  ( ( e 
.+  x )  =  x  /\  ( x 
.+  e )  =  x ) ) )
144, 13raleqbidv 3013 . . . . . 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 722 . . . . 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 5586 . . . 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 15389 . . . 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 5582 . . . 4  |-  ( iota e ( e  e.  B  /\  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) ) )  e.  _V
1916, 17, 18fvmpt 5971 . . 3  |-  ( G  e.  _V  ->  ( 0g `  G )  =  ( iota e ( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) ) )
20 fvprc 5882 . . . 4  |-  ( -.  G  e.  _V  ->  ( 0g `  G )  =  (/) )
21 euex 2334 . . . . . . 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 3748 . . . . . . . . . 10  |-  ( e  e.  B  ->  -.  B  =  (/) )
23 fvprc 5882 . . . . . . . . . . 11  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
243, 23syl5eq 2508 . . . . . . . . . 10  |-  ( -.  G  e.  _V  ->  B  =  (/) )
2522, 24nsyl2 132 . . . . . . . . 9  |-  ( e  e.  B  ->  G  e.  _V )
2625adantr 471 . . . . . . . 8  |-  ( ( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) )  ->  G  e.  _V )
2726exlimiv 1787 . . . . . . 7  |-  ( E. e ( e  e.  B  /\  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) )  ->  G  e.  _V )
2821, 27syl 17 . . . . . 6  |-  ( E! e ( e  e.  B  /\  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) )  ->  G  e.  _V )
2928con3i 142 . . . . 5  |-  ( -.  G  e.  _V  ->  -.  E! e ( e  e.  B  /\  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) )
30 iotanul 5580 . . . . 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 17 . . . 4  |-  ( -.  G  e.  _V  ->  ( iota e ( e  e.  B  /\  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) )  =  (/) )
3220, 31eqtr4d 2499 . . 3  |-  ( -.  G  e.  _V  ->  ( 0g `  G )  =  ( iota e
( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) ) ) )
3319, 32pm2.61i 169 . 2  |-  ( 0g
`  G )  =  ( iota e ( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) )
341, 33eqtri 2484 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 375    = wceq 1455   E.wex 1674    e. wcel 1898   E!weu 2310   A.wral 2749   _Vcvv 3057   (/)c0 3743   iotacio 5563   ` cfv 5601  (class class class)co 6315   Basecbs 15170   +g cplusg 15239   0gc0g 15387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6318  df-0g 15389
This theorem is referenced by:  grpidpropd  16553  0g0  16555  ismgmid  16556  oppgid  17056  dfur2  17787  oppr0  17910  oppr1  17911
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