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

Theorem grpidval 15453
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 5712 . . . . . . . 8  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 grpidval.b . . . . . . . 8  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2493 . . . . . . 7  |-  ( g  =  G  ->  ( Base `  g )  =  B )
54eleq2d 2510 . . . . . 6  |-  ( g  =  G  ->  (
e  e.  ( Base `  g )  <->  e  e.  B ) )
6 fveq2 5712 . . . . . . . . . . 11  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
7 grpidval.p . . . . . . . . . . 11  |-  .+  =  ( +g  `  G )
86, 7syl6eqr 2493 . . . . . . . . . 10  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
98oveqd 6129 . . . . . . . . 9  |-  ( g  =  G  ->  (
e ( +g  `  g
) x )  =  ( e  .+  x
) )
109eqeq1d 2451 . . . . . . . 8  |-  ( g  =  G  ->  (
( e ( +g  `  g ) x )  =  x  <->  ( e  .+  x )  =  x ) )
118oveqd 6129 . . . . . . . . 9  |-  ( g  =  G  ->  (
x ( +g  `  g
) e )  =  ( x  .+  e
) )
1211eqeq1d 2451 . . . . . . . 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 2952 . . . . . 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 5423 . . . 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 14401 . . . 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 5419 . . . 4  |-  ( iota e ( e  e.  B  /\  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) ) )  e.  _V
1916, 17, 18fvmpt 5795 . . 3  |-  ( G  e.  _V  ->  ( 0g `  G )  =  ( iota e ( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) ) )
20 fvprc 5706 . . . 4  |-  ( -.  G  e.  _V  ->  ( 0g `  G )  =  (/) )
21 euex 2281 . . . . . . 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 3663 . . . . . . . . . 10  |-  ( e  e.  B  ->  -.  B  =  (/) )
23 fvprc 5706 . . . . . . . . . . 11  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
243, 23syl5eq 2487 . . . . . . . . . 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 1688 . . . . . . 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 5417 . . . . 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 2478 . . 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 2463 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 1369   E.wex 1586    e. wcel 1756   E!weu 2253   A.wral 2736   _Vcvv 2993   (/)c0 3658   iotacio 5400   ` cfv 5439  (class class class)co 6112   Basecbs 14195   +g cplusg 14259   0gc0g 14399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-iota 5402  df-fun 5441  df-fv 5447  df-ov 6115  df-0g 14401
This theorem is referenced by:  0g0  15455  ismgmid  15456  grpidpropd  15468  oppgid  15892  dfur2  16628  oppr0  16747  oppr1  16748
  Copyright terms: Public domain W3C validator