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Theorem grpidvalg 12810
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
grpidvalg  |-  ( G  e.  V  ->  .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)    V( x, e)    .0. ( x, e)

Proof of Theorem grpidvalg
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 grpidval.o . 2  |-  .0.  =  ( 0g `  G )
2 df-0g 12724 . . 3  |-  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 ) ) ) )
3 fveq2 5527 . . . . . . 7  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
4 grpidval.b . . . . . . 7  |-  B  =  ( Base `  G
)
53, 4eqtr4di 2238 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  B )
65eleq2d 2257 . . . . 5  |-  ( g  =  G  ->  (
e  e.  ( Base `  g )  <->  e  e.  B ) )
7 fveq2 5527 . . . . . . . . . 10  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
8 grpidval.p . . . . . . . . . 10  |-  .+  =  ( +g  `  G )
97, 8eqtr4di 2238 . . . . . . . . 9  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
109oveqd 5905 . . . . . . . 8  |-  ( g  =  G  ->  (
e ( +g  `  g
) x )  =  ( e  .+  x
) )
1110eqeq1d 2196 . . . . . . 7  |-  ( g  =  G  ->  (
( e ( +g  `  g ) x )  =  x  <->  ( e  .+  x )  =  x ) )
129oveqd 5905 . . . . . . . 8  |-  ( g  =  G  ->  (
x ( +g  `  g
) e )  =  ( x  .+  e
) )
1312eqeq1d 2196 . . . . . . 7  |-  ( g  =  G  ->  (
( x ( +g  `  g ) e )  =  x  <->  ( x  .+  e )  =  x ) )
1411, 13anbi12d 473 . . . . . 6  |-  ( g  =  G  ->  (
( ( e ( +g  `  g ) x )  =  x  /\  ( x ( +g  `  g ) e )  =  x )  <->  ( ( e 
.+  x )  =  x  /\  ( x 
.+  e )  =  x ) ) )
155, 14raleqbidv 2695 . . . . 5  |-  ( 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 ) ) )
166, 15anbi12d 473 . . . 4  |-  ( 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 ) ) ) )
1716iotabidv 5211 . . 3  |-  ( 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 ) ) ) )
18 elex 2760 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
19 df-riota 5844 . . . 4  |-  ( iota_ 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 ) ) )
20 basfn 12533 . . . . . . 7  |-  Base  Fn  _V
21 funfvex 5544 . . . . . . . 8  |-  ( ( Fun  Base  /\  G  e. 
dom  Base )  ->  ( Base `  G )  e. 
_V )
2221funfni 5328 . . . . . . 7  |-  ( (
Base  Fn  _V  /\  G  e.  _V )  ->  ( Base `  G )  e. 
_V )
2320, 18, 22sylancr 414 . . . . . 6  |-  ( G  e.  V  ->  ( Base `  G )  e. 
_V )
244, 23eqeltrid 2274 . . . . 5  |-  ( G  e.  V  ->  B  e.  _V )
25 riotaexg 5848 . . . . 5  |-  ( B  e.  _V  ->  ( iota_ e  e.  B  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) )  e.  _V )
2624, 25syl 14 . . . 4  |-  ( G  e.  V  ->  ( iota_ e  e.  B  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) )  e.  _V )
2719, 26eqeltrrid 2275 . . 3  |-  ( G  e.  V  ->  ( iota e ( e  e.  B  /\  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) ) )  e.  _V )
282, 17, 18, 27fvmptd3 5622 . 2  |-  ( G  e.  V  ->  ( 0g `  G )  =  ( iota e ( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) ) )
291, 28eqtrid 2232 1  |-  ( G  e.  V  ->  .0.  =  ( iota e
( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1363    e. wcel 2158   A.wral 2465   _Vcvv 2749   iotacio 5188    Fn wfn 5223   ` cfv 5228   iota_crio 5843  (class class class)co 5888   Basecbs 12475   +g cplusg 12550   0gc0g 12722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-riota 5844  df-ov 5891  df-inn 8933  df-ndx 12478  df-slot 12479  df-base 12481  df-0g 12724
This theorem is referenced by:  grpidpropdg  12811  0g0  12813  ismgmid  12814  sgrpidmndm  12840  dfur2g  13199  oppr0g  13313  oppr1g  13314
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