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

Theorem gidval 20986
Description: The value of the identity element of a group. (Contributed by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
gidval.1  |-  X  =  ran  G
Assertion
Ref Expression
gidval  |-  ( G  e.  V  ->  (GId `  G )  =  (
iota_ u  e.  X A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
Distinct variable groups:    x, u, G    u, X, x
Allowed substitution hints:    V( x, u)

Proof of Theorem gidval
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 elex 2872 . 2  |-  ( G  e.  V  ->  G  e.  _V )
2 rneq 4983 . . . . 5  |-  ( g  =  G  ->  ran  g  =  ran  G )
3 gidval.1 . . . . 5  |-  X  =  ran  G
42, 3syl6eqr 2408 . . . 4  |-  ( g  =  G  ->  ran  g  =  X )
5 oveq 5948 . . . . . . 7  |-  ( g  =  G  ->  (
u g x )  =  ( u G x ) )
65eqeq1d 2366 . . . . . 6  |-  ( g  =  G  ->  (
( u g x )  =  x  <->  ( u G x )  =  x ) )
7 oveq 5948 . . . . . . 7  |-  ( g  =  G  ->  (
x g u )  =  ( x G u ) )
87eqeq1d 2366 . . . . . 6  |-  ( g  =  G  ->  (
( x g u )  =  x  <->  ( x G u )  =  x ) )
96, 8anbi12d 691 . . . . 5  |-  ( g  =  G  ->  (
( ( u g x )  =  x  /\  ( x g u )  =  x )  <->  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
104, 9raleqbidv 2824 . . . 4  |-  ( g  =  G  ->  ( A. x  e.  ran  g ( ( u g x )  =  x  /\  ( x g u )  =  x )  <->  A. x  e.  X  ( (
u G x )  =  x  /\  (
x G u )  =  x ) ) )
114, 10riotaeqbidv 6391 . . 3  |-  ( g  =  G  ->  ( iota_ u  e.  ran  g A. x  e.  ran  g ( ( u g x )  =  x  /\  ( x g u )  =  x ) )  =  ( iota_ u  e.  X A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
12 df-gid 20965 . . 3  |- GId  =  ( g  e.  _V  |->  (
iota_ u  e.  ran  g A. x  e.  ran  g ( ( u g x )  =  x  /\  ( x g u )  =  x ) ) )
13 riotaex 6392 . . 3  |-  ( iota_ u  e.  X A. x  e.  X  ( (
u G x )  =  x  /\  (
x G u )  =  x ) )  e.  _V
1411, 12, 13fvmpt 5682 . 2  |-  ( G  e.  _V  ->  (GId `  G )  =  (
iota_ u  e.  X A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
151, 14syl 15 1  |-  ( G  e.  V  ->  (GId `  G )  =  (
iota_ u  e.  X A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619   _Vcvv 2864   ran crn 4769   ` cfv 5334  (class class class)co 5942   iota_crio 6381  GIdcgi 20960
This theorem is referenced by:  grpoidval  20989  idrval  21100  exidresid  25892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-iota 5298  df-fun 5336  df-fv 5342  df-ov 5945  df-riota 6388  df-gid 20965
  Copyright terms: Public domain W3C validator