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

Theorem symgval 15021
Description: The value of the symmetry group function at  A. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.)
Hypotheses
Ref Expression
symgval.1  |-  G  =  ( SymGrp `  A )
symgval.2  |-  B  =  { x  |  x : A -1-1-onto-> A }
symgval.3  |-  .+  =  ( f  e.  B ,  g  e.  B  |->  ( f  o.  g
) )
symgval.4  |-  J  =  ( Xt_ `  ( A  X.  { ~P A } ) )
Assertion
Ref Expression
symgval  |-  ( A  e.  V  ->  G  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. } )
Distinct variable group:    f, g, x, A
Allowed substitution hints:    B( x, f, g)    .+ ( x, f, g)    G( x, f, g)    J( x, f, g)    V( x, f, g)

Proof of Theorem symgval
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 symgval.1 . 2  |-  G  =  ( SymGrp `  A )
2 elex 2907 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
3 ovex 6045 . . . . . . 7  |-  ( a  ^m  a )  e. 
_V
4 f1of 5614 . . . . . . . . 9  |-  ( x : a -1-1-onto-> a  ->  x : a --> a )
5 vex 2902 . . . . . . . . . 10  |-  a  e. 
_V
65, 5elmap 6978 . . . . . . . . 9  |-  ( x  e.  ( a  ^m  a )  <->  x :
a --> a )
74, 6sylibr 204 . . . . . . . 8  |-  ( x : a -1-1-onto-> a  ->  x  e.  ( a  ^m  a
) )
87abssi 3361 . . . . . . 7  |-  { x  |  x : a -1-1-onto-> a } 
C_  ( a  ^m  a )
93, 8ssexi 4289 . . . . . 6  |-  { x  |  x : a -1-1-onto-> a }  e.  _V
109a1i 11 . . . . 5  |-  ( a  =  A  ->  { x  |  x : a -1-1-onto-> a }  e.  _V )
11 id 20 . . . . . . . 8  |-  ( b  =  { x  |  x : a -1-1-onto-> a }  ->  b  =  {
x  |  x : a -1-1-onto-> a } )
12 f1oeq23 5608 . . . . . . . . . . 11  |-  ( ( a  =  A  /\  a  =  A )  ->  ( x : a -1-1-onto-> a  <-> 
x : A -1-1-onto-> A ) )
1312anidms 627 . . . . . . . . . 10  |-  ( a  =  A  ->  (
x : a -1-1-onto-> a  <->  x : A
-1-1-onto-> A ) )
1413abbidv 2501 . . . . . . . . 9  |-  ( a  =  A  ->  { x  |  x : a -1-1-onto-> a }  =  { x  |  x : A -1-1-onto-> A }
)
15 symgval.2 . . . . . . . . 9  |-  B  =  { x  |  x : A -1-1-onto-> A }
1614, 15syl6eqr 2437 . . . . . . . 8  |-  ( a  =  A  ->  { x  |  x : a -1-1-onto-> a }  =  B )
1711, 16sylan9eqr 2441 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  b  =  B )
1817opeq2d 3933 . . . . . 6  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  <. ( Base `  ndx ) ,  b
>.  =  <. ( Base `  ndx ) ,  B >. )
19 eqidd 2388 . . . . . . . . 9  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ( f  o.  g )  =  ( f  o.  g ) )
2017, 17, 19mpt2eq123dv 6075 . . . . . . . 8  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ( f  e.  b ,  g  e.  b  |->  ( f  o.  g ) )  =  ( f  e.  B ,  g  e.  B  |->  ( f  o.  g
) ) )
21 symgval.3 . . . . . . . 8  |-  .+  =  ( f  e.  B ,  g  e.  B  |->  ( f  o.  g
) )
2220, 21syl6eqr 2437 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ( f  e.  b ,  g  e.  b  |->  ( f  o.  g ) )  = 
.+  )
2322opeq2d 3933 . . . . . 6  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  <. ( +g  ` 
ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( f  o.  g ) ) >.  =  <. ( +g  `  ndx ) ,  .+  >. )
24 simpl 444 . . . . . . . . . 10  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  a  =  A )
2524pweqd 3747 . . . . . . . . . . 11  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ~P a  =  ~P A )
2625sneqd 3770 . . . . . . . . . 10  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  { ~P a }  =  { ~P A } )
2724, 26xpeq12d 4843 . . . . . . . . 9  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ( a  X.  { ~P a } )  =  ( A  X.  { ~P A } ) )
2827fveq2d 5672 . . . . . . . 8  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ( Xt_ `  ( a  X.  { ~P a } ) )  =  ( Xt_ `  ( A  X.  { ~P A } ) ) )
29 symgval.4 . . . . . . . 8  |-  J  =  ( Xt_ `  ( A  X.  { ~P A } ) )
3028, 29syl6eqr 2437 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ( Xt_ `  ( a  X.  { ~P a } ) )  =  J )
3130opeq2d 3933 . . . . . 6  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  <. (TopSet `  ndx ) ,  ( Xt_ `  ( a  X.  { ~P a } ) )
>.  =  <. (TopSet `  ndx ) ,  J >. )
3218, 23, 31tpeq123d 3841 . . . . 5  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  { <. ( Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  ( f  e.  b ,  g  e.  b 
|->  ( f  o.  g
) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( a  X.  { ~P a } ) ) >. }  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. } )
3310, 32csbied 3236 . . . 4  |-  ( a  =  A  ->  [_ {
x  |  x : a -1-1-onto-> a }  /  b ]_ { <. ( Base `  ndx ) ,  b >. , 
<. ( +g  `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( f  o.  g ) ) >. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( a  X.  { ~P a } ) )
>. }  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (TopSet ` 
ndx ) ,  J >. } )
34 df-symg 15020 . . . 4  |-  SymGrp  =  ( a  e.  _V  |->  [_ { x  |  x : a -1-1-onto-> a }  /  b ]_ { <. ( Base `  ndx ) ,  b >. , 
<. ( +g  `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( f  o.  g ) ) >. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( a  X.  { ~P a } ) )
>. } )
35 tpex 4648 . . . 4  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (TopSet ` 
ndx ) ,  J >. }  e.  _V
3633, 34, 35fvmpt 5745 . . 3  |-  ( A  e.  _V  ->  ( SymGrp `
 A )  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. } )
372, 36syl 16 . 2  |-  ( A  e.  V  ->  ( SymGrp `
 A )  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. } )
381, 37syl5eq 2431 1  |-  ( A  e.  V  ->  G  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2373   _Vcvv 2899   [_csb 3194   ~Pcpw 3742   {csn 3757   {ctp 3759   <.cop 3760    X. cxp 4816    o. ccom 4822   -->wf 5390   -1-1-onto->wf1o 5393   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022    ^m cmap 6954   ndxcnx 13393   Basecbs 13396   +g cplusg 13456  TopSetcts 13462   Xt_cpt 13593   SymGrpcsymg 15019
This theorem is referenced by:  symgbas  15022  symgplusg  15026  symgtset  15029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-map 6956  df-symg 15020
  Copyright terms: Public domain W3C validator