Metamath Proof Explorer


Theorem psgnvali

Description: A finitary permutation has at least one representation for its parity. (Contributed by Stefan O'Rear, 28-Aug-2015)

Ref Expression
Hypotheses psgnval.g
|- G = ( SymGrp ` D )
psgnval.t
|- T = ran ( pmTrsp ` D )
psgnval.n
|- N = ( pmSgn ` D )
Assertion psgnvali
|- ( P e. dom N -> E. w e. Word T ( P = ( G gsum w ) /\ ( N ` P ) = ( -u 1 ^ ( # ` w ) ) ) )

Proof

Step Hyp Ref Expression
1 psgnval.g
 |-  G = ( SymGrp ` D )
2 psgnval.t
 |-  T = ran ( pmTrsp ` D )
3 psgnval.n
 |-  N = ( pmSgn ` D )
4 1 2 3 psgnval
 |-  ( P e. dom N -> ( N ` P ) = ( iota s E. w e. Word T ( P = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) )
5 1 2 3 psgneu
 |-  ( P e. dom N -> E! s E. w e. Word T ( P = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) )
6 iotacl
 |-  ( E! s E. w e. Word T ( P = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) -> ( iota s E. w e. Word T ( P = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) e. { s | E. w e. Word T ( P = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) } )
7 5 6 syl
 |-  ( P e. dom N -> ( iota s E. w e. Word T ( P = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) e. { s | E. w e. Word T ( P = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) } )
8 4 7 eqeltrd
 |-  ( P e. dom N -> ( N ` P ) e. { s | E. w e. Word T ( P = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) } )
9 fvex
 |-  ( N ` P ) e. _V
10 eqeq1
 |-  ( s = ( N ` P ) -> ( s = ( -u 1 ^ ( # ` w ) ) <-> ( N ` P ) = ( -u 1 ^ ( # ` w ) ) ) )
11 10 anbi2d
 |-  ( s = ( N ` P ) -> ( ( P = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) <-> ( P = ( G gsum w ) /\ ( N ` P ) = ( -u 1 ^ ( # ` w ) ) ) ) )
12 11 rexbidv
 |-  ( s = ( N ` P ) -> ( E. w e. Word T ( P = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) <-> E. w e. Word T ( P = ( G gsum w ) /\ ( N ` P ) = ( -u 1 ^ ( # ` w ) ) ) ) )
13 9 12 elab
 |-  ( ( N ` P ) e. { s | E. w e. Word T ( P = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) } <-> E. w e. Word T ( P = ( G gsum w ) /\ ( N ` P ) = ( -u 1 ^ ( # ` w ) ) ) )
14 8 13 sylib
 |-  ( P e. dom N -> E. w e. Word T ( P = ( G gsum w ) /\ ( N ` P ) = ( -u 1 ^ ( # ` w ) ) ) )