Metamath Proof Explorer


Theorem psgnfieu

Description: A permutation of a finite set has exactly one parity. (Contributed by AV, 13-Jan-2019)

Ref Expression
Hypotheses psgnfitr.g
|- G = ( SymGrp ` N )
psgnfitr.p
|- B = ( Base ` G )
psgnfitr.t
|- T = ran ( pmTrsp ` N )
Assertion psgnfieu
|- ( ( N e. Fin /\ Q e. B ) -> E! s E. w e. Word T ( Q = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) )

Proof

Step Hyp Ref Expression
1 psgnfitr.g
 |-  G = ( SymGrp ` N )
2 psgnfitr.p
 |-  B = ( Base ` G )
3 psgnfitr.t
 |-  T = ran ( pmTrsp ` N )
4 simpr
 |-  ( ( N e. Fin /\ Q e. B ) -> Q e. B )
5 1 2 sygbasnfpfi
 |-  ( ( N e. Fin /\ Q e. B ) -> dom ( Q \ _I ) e. Fin )
6 eqid
 |-  ( pmSgn ` N ) = ( pmSgn ` N )
7 1 6 2 psgneldm
 |-  ( Q e. dom ( pmSgn ` N ) <-> ( Q e. B /\ dom ( Q \ _I ) e. Fin ) )
8 4 5 7 sylanbrc
 |-  ( ( N e. Fin /\ Q e. B ) -> Q e. dom ( pmSgn ` N ) )
9 1 3 6 psgneu
 |-  ( Q e. dom ( pmSgn ` N ) -> E! s E. w e. Word T ( Q = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) )
10 8 9 syl
 |-  ( ( N e. Fin /\ Q e. B ) -> E! s E. w e. Word T ( Q = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) )