Theorem pmtrodpm

Description: A transposition is an odd permutation. (Contributed by SO, 9-Jul-2018)

	Ref	Expression
Hypotheses	evpmodpmf1o.s	|- S = ( SymGrp ` D )
	evpmodpmf1o.p	|- P = ( Base ` S )
	pmtrodpm.t	|- T = ran ( pmTrsp ` D )
Assertion	pmtrodpm	|- ( ( D e. Fin /\ F e. T ) -> F e. ( P \ ( pmEven ` D ) ) )

Proof

Step	Hyp	Ref	Expression
1		evpmodpmf1o.s	|- S = ( SymGrp ` D )
2		evpmodpmf1o.p	|- P = ( Base ` S )
3		pmtrodpm.t	|- T = ran ( pmTrsp ` D )
4		simpl	|- ( ( D e. Fin /\ F e. T ) -> D e. Fin )
5	3 1 2	symgtrf	|- T C_ P
6	5	sseli	|- ( F e. T -> F e. P )
7	6	adantl	|- ( ( D e. Fin /\ F e. T ) -> F e. P )
8		eqid	|- ( pmSgn ` D ) = ( pmSgn ` D )
9	1 3 8	psgnpmtr	|- ( F e. T -> ( ( pmSgn ` D ) ` F ) = -u 1 )
10	9	adantl	|- ( ( D e. Fin /\ F e. T ) -> ( ( pmSgn ` D ) ` F ) = -u 1 )
11	1 2 8	psgnodpmr	|- ( ( D e. Fin /\ F e. P /\ ( ( pmSgn ` D ) ` F ) = -u 1 ) -> F e. ( P \ ( pmEven ` D ) ) )
12	4 7 10 11	syl3anc	|- ( ( D e. Fin /\ F e. T ) -> F e. ( P \ ( pmEven ` D ) ) )