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Metamath Proof Explorer

Theorem gsmtrcl

Description: The group sum of transpositions of a finite set is a permutation, see also psgneldm2i . (Contributed by AV, 19-Jan-2019)

Ref Expression
Hypotheses gsmtrcl.s 
|- S = ( SymGrp ` N )
gsmtrcl.b 
|- B = ( Base ` S )
gsmtrcl.t 
|- T = ran ( pmTrsp ` N )
Assertion gsmtrcl 
|- ( ( N e. Fin /\ W e. Word T ) -> ( S gsum W ) e. B )

Proof

Step Hyp Ref Expression
1 gsmtrcl.s 
 |-  S = ( SymGrp ` N )
2 gsmtrcl.b 
 |-  B = ( Base ` S )
3 gsmtrcl.t 
 |-  T = ran ( pmTrsp ` N )
4 eqid 
 |-  ( pmSgn ` N ) = ( pmSgn ` N )
5 1 3 4 psgneldm2i 
 |-  ( ( N e. Fin /\ W e. Word T ) -> ( S gsum W ) e. dom ( pmSgn ` N ) )
6 1 4 2 psgneldm 
 |-  ( ( S gsum W ) e. dom ( pmSgn ` N ) <-> ( ( S gsum W ) e. B /\ dom ( ( S gsum W ) \ _I ) e. Fin ) )
7 ax-1 
 |-  ( ( S gsum W ) e. B -> ( ( N e. Fin /\ W e. Word T ) -> ( S gsum W ) e. B ) )
8 7 adantr 
 |-  ( ( ( S gsum W ) e. B /\ dom ( ( S gsum W ) \ _I ) e. Fin ) -> ( ( N e. Fin /\ W e. Word T ) -> ( S gsum W ) e. B ) )
9 6 8 sylbi 
 |-  ( ( S gsum W ) e. dom ( pmSgn ` N ) -> ( ( N e. Fin /\ W e. Word T ) -> ( S gsum W ) e. B ) )
10 5 9 mpcom 
 |-  ( ( N e. Fin /\ W e. Word T ) -> ( S gsum W ) e. B )