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 ) ) ) ) |