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 ) ) ) ) |