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
|
psgneldm2i |
|- ( ( D e. V /\ W e. Word T ) -> ( G gsum W ) e. dom N ) |
5 |
1 2 3
|
psgnval |
|- ( ( G gsum W ) e. dom N -> ( N ` ( G gsum W ) ) = ( iota s E. w e. Word T ( ( G gsum W ) = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) ) |
6 |
4 5
|
syl |
|- ( ( D e. V /\ W e. Word T ) -> ( N ` ( G gsum W ) ) = ( iota s E. w e. Word T ( ( G gsum W ) = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) ) |
7 |
|
simpr |
|- ( ( D e. V /\ W e. Word T ) -> W e. Word T ) |
8 |
|
eqidd |
|- ( ( D e. V /\ W e. Word T ) -> ( G gsum W ) = ( G gsum W ) ) |
9 |
|
eqidd |
|- ( ( D e. V /\ W e. Word T ) -> ( -u 1 ^ ( # ` W ) ) = ( -u 1 ^ ( # ` W ) ) ) |
10 |
|
oveq2 |
|- ( w = W -> ( G gsum w ) = ( G gsum W ) ) |
11 |
10
|
eqeq2d |
|- ( w = W -> ( ( G gsum W ) = ( G gsum w ) <-> ( G gsum W ) = ( G gsum W ) ) ) |
12 |
|
fveq2 |
|- ( w = W -> ( # ` w ) = ( # ` W ) ) |
13 |
12
|
oveq2d |
|- ( w = W -> ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ ( # ` W ) ) ) |
14 |
13
|
eqeq2d |
|- ( w = W -> ( ( -u 1 ^ ( # ` W ) ) = ( -u 1 ^ ( # ` w ) ) <-> ( -u 1 ^ ( # ` W ) ) = ( -u 1 ^ ( # ` W ) ) ) ) |
15 |
11 14
|
anbi12d |
|- ( w = W -> ( ( ( G gsum W ) = ( G gsum w ) /\ ( -u 1 ^ ( # ` W ) ) = ( -u 1 ^ ( # ` w ) ) ) <-> ( ( G gsum W ) = ( G gsum W ) /\ ( -u 1 ^ ( # ` W ) ) = ( -u 1 ^ ( # ` W ) ) ) ) ) |
16 |
15
|
rspcev |
|- ( ( W e. Word T /\ ( ( G gsum W ) = ( G gsum W ) /\ ( -u 1 ^ ( # ` W ) ) = ( -u 1 ^ ( # ` W ) ) ) ) -> E. w e. Word T ( ( G gsum W ) = ( G gsum w ) /\ ( -u 1 ^ ( # ` W ) ) = ( -u 1 ^ ( # ` w ) ) ) ) |
17 |
7 8 9 16
|
syl12anc |
|- ( ( D e. V /\ W e. Word T ) -> E. w e. Word T ( ( G gsum W ) = ( G gsum w ) /\ ( -u 1 ^ ( # ` W ) ) = ( -u 1 ^ ( # ` w ) ) ) ) |
18 |
|
ovexd |
|- ( ( D e. V /\ W e. Word T ) -> ( -u 1 ^ ( # ` W ) ) e. _V ) |
19 |
1 2 3
|
psgneu |
|- ( ( G gsum W ) e. dom N -> E! s E. w e. Word T ( ( G gsum W ) = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) |
20 |
4 19
|
syl |
|- ( ( D e. V /\ W e. Word T ) -> E! s E. w e. Word T ( ( G gsum W ) = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) |
21 |
|
eqeq1 |
|- ( s = ( -u 1 ^ ( # ` W ) ) -> ( s = ( -u 1 ^ ( # ` w ) ) <-> ( -u 1 ^ ( # ` W ) ) = ( -u 1 ^ ( # ` w ) ) ) ) |
22 |
21
|
anbi2d |
|- ( s = ( -u 1 ^ ( # ` W ) ) -> ( ( ( G gsum W ) = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) <-> ( ( G gsum W ) = ( G gsum w ) /\ ( -u 1 ^ ( # ` W ) ) = ( -u 1 ^ ( # ` w ) ) ) ) ) |
23 |
22
|
rexbidv |
|- ( s = ( -u 1 ^ ( # ` W ) ) -> ( E. w e. Word T ( ( G gsum W ) = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) <-> E. w e. Word T ( ( G gsum W ) = ( G gsum w ) /\ ( -u 1 ^ ( # ` W ) ) = ( -u 1 ^ ( # ` w ) ) ) ) ) |
24 |
23
|
adantl |
|- ( ( ( D e. V /\ W e. Word T ) /\ s = ( -u 1 ^ ( # ` W ) ) ) -> ( E. w e. Word T ( ( G gsum W ) = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) <-> E. w e. Word T ( ( G gsum W ) = ( G gsum w ) /\ ( -u 1 ^ ( # ` W ) ) = ( -u 1 ^ ( # ` w ) ) ) ) ) |
25 |
18 20 24
|
iota2d |
|- ( ( D e. V /\ W e. Word T ) -> ( E. w e. Word T ( ( G gsum W ) = ( G gsum w ) /\ ( -u 1 ^ ( # ` W ) ) = ( -u 1 ^ ( # ` w ) ) ) <-> ( iota s E. w e. Word T ( ( G gsum W ) = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) = ( -u 1 ^ ( # ` W ) ) ) ) |
26 |
17 25
|
mpbid |
|- ( ( D e. V /\ W e. Word T ) -> ( iota s E. w e. Word T ( ( G gsum W ) = ( G gsum w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) = ( -u 1 ^ ( # ` W ) ) ) |
27 |
6 26
|
eqtrd |
|- ( ( D e. V /\ W e. Word T ) -> ( N ` ( G gsum W ) ) = ( -u 1 ^ ( # ` W ) ) ) |