Step |
Hyp |
Ref |
Expression |
1 |
|
psgnuni.g |
|- G = ( SymGrp ` D ) |
2 |
|
psgnuni.t |
|- T = ran ( pmTrsp ` D ) |
3 |
|
psgnuni.d |
|- ( ph -> D e. V ) |
4 |
|
psgnuni.w |
|- ( ph -> W e. Word T ) |
5 |
|
psgnuni.x |
|- ( ph -> X e. Word T ) |
6 |
|
psgnuni.e |
|- ( ph -> ( G gsum W ) = ( G gsum X ) ) |
7 |
|
lencl |
|- ( W e. Word T -> ( # ` W ) e. NN0 ) |
8 |
4 7
|
syl |
|- ( ph -> ( # ` W ) e. NN0 ) |
9 |
8
|
nn0zd |
|- ( ph -> ( # ` W ) e. ZZ ) |
10 |
|
m1expcl |
|- ( ( # ` W ) e. ZZ -> ( -u 1 ^ ( # ` W ) ) e. ZZ ) |
11 |
9 10
|
syl |
|- ( ph -> ( -u 1 ^ ( # ` W ) ) e. ZZ ) |
12 |
11
|
zcnd |
|- ( ph -> ( -u 1 ^ ( # ` W ) ) e. CC ) |
13 |
|
lencl |
|- ( X e. Word T -> ( # ` X ) e. NN0 ) |
14 |
5 13
|
syl |
|- ( ph -> ( # ` X ) e. NN0 ) |
15 |
14
|
nn0zd |
|- ( ph -> ( # ` X ) e. ZZ ) |
16 |
|
m1expcl |
|- ( ( # ` X ) e. ZZ -> ( -u 1 ^ ( # ` X ) ) e. ZZ ) |
17 |
15 16
|
syl |
|- ( ph -> ( -u 1 ^ ( # ` X ) ) e. ZZ ) |
18 |
17
|
zcnd |
|- ( ph -> ( -u 1 ^ ( # ` X ) ) e. CC ) |
19 |
|
neg1cn |
|- -u 1 e. CC |
20 |
|
neg1ne0 |
|- -u 1 =/= 0 |
21 |
|
expne0i |
|- ( ( -u 1 e. CC /\ -u 1 =/= 0 /\ ( # ` X ) e. ZZ ) -> ( -u 1 ^ ( # ` X ) ) =/= 0 ) |
22 |
19 20 15 21
|
mp3an12i |
|- ( ph -> ( -u 1 ^ ( # ` X ) ) =/= 0 ) |
23 |
|
m1expaddsub |
|- ( ( ( # ` W ) e. ZZ /\ ( # ` X ) e. ZZ ) -> ( -u 1 ^ ( ( # ` W ) - ( # ` X ) ) ) = ( -u 1 ^ ( ( # ` W ) + ( # ` X ) ) ) ) |
24 |
9 15 23
|
syl2anc |
|- ( ph -> ( -u 1 ^ ( ( # ` W ) - ( # ` X ) ) ) = ( -u 1 ^ ( ( # ` W ) + ( # ` X ) ) ) ) |
25 |
|
expsub |
|- ( ( ( -u 1 e. CC /\ -u 1 =/= 0 ) /\ ( ( # ` W ) e. ZZ /\ ( # ` X ) e. ZZ ) ) -> ( -u 1 ^ ( ( # ` W ) - ( # ` X ) ) ) = ( ( -u 1 ^ ( # ` W ) ) / ( -u 1 ^ ( # ` X ) ) ) ) |
26 |
19 20 25
|
mpanl12 |
|- ( ( ( # ` W ) e. ZZ /\ ( # ` X ) e. ZZ ) -> ( -u 1 ^ ( ( # ` W ) - ( # ` X ) ) ) = ( ( -u 1 ^ ( # ` W ) ) / ( -u 1 ^ ( # ` X ) ) ) ) |
27 |
9 15 26
|
syl2anc |
|- ( ph -> ( -u 1 ^ ( ( # ` W ) - ( # ` X ) ) ) = ( ( -u 1 ^ ( # ` W ) ) / ( -u 1 ^ ( # ` X ) ) ) ) |
28 |
|
revcl |
|- ( X e. Word T -> ( reverse ` X ) e. Word T ) |
29 |
5 28
|
syl |
|- ( ph -> ( reverse ` X ) e. Word T ) |
30 |
|
ccatlen |
|- ( ( W e. Word T /\ ( reverse ` X ) e. Word T ) -> ( # ` ( W ++ ( reverse ` X ) ) ) = ( ( # ` W ) + ( # ` ( reverse ` X ) ) ) ) |
31 |
4 29 30
|
syl2anc |
|- ( ph -> ( # ` ( W ++ ( reverse ` X ) ) ) = ( ( # ` W ) + ( # ` ( reverse ` X ) ) ) ) |
32 |
|
revlen |
|- ( X e. Word T -> ( # ` ( reverse ` X ) ) = ( # ` X ) ) |
33 |
5 32
|
syl |
|- ( ph -> ( # ` ( reverse ` X ) ) = ( # ` X ) ) |
34 |
33
|
oveq2d |
|- ( ph -> ( ( # ` W ) + ( # ` ( reverse ` X ) ) ) = ( ( # ` W ) + ( # ` X ) ) ) |
35 |
31 34
|
eqtr2d |
|- ( ph -> ( ( # ` W ) + ( # ` X ) ) = ( # ` ( W ++ ( reverse ` X ) ) ) ) |
36 |
35
|
oveq2d |
|- ( ph -> ( -u 1 ^ ( ( # ` W ) + ( # ` X ) ) ) = ( -u 1 ^ ( # ` ( W ++ ( reverse ` X ) ) ) ) ) |
37 |
|
ccatcl |
|- ( ( W e. Word T /\ ( reverse ` X ) e. Word T ) -> ( W ++ ( reverse ` X ) ) e. Word T ) |
38 |
4 29 37
|
syl2anc |
|- ( ph -> ( W ++ ( reverse ` X ) ) e. Word T ) |
39 |
6
|
fveq2d |
|- ( ph -> ( ( invg ` G ) ` ( G gsum W ) ) = ( ( invg ` G ) ` ( G gsum X ) ) ) |
40 |
|
eqid |
|- ( invg ` G ) = ( invg ` G ) |
41 |
2 1 40
|
symgtrinv |
|- ( ( D e. V /\ X e. Word T ) -> ( ( invg ` G ) ` ( G gsum X ) ) = ( G gsum ( reverse ` X ) ) ) |
42 |
3 5 41
|
syl2anc |
|- ( ph -> ( ( invg ` G ) ` ( G gsum X ) ) = ( G gsum ( reverse ` X ) ) ) |
43 |
39 42
|
eqtr2d |
|- ( ph -> ( G gsum ( reverse ` X ) ) = ( ( invg ` G ) ` ( G gsum W ) ) ) |
44 |
43
|
oveq2d |
|- ( ph -> ( ( G gsum W ) ( +g ` G ) ( G gsum ( reverse ` X ) ) ) = ( ( G gsum W ) ( +g ` G ) ( ( invg ` G ) ` ( G gsum W ) ) ) ) |
45 |
1
|
symggrp |
|- ( D e. V -> G e. Grp ) |
46 |
3 45
|
syl |
|- ( ph -> G e. Grp ) |
47 |
|
grpmnd |
|- ( G e. Grp -> G e. Mnd ) |
48 |
3 45 47
|
3syl |
|- ( ph -> G e. Mnd ) |
49 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
50 |
2 1 49
|
symgtrf |
|- T C_ ( Base ` G ) |
51 |
|
sswrd |
|- ( T C_ ( Base ` G ) -> Word T C_ Word ( Base ` G ) ) |
52 |
50 51
|
ax-mp |
|- Word T C_ Word ( Base ` G ) |
53 |
52 4
|
sselid |
|- ( ph -> W e. Word ( Base ` G ) ) |
54 |
49
|
gsumwcl |
|- ( ( G e. Mnd /\ W e. Word ( Base ` G ) ) -> ( G gsum W ) e. ( Base ` G ) ) |
55 |
48 53 54
|
syl2anc |
|- ( ph -> ( G gsum W ) e. ( Base ` G ) ) |
56 |
|
eqid |
|- ( +g ` G ) = ( +g ` G ) |
57 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
58 |
49 56 57 40
|
grprinv |
|- ( ( G e. Grp /\ ( G gsum W ) e. ( Base ` G ) ) -> ( ( G gsum W ) ( +g ` G ) ( ( invg ` G ) ` ( G gsum W ) ) ) = ( 0g ` G ) ) |
59 |
46 55 58
|
syl2anc |
|- ( ph -> ( ( G gsum W ) ( +g ` G ) ( ( invg ` G ) ` ( G gsum W ) ) ) = ( 0g ` G ) ) |
60 |
44 59
|
eqtrd |
|- ( ph -> ( ( G gsum W ) ( +g ` G ) ( G gsum ( reverse ` X ) ) ) = ( 0g ` G ) ) |
61 |
52 29
|
sselid |
|- ( ph -> ( reverse ` X ) e. Word ( Base ` G ) ) |
62 |
49 56
|
gsumccat |
|- ( ( G e. Mnd /\ W e. Word ( Base ` G ) /\ ( reverse ` X ) e. Word ( Base ` G ) ) -> ( G gsum ( W ++ ( reverse ` X ) ) ) = ( ( G gsum W ) ( +g ` G ) ( G gsum ( reverse ` X ) ) ) ) |
63 |
48 53 61 62
|
syl3anc |
|- ( ph -> ( G gsum ( W ++ ( reverse ` X ) ) ) = ( ( G gsum W ) ( +g ` G ) ( G gsum ( reverse ` X ) ) ) ) |
64 |
1
|
symgid |
|- ( D e. V -> ( _I |` D ) = ( 0g ` G ) ) |
65 |
3 64
|
syl |
|- ( ph -> ( _I |` D ) = ( 0g ` G ) ) |
66 |
60 63 65
|
3eqtr4d |
|- ( ph -> ( G gsum ( W ++ ( reverse ` X ) ) ) = ( _I |` D ) ) |
67 |
1 2 3 38 66
|
psgnunilem4 |
|- ( ph -> ( -u 1 ^ ( # ` ( W ++ ( reverse ` X ) ) ) ) = 1 ) |
68 |
36 67
|
eqtrd |
|- ( ph -> ( -u 1 ^ ( ( # ` W ) + ( # ` X ) ) ) = 1 ) |
69 |
24 27 68
|
3eqtr3d |
|- ( ph -> ( ( -u 1 ^ ( # ` W ) ) / ( -u 1 ^ ( # ` X ) ) ) = 1 ) |
70 |
12 18 22 69
|
diveq1d |
|- ( ph -> ( -u 1 ^ ( # ` W ) ) = ( -u 1 ^ ( # ` X ) ) ) |