Step |
Hyp |
Ref |
Expression |
1 |
|
2wlkd.p |
|- P = <" A B C "> |
2 |
|
2wlkd.f |
|- F = <" J K "> |
3 |
|
2wlkd.s |
|- ( ph -> ( A e. V /\ B e. V /\ C e. V ) ) |
4 |
|
2wlkd.n |
|- ( ph -> ( A =/= B /\ B =/= C ) ) |
5 |
|
2wlkd.e |
|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) ) |
6 |
|
2wlkd.v |
|- V = ( Vtx ` G ) |
7 |
|
2wlkd.i |
|- I = ( iEdg ` G ) |
8 |
|
2trld.n |
|- ( ph -> J =/= K ) |
9 |
|
s3cli |
|- <" A B C "> e. Word _V |
10 |
1 9
|
eqeltri |
|- P e. Word _V |
11 |
10
|
a1i |
|- ( ph -> P e. Word _V ) |
12 |
2
|
fveq2i |
|- ( # ` F ) = ( # ` <" J K "> ) |
13 |
|
s2len |
|- ( # ` <" J K "> ) = 2 |
14 |
12 13
|
eqtri |
|- ( # ` F ) = 2 |
15 |
|
3m1e2 |
|- ( 3 - 1 ) = 2 |
16 |
1
|
fveq2i |
|- ( # ` P ) = ( # ` <" A B C "> ) |
17 |
|
s3len |
|- ( # ` <" A B C "> ) = 3 |
18 |
16 17
|
eqtr2i |
|- 3 = ( # ` P ) |
19 |
18
|
oveq1i |
|- ( 3 - 1 ) = ( ( # ` P ) - 1 ) |
20 |
14 15 19
|
3eqtr2i |
|- ( # ` F ) = ( ( # ` P ) - 1 ) |
21 |
1 2 3 4
|
2pthdlem1 |
|- ( ph -> A. k e. ( 0 ..^ ( # ` P ) ) A. j e. ( 1 ..^ ( # ` F ) ) ( k =/= j -> ( P ` k ) =/= ( P ` j ) ) ) |
22 |
|
eqid |
|- ( # ` F ) = ( # ` F ) |
23 |
1 2 3 4 5 6 7 8
|
2trld |
|- ( ph -> F ( Trails ` G ) P ) |
24 |
11 20 21 22 23
|
pthd |
|- ( ph -> F ( Paths ` G ) P ) |