Step |
Hyp |
Ref |
Expression |
1 |
|
3wlkd.p |
|- P = <" A B C D "> |
2 |
|
3wlkd.f |
|- F = <" J K L "> |
3 |
|
3wlkd.s |
|- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) ) |
4 |
|
3wlkd.n |
|- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) |
5 |
|
3wlkd.e |
|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) ) |
6 |
|
3wlkd.v |
|- V = ( Vtx ` G ) |
7 |
|
3wlkd.i |
|- I = ( iEdg ` G ) |
8 |
|
s4cli |
|- <" A B C D "> e. Word _V |
9 |
1 8
|
eqeltri |
|- P e. Word _V |
10 |
9
|
a1i |
|- ( ph -> P e. Word _V ) |
11 |
|
s3cli |
|- <" J K L "> e. Word _V |
12 |
2 11
|
eqeltri |
|- F e. Word _V |
13 |
12
|
a1i |
|- ( ph -> F e. Word _V ) |
14 |
1 2
|
3wlkdlem1 |
|- ( # ` P ) = ( ( # ` F ) + 1 ) |
15 |
14
|
a1i |
|- ( ph -> ( # ` P ) = ( ( # ` F ) + 1 ) ) |
16 |
1 2 3 4 5
|
3wlkdlem10 |
|- ( ph -> A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) |
17 |
1 2 3 4
|
3wlkdlem5 |
|- ( ph -> A. k e. ( 0 ..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) ) |
18 |
6
|
1vgrex |
|- ( A e. V -> G e. _V ) |
19 |
18
|
ad2antrr |
|- ( ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> G e. _V ) |
20 |
3 19
|
syl |
|- ( ph -> G e. _V ) |
21 |
1 2 3
|
3wlkdlem4 |
|- ( ph -> A. k e. ( 0 ... ( # ` F ) ) ( P ` k ) e. V ) |
22 |
10 13 15 16 17 20 6 7 21
|
wlkd |
|- ( ph -> F ( Walks ` G ) P ) |