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 |
|
3trld.n |
|- ( ph -> ( J =/= K /\ J =/= L /\ K =/= L ) ) |
9 |
|
3cycld.e |
|- ( ph -> A = D ) |
10 |
1 2 3 4 5 6 7 8
|
3pthd |
|- ( ph -> F ( Paths ` G ) P ) |
11 |
1
|
fveq1i |
|- ( P ` 0 ) = ( <" A B C D "> ` 0 ) |
12 |
|
s4fv0 |
|- ( A e. V -> ( <" A B C D "> ` 0 ) = A ) |
13 |
11 12
|
syl5eq |
|- ( A e. V -> ( P ` 0 ) = A ) |
14 |
13
|
ad3antrrr |
|- ( ( ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) /\ A = D ) -> ( P ` 0 ) = A ) |
15 |
|
simpr |
|- ( ( ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) /\ A = D ) -> A = D ) |
16 |
2
|
fveq2i |
|- ( # ` F ) = ( # ` <" J K L "> ) |
17 |
|
s3len |
|- ( # ` <" J K L "> ) = 3 |
18 |
16 17
|
eqtri |
|- ( # ` F ) = 3 |
19 |
1 18
|
fveq12i |
|- ( P ` ( # ` F ) ) = ( <" A B C D "> ` 3 ) |
20 |
|
s4fv3 |
|- ( D e. V -> ( <" A B C D "> ` 3 ) = D ) |
21 |
19 20
|
eqtr2id |
|- ( D e. V -> D = ( P ` ( # ` F ) ) ) |
22 |
21
|
adantl |
|- ( ( C e. V /\ D e. V ) -> D = ( P ` ( # ` F ) ) ) |
23 |
22
|
ad2antlr |
|- ( ( ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) /\ A = D ) -> D = ( P ` ( # ` F ) ) ) |
24 |
14 15 23
|
3eqtrd |
|- ( ( ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) /\ A = D ) -> ( P ` 0 ) = ( P ` ( # ` F ) ) ) |
25 |
3 9 24
|
syl2anc |
|- ( ph -> ( P ` 0 ) = ( P ` ( # ` F ) ) ) |
26 |
|
iscycl |
|- ( F ( Cycles ` G ) P <-> ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) |
27 |
10 25 26
|
sylanbrc |
|- ( ph -> F ( Cycles ` G ) P ) |