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 |
|
3spthd.n |
|- ( ph -> A =/= D ) |
10 |
1 2 3 4 5 6 7 8
|
3trld |
|- ( ph -> F ( Trails ` G ) P ) |
11 |
|
simpr |
|- ( ( ph /\ F ( Trails ` G ) P ) -> F ( Trails ` G ) P ) |
12 |
|
df-3an |
|- ( ( A =/= B /\ A =/= C /\ A =/= D ) <-> ( ( A =/= B /\ A =/= C ) /\ A =/= D ) ) |
13 |
12
|
simplbi2 |
|- ( ( A =/= B /\ A =/= C ) -> ( A =/= D -> ( A =/= B /\ A =/= C /\ A =/= D ) ) ) |
14 |
13
|
3ad2ant1 |
|- ( ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) -> ( A =/= D -> ( A =/= B /\ A =/= C /\ A =/= D ) ) ) |
15 |
9 14
|
mpan9 |
|- ( ( ph /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( A =/= B /\ A =/= C /\ A =/= D ) ) |
16 |
|
simpr2 |
|- ( ( ph /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( B =/= C /\ B =/= D ) ) |
17 |
|
simpr3 |
|- ( ( ph /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> C =/= D ) |
18 |
15 16 17
|
3jca |
|- ( ( ph /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) |
19 |
4 18
|
mpdan |
|- ( ph -> ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) |
20 |
|
funcnvs4 |
|- ( ( ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) /\ ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> Fun `' <" A B C D "> ) |
21 |
3 19 20
|
syl2anc |
|- ( ph -> Fun `' <" A B C D "> ) |
22 |
21
|
adantr |
|- ( ( ph /\ F ( Trails ` G ) P ) -> Fun `' <" A B C D "> ) |
23 |
1
|
a1i |
|- ( ( ph /\ F ( Trails ` G ) P ) -> P = <" A B C D "> ) |
24 |
23
|
cnveqd |
|- ( ( ph /\ F ( Trails ` G ) P ) -> `' P = `' <" A B C D "> ) |
25 |
24
|
funeqd |
|- ( ( ph /\ F ( Trails ` G ) P ) -> ( Fun `' P <-> Fun `' <" A B C D "> ) ) |
26 |
22 25
|
mpbird |
|- ( ( ph /\ F ( Trails ` G ) P ) -> Fun `' P ) |
27 |
|
isspth |
|- ( F ( SPaths ` G ) P <-> ( F ( Trails ` G ) P /\ Fun `' P ) ) |
28 |
11 26 27
|
sylanbrc |
|- ( ( ph /\ F ( Trails ` G ) P ) -> F ( SPaths ` G ) P ) |
29 |
10 28
|
mpdan |
|- ( ph -> F ( SPaths ` G ) P ) |