Step |
Hyp |
Ref |
Expression |
1 |
|
umgr2adedgwlk.e |
|- E = ( Edg ` G ) |
2 |
|
umgr2adedgwlk.i |
|- I = ( iEdg ` G ) |
3 |
|
umgr2adedgwlk.f |
|- F = <" J K "> |
4 |
|
umgr2adedgwlk.p |
|- P = <" A B C "> |
5 |
|
umgr2adedgwlk.g |
|- ( ph -> G e. UMGraph ) |
6 |
|
umgr2adedgwlk.a |
|- ( ph -> ( { A , B } e. E /\ { B , C } e. E ) ) |
7 |
|
umgr2adedgwlk.j |
|- ( ph -> ( I ` J ) = { A , B } ) |
8 |
|
umgr2adedgwlk.k |
|- ( ph -> ( I ` K ) = { B , C } ) |
9 |
1 2 3 4 5 6 7 8
|
umgr2adedgwlk |
|- ( ph -> ( F ( Walks ` G ) P /\ ( # ` F ) = 2 /\ ( A = ( P ` 0 ) /\ B = ( P ` 1 ) /\ C = ( P ` 2 ) ) ) ) |
10 |
|
simp1 |
|- ( ( F ( Walks ` G ) P /\ ( # ` F ) = 2 /\ ( A = ( P ` 0 ) /\ B = ( P ` 1 ) /\ C = ( P ` 2 ) ) ) -> F ( Walks ` G ) P ) |
11 |
|
id |
|- ( ( P ` 0 ) = A -> ( P ` 0 ) = A ) |
12 |
11
|
eqcoms |
|- ( A = ( P ` 0 ) -> ( P ` 0 ) = A ) |
13 |
12
|
3ad2ant1 |
|- ( ( A = ( P ` 0 ) /\ B = ( P ` 1 ) /\ C = ( P ` 2 ) ) -> ( P ` 0 ) = A ) |
14 |
13
|
3ad2ant3 |
|- ( ( F ( Walks ` G ) P /\ ( # ` F ) = 2 /\ ( A = ( P ` 0 ) /\ B = ( P ` 1 ) /\ C = ( P ` 2 ) ) ) -> ( P ` 0 ) = A ) |
15 |
|
fveq2 |
|- ( 2 = ( # ` F ) -> ( P ` 2 ) = ( P ` ( # ` F ) ) ) |
16 |
15
|
eqcoms |
|- ( ( # ` F ) = 2 -> ( P ` 2 ) = ( P ` ( # ` F ) ) ) |
17 |
16
|
eqeq1d |
|- ( ( # ` F ) = 2 -> ( ( P ` 2 ) = C <-> ( P ` ( # ` F ) ) = C ) ) |
18 |
17
|
biimpcd |
|- ( ( P ` 2 ) = C -> ( ( # ` F ) = 2 -> ( P ` ( # ` F ) ) = C ) ) |
19 |
18
|
eqcoms |
|- ( C = ( P ` 2 ) -> ( ( # ` F ) = 2 -> ( P ` ( # ` F ) ) = C ) ) |
20 |
19
|
3ad2ant3 |
|- ( ( A = ( P ` 0 ) /\ B = ( P ` 1 ) /\ C = ( P ` 2 ) ) -> ( ( # ` F ) = 2 -> ( P ` ( # ` F ) ) = C ) ) |
21 |
20
|
com12 |
|- ( ( # ` F ) = 2 -> ( ( A = ( P ` 0 ) /\ B = ( P ` 1 ) /\ C = ( P ` 2 ) ) -> ( P ` ( # ` F ) ) = C ) ) |
22 |
21
|
a1i |
|- ( F ( Walks ` G ) P -> ( ( # ` F ) = 2 -> ( ( A = ( P ` 0 ) /\ B = ( P ` 1 ) /\ C = ( P ` 2 ) ) -> ( P ` ( # ` F ) ) = C ) ) ) |
23 |
22
|
3imp |
|- ( ( F ( Walks ` G ) P /\ ( # ` F ) = 2 /\ ( A = ( P ` 0 ) /\ B = ( P ` 1 ) /\ C = ( P ` 2 ) ) ) -> ( P ` ( # ` F ) ) = C ) |
24 |
10 14 23
|
3jca |
|- ( ( F ( Walks ` G ) P /\ ( # ` F ) = 2 /\ ( A = ( P ` 0 ) /\ B = ( P ` 1 ) /\ C = ( P ` 2 ) ) ) -> ( F ( Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = C ) ) |
25 |
9 24
|
syl |
|- ( ph -> ( F ( Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = C ) ) |
26 |
|
3anass |
|- ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) <-> ( G e. UMGraph /\ ( { A , B } e. E /\ { B , C } e. E ) ) ) |
27 |
5 6 26
|
sylanbrc |
|- ( ph -> ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) ) |
28 |
1
|
umgr2adedgwlklem |
|- ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) -> ( ( A =/= B /\ B =/= C ) /\ ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) ) ) |
29 |
|
3simpb |
|- ( ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) -> ( A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) ) |
30 |
29
|
adantl |
|- ( ( ( A =/= B /\ B =/= C ) /\ ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) ) -> ( A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) ) |
31 |
27 28 30
|
3syl |
|- ( ph -> ( A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) ) |
32 |
|
3anass |
|- ( ( G e. UMGraph /\ A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) <-> ( G e. UMGraph /\ ( A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) ) ) |
33 |
5 31 32
|
sylanbrc |
|- ( ph -> ( G e. UMGraph /\ A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) ) |
34 |
|
s2cli |
|- <" J K "> e. Word _V |
35 |
3 34
|
eqeltri |
|- F e. Word _V |
36 |
|
s3cli |
|- <" A B C "> e. Word _V |
37 |
4 36
|
eqeltri |
|- P e. Word _V |
38 |
35 37
|
pm3.2i |
|- ( F e. Word _V /\ P e. Word _V ) |
39 |
|
id |
|- ( ( A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) -> ( A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) ) |
40 |
39
|
3adant1 |
|- ( ( G e. UMGraph /\ A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) -> ( A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) ) |
41 |
40
|
anim1i |
|- ( ( ( G e. UMGraph /\ A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) /\ ( F e. Word _V /\ P e. Word _V ) ) -> ( ( A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) /\ ( F e. Word _V /\ P e. Word _V ) ) ) |
42 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
43 |
42
|
iswlkon |
|- ( ( ( A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) /\ ( F e. Word _V /\ P e. Word _V ) ) -> ( F ( A ( WalksOn ` G ) C ) P <-> ( F ( Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = C ) ) ) |
44 |
41 43
|
syl |
|- ( ( ( G e. UMGraph /\ A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) /\ ( F e. Word _V /\ P e. Word _V ) ) -> ( F ( A ( WalksOn ` G ) C ) P <-> ( F ( Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = C ) ) ) |
45 |
33 38 44
|
sylancl |
|- ( ph -> ( F ( A ( WalksOn ` G ) C ) P <-> ( F ( Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = C ) ) ) |
46 |
25 45
|
mpbird |
|- ( ph -> F ( A ( WalksOn ` G ) C ) P ) |