Step |
Hyp |
Ref |
Expression |
1 |
|
wlkp1.v |
|- V = ( Vtx ` G ) |
2 |
|
wlkp1.i |
|- I = ( iEdg ` G ) |
3 |
|
wlkp1.f |
|- ( ph -> Fun I ) |
4 |
|
wlkp1.a |
|- ( ph -> I e. Fin ) |
5 |
|
wlkp1.b |
|- ( ph -> B e. W ) |
6 |
|
wlkp1.c |
|- ( ph -> C e. V ) |
7 |
|
wlkp1.d |
|- ( ph -> -. B e. dom I ) |
8 |
|
wlkp1.w |
|- ( ph -> F ( Walks ` G ) P ) |
9 |
|
wlkp1.n |
|- N = ( # ` F ) |
10 |
|
wlkp1.e |
|- ( ph -> E e. ( Edg ` G ) ) |
11 |
|
wlkp1.x |
|- ( ph -> { ( P ` N ) , C } C_ E ) |
12 |
|
wlkp1.u |
|- ( ph -> ( iEdg ` S ) = ( I u. { <. B , E >. } ) ) |
13 |
|
wlkp1.h |
|- H = ( F u. { <. N , B >. } ) |
14 |
|
wlkp1.q |
|- Q = ( P u. { <. ( N + 1 ) , C >. } ) |
15 |
|
wlkp1.s |
|- ( ph -> ( Vtx ` S ) = V ) |
16 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
|
wlkp1lem5 |
|- ( ph -> A. x e. ( 0 ... N ) ( Q ` x ) = ( P ` x ) ) |
17 |
|
elfzofz |
|- ( k e. ( 0 ..^ N ) -> k e. ( 0 ... N ) ) |
18 |
17
|
adantl |
|- ( ( ph /\ k e. ( 0 ..^ N ) ) -> k e. ( 0 ... N ) ) |
19 |
|
fveq2 |
|- ( x = k -> ( Q ` x ) = ( Q ` k ) ) |
20 |
|
fveq2 |
|- ( x = k -> ( P ` x ) = ( P ` k ) ) |
21 |
19 20
|
eqeq12d |
|- ( x = k -> ( ( Q ` x ) = ( P ` x ) <-> ( Q ` k ) = ( P ` k ) ) ) |
22 |
21
|
rspcv |
|- ( k e. ( 0 ... N ) -> ( A. x e. ( 0 ... N ) ( Q ` x ) = ( P ` x ) -> ( Q ` k ) = ( P ` k ) ) ) |
23 |
18 22
|
syl |
|- ( ( ph /\ k e. ( 0 ..^ N ) ) -> ( A. x e. ( 0 ... N ) ( Q ` x ) = ( P ` x ) -> ( Q ` k ) = ( P ` k ) ) ) |
24 |
23
|
imp |
|- ( ( ( ph /\ k e. ( 0 ..^ N ) ) /\ A. x e. ( 0 ... N ) ( Q ` x ) = ( P ` x ) ) -> ( Q ` k ) = ( P ` k ) ) |
25 |
|
fzofzp1 |
|- ( k e. ( 0 ..^ N ) -> ( k + 1 ) e. ( 0 ... N ) ) |
26 |
25
|
adantl |
|- ( ( ph /\ k e. ( 0 ..^ N ) ) -> ( k + 1 ) e. ( 0 ... N ) ) |
27 |
|
fveq2 |
|- ( x = ( k + 1 ) -> ( Q ` x ) = ( Q ` ( k + 1 ) ) ) |
28 |
|
fveq2 |
|- ( x = ( k + 1 ) -> ( P ` x ) = ( P ` ( k + 1 ) ) ) |
29 |
27 28
|
eqeq12d |
|- ( x = ( k + 1 ) -> ( ( Q ` x ) = ( P ` x ) <-> ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) ) ) |
30 |
29
|
rspcv |
|- ( ( k + 1 ) e. ( 0 ... N ) -> ( A. x e. ( 0 ... N ) ( Q ` x ) = ( P ` x ) -> ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) ) ) |
31 |
26 30
|
syl |
|- ( ( ph /\ k e. ( 0 ..^ N ) ) -> ( A. x e. ( 0 ... N ) ( Q ` x ) = ( P ` x ) -> ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) ) ) |
32 |
31
|
imp |
|- ( ( ( ph /\ k e. ( 0 ..^ N ) ) /\ A. x e. ( 0 ... N ) ( Q ` x ) = ( P ` x ) ) -> ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) ) |
33 |
12
|
adantr |
|- ( ( ph /\ k e. ( 0 ..^ N ) ) -> ( iEdg ` S ) = ( I u. { <. B , E >. } ) ) |
34 |
13
|
fveq1i |
|- ( H ` k ) = ( ( F u. { <. N , B >. } ) ` k ) |
35 |
|
fzonel |
|- -. N e. ( 0 ..^ N ) |
36 |
|
eleq1 |
|- ( N = k -> ( N e. ( 0 ..^ N ) <-> k e. ( 0 ..^ N ) ) ) |
37 |
35 36
|
mtbii |
|- ( N = k -> -. k e. ( 0 ..^ N ) ) |
38 |
37
|
a1i |
|- ( ph -> ( N = k -> -. k e. ( 0 ..^ N ) ) ) |
39 |
38
|
con2d |
|- ( ph -> ( k e. ( 0 ..^ N ) -> -. N = k ) ) |
40 |
39
|
imp |
|- ( ( ph /\ k e. ( 0 ..^ N ) ) -> -. N = k ) |
41 |
40
|
neqned |
|- ( ( ph /\ k e. ( 0 ..^ N ) ) -> N =/= k ) |
42 |
|
fvunsn |
|- ( N =/= k -> ( ( F u. { <. N , B >. } ) ` k ) = ( F ` k ) ) |
43 |
41 42
|
syl |
|- ( ( ph /\ k e. ( 0 ..^ N ) ) -> ( ( F u. { <. N , B >. } ) ` k ) = ( F ` k ) ) |
44 |
34 43
|
syl5eq |
|- ( ( ph /\ k e. ( 0 ..^ N ) ) -> ( H ` k ) = ( F ` k ) ) |
45 |
33 44
|
fveq12d |
|- ( ( ph /\ k e. ( 0 ..^ N ) ) -> ( ( iEdg ` S ) ` ( H ` k ) ) = ( ( I u. { <. B , E >. } ) ` ( F ` k ) ) ) |
46 |
9
|
oveq2i |
|- ( 0 ..^ N ) = ( 0 ..^ ( # ` F ) ) |
47 |
46
|
eleq2i |
|- ( k e. ( 0 ..^ N ) <-> k e. ( 0 ..^ ( # ` F ) ) ) |
48 |
2
|
wlkf |
|- ( F ( Walks ` G ) P -> F e. Word dom I ) |
49 |
8 48
|
syl |
|- ( ph -> F e. Word dom I ) |
50 |
|
wrdsymbcl |
|- ( ( F e. Word dom I /\ k e. ( 0 ..^ ( # ` F ) ) ) -> ( F ` k ) e. dom I ) |
51 |
50
|
ex |
|- ( F e. Word dom I -> ( k e. ( 0 ..^ ( # ` F ) ) -> ( F ` k ) e. dom I ) ) |
52 |
49 51
|
syl |
|- ( ph -> ( k e. ( 0 ..^ ( # ` F ) ) -> ( F ` k ) e. dom I ) ) |
53 |
47 52
|
syl5bi |
|- ( ph -> ( k e. ( 0 ..^ N ) -> ( F ` k ) e. dom I ) ) |
54 |
53
|
imp |
|- ( ( ph /\ k e. ( 0 ..^ N ) ) -> ( F ` k ) e. dom I ) |
55 |
|
eleq1 |
|- ( B = ( F ` k ) -> ( B e. dom I <-> ( F ` k ) e. dom I ) ) |
56 |
54 55
|
syl5ibrcom |
|- ( ( ph /\ k e. ( 0 ..^ N ) ) -> ( B = ( F ` k ) -> B e. dom I ) ) |
57 |
56
|
con3d |
|- ( ( ph /\ k e. ( 0 ..^ N ) ) -> ( -. B e. dom I -> -. B = ( F ` k ) ) ) |
58 |
57
|
ex |
|- ( ph -> ( k e. ( 0 ..^ N ) -> ( -. B e. dom I -> -. B = ( F ` k ) ) ) ) |
59 |
7 58
|
mpid |
|- ( ph -> ( k e. ( 0 ..^ N ) -> -. B = ( F ` k ) ) ) |
60 |
59
|
imp |
|- ( ( ph /\ k e. ( 0 ..^ N ) ) -> -. B = ( F ` k ) ) |
61 |
60
|
neqned |
|- ( ( ph /\ k e. ( 0 ..^ N ) ) -> B =/= ( F ` k ) ) |
62 |
|
fvunsn |
|- ( B =/= ( F ` k ) -> ( ( I u. { <. B , E >. } ) ` ( F ` k ) ) = ( I ` ( F ` k ) ) ) |
63 |
61 62
|
syl |
|- ( ( ph /\ k e. ( 0 ..^ N ) ) -> ( ( I u. { <. B , E >. } ) ` ( F ` k ) ) = ( I ` ( F ` k ) ) ) |
64 |
45 63
|
eqtrd |
|- ( ( ph /\ k e. ( 0 ..^ N ) ) -> ( ( iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) ) |
65 |
64
|
adantr |
|- ( ( ( ph /\ k e. ( 0 ..^ N ) ) /\ A. x e. ( 0 ... N ) ( Q ` x ) = ( P ` x ) ) -> ( ( iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) ) |
66 |
24 32 65
|
3jca |
|- ( ( ( ph /\ k e. ( 0 ..^ N ) ) /\ A. x e. ( 0 ... N ) ( Q ` x ) = ( P ` x ) ) -> ( ( Q ` k ) = ( P ` k ) /\ ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) /\ ( ( iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) ) ) |
67 |
16 66
|
mpidan |
|- ( ( ph /\ k e. ( 0 ..^ N ) ) -> ( ( Q ` k ) = ( P ` k ) /\ ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) /\ ( ( iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) ) ) |
68 |
67
|
ralrimiva |
|- ( ph -> A. k e. ( 0 ..^ N ) ( ( Q ` k ) = ( P ` k ) /\ ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) /\ ( ( iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) ) ) |