Step |
Hyp |
Ref |
Expression |
1 |
|
clwwlkext2edg.v |
|- V = ( Vtx ` G ) |
2 |
|
clwwlkext2edg.e |
|- E = ( Edg ` G ) |
3 |
|
clwwlknnn |
|- ( ( W ++ <" Z "> ) e. ( N ClWWalksN G ) -> N e. NN ) |
4 |
1 2
|
isclwwlknx |
|- ( N e. NN -> ( ( W ++ <" Z "> ) e. ( N ClWWalksN G ) <-> ( ( ( W ++ <" Z "> ) e. Word V /\ A. i e. ( 0 ..^ ( ( # ` ( W ++ <" Z "> ) ) - 1 ) ) { ( ( W ++ <" Z "> ) ` i ) , ( ( W ++ <" Z "> ) ` ( i + 1 ) ) } e. E /\ { ( lastS ` ( W ++ <" Z "> ) ) , ( ( W ++ <" Z "> ) ` 0 ) } e. E ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) ) ) |
5 |
|
ige2m2fzo |
|- ( N e. ( ZZ>= ` 2 ) -> ( N - 2 ) e. ( 0 ..^ ( N - 1 ) ) ) |
6 |
5
|
3ad2ant3 |
|- ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( N - 2 ) e. ( 0 ..^ ( N - 1 ) ) ) |
7 |
6
|
adantr |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) -> ( N - 2 ) e. ( 0 ..^ ( N - 1 ) ) ) |
8 |
|
oveq1 |
|- ( ( # ` ( W ++ <" Z "> ) ) = N -> ( ( # ` ( W ++ <" Z "> ) ) - 1 ) = ( N - 1 ) ) |
9 |
8
|
oveq2d |
|- ( ( # ` ( W ++ <" Z "> ) ) = N -> ( 0 ..^ ( ( # ` ( W ++ <" Z "> ) ) - 1 ) ) = ( 0 ..^ ( N - 1 ) ) ) |
10 |
9
|
eleq2d |
|- ( ( # ` ( W ++ <" Z "> ) ) = N -> ( ( N - 2 ) e. ( 0 ..^ ( ( # ` ( W ++ <" Z "> ) ) - 1 ) ) <-> ( N - 2 ) e. ( 0 ..^ ( N - 1 ) ) ) ) |
11 |
10
|
adantl |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) -> ( ( N - 2 ) e. ( 0 ..^ ( ( # ` ( W ++ <" Z "> ) ) - 1 ) ) <-> ( N - 2 ) e. ( 0 ..^ ( N - 1 ) ) ) ) |
12 |
7 11
|
mpbird |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) -> ( N - 2 ) e. ( 0 ..^ ( ( # ` ( W ++ <" Z "> ) ) - 1 ) ) ) |
13 |
|
fveq2 |
|- ( i = ( N - 2 ) -> ( ( W ++ <" Z "> ) ` i ) = ( ( W ++ <" Z "> ) ` ( N - 2 ) ) ) |
14 |
|
fvoveq1 |
|- ( i = ( N - 2 ) -> ( ( W ++ <" Z "> ) ` ( i + 1 ) ) = ( ( W ++ <" Z "> ) ` ( ( N - 2 ) + 1 ) ) ) |
15 |
13 14
|
preq12d |
|- ( i = ( N - 2 ) -> { ( ( W ++ <" Z "> ) ` i ) , ( ( W ++ <" Z "> ) ` ( i + 1 ) ) } = { ( ( W ++ <" Z "> ) ` ( N - 2 ) ) , ( ( W ++ <" Z "> ) ` ( ( N - 2 ) + 1 ) ) } ) |
16 |
15
|
eleq1d |
|- ( i = ( N - 2 ) -> ( { ( ( W ++ <" Z "> ) ` i ) , ( ( W ++ <" Z "> ) ` ( i + 1 ) ) } e. E <-> { ( ( W ++ <" Z "> ) ` ( N - 2 ) ) , ( ( W ++ <" Z "> ) ` ( ( N - 2 ) + 1 ) ) } e. E ) ) |
17 |
16
|
rspcv |
|- ( ( N - 2 ) e. ( 0 ..^ ( ( # ` ( W ++ <" Z "> ) ) - 1 ) ) -> ( A. i e. ( 0 ..^ ( ( # ` ( W ++ <" Z "> ) ) - 1 ) ) { ( ( W ++ <" Z "> ) ` i ) , ( ( W ++ <" Z "> ) ` ( i + 1 ) ) } e. E -> { ( ( W ++ <" Z "> ) ` ( N - 2 ) ) , ( ( W ++ <" Z "> ) ` ( ( N - 2 ) + 1 ) ) } e. E ) ) |
18 |
12 17
|
syl |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) -> ( A. i e. ( 0 ..^ ( ( # ` ( W ++ <" Z "> ) ) - 1 ) ) { ( ( W ++ <" Z "> ) ` i ) , ( ( W ++ <" Z "> ) ` ( i + 1 ) ) } e. E -> { ( ( W ++ <" Z "> ) ` ( N - 2 ) ) , ( ( W ++ <" Z "> ) ` ( ( N - 2 ) + 1 ) ) } e. E ) ) |
19 |
|
wrdlenccats1lenm1 |
|- ( W e. Word V -> ( ( # ` ( W ++ <" Z "> ) ) - 1 ) = ( # ` W ) ) |
20 |
19
|
eqcomd |
|- ( W e. Word V -> ( # ` W ) = ( ( # ` ( W ++ <" Z "> ) ) - 1 ) ) |
21 |
20
|
adantr |
|- ( ( W e. Word V /\ Z e. V ) -> ( # ` W ) = ( ( # ` ( W ++ <" Z "> ) ) - 1 ) ) |
22 |
21 8
|
sylan9eq |
|- ( ( ( W e. Word V /\ Z e. V ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) -> ( # ` W ) = ( N - 1 ) ) |
23 |
22
|
ex |
|- ( ( W e. Word V /\ Z e. V ) -> ( ( # ` ( W ++ <" Z "> ) ) = N -> ( # ` W ) = ( N - 1 ) ) ) |
24 |
23
|
3adant3 |
|- ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( ( # ` ( W ++ <" Z "> ) ) = N -> ( # ` W ) = ( N - 1 ) ) ) |
25 |
|
eluzelcn |
|- ( N e. ( ZZ>= ` 2 ) -> N e. CC ) |
26 |
|
1cnd |
|- ( N e. ( ZZ>= ` 2 ) -> 1 e. CC ) |
27 |
25 26 26
|
subsub4d |
|- ( N e. ( ZZ>= ` 2 ) -> ( ( N - 1 ) - 1 ) = ( N - ( 1 + 1 ) ) ) |
28 |
|
1p1e2 |
|- ( 1 + 1 ) = 2 |
29 |
28
|
a1i |
|- ( N e. ( ZZ>= ` 2 ) -> ( 1 + 1 ) = 2 ) |
30 |
29
|
oveq2d |
|- ( N e. ( ZZ>= ` 2 ) -> ( N - ( 1 + 1 ) ) = ( N - 2 ) ) |
31 |
27 30
|
eqtr2d |
|- ( N e. ( ZZ>= ` 2 ) -> ( N - 2 ) = ( ( N - 1 ) - 1 ) ) |
32 |
31
|
3ad2ant3 |
|- ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( N - 2 ) = ( ( N - 1 ) - 1 ) ) |
33 |
|
oveq1 |
|- ( ( # ` W ) = ( N - 1 ) -> ( ( # ` W ) - 1 ) = ( ( N - 1 ) - 1 ) ) |
34 |
33
|
eqcomd |
|- ( ( # ` W ) = ( N - 1 ) -> ( ( N - 1 ) - 1 ) = ( ( # ` W ) - 1 ) ) |
35 |
32 34
|
sylan9eq |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` W ) = ( N - 1 ) ) -> ( N - 2 ) = ( ( # ` W ) - 1 ) ) |
36 |
35
|
ex |
|- ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( ( # ` W ) = ( N - 1 ) -> ( N - 2 ) = ( ( # ` W ) - 1 ) ) ) |
37 |
24 36
|
syld |
|- ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( ( # ` ( W ++ <" Z "> ) ) = N -> ( N - 2 ) = ( ( # ` W ) - 1 ) ) ) |
38 |
37
|
imp |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) -> ( N - 2 ) = ( ( # ` W ) - 1 ) ) |
39 |
38
|
fveq2d |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) -> ( ( W ++ <" Z "> ) ` ( N - 2 ) ) = ( ( W ++ <" Z "> ) ` ( ( # ` W ) - 1 ) ) ) |
40 |
|
simpl1 |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` W ) = ( N - 1 ) ) -> W e. Word V ) |
41 |
|
s1cl |
|- ( Z e. V -> <" Z "> e. Word V ) |
42 |
41
|
3ad2ant2 |
|- ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> <" Z "> e. Word V ) |
43 |
42
|
adantr |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` W ) = ( N - 1 ) ) -> <" Z "> e. Word V ) |
44 |
|
eluz2 |
|- ( N e. ( ZZ>= ` 2 ) <-> ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) ) |
45 |
|
zre |
|- ( N e. ZZ -> N e. RR ) |
46 |
|
1red |
|- ( ( N e. RR /\ 2 <_ N ) -> 1 e. RR ) |
47 |
|
2re |
|- 2 e. RR |
48 |
47
|
a1i |
|- ( ( N e. RR /\ 2 <_ N ) -> 2 e. RR ) |
49 |
|
simpl |
|- ( ( N e. RR /\ 2 <_ N ) -> N e. RR ) |
50 |
|
1lt2 |
|- 1 < 2 |
51 |
50
|
a1i |
|- ( ( N e. RR /\ 2 <_ N ) -> 1 < 2 ) |
52 |
|
simpr |
|- ( ( N e. RR /\ 2 <_ N ) -> 2 <_ N ) |
53 |
46 48 49 51 52
|
ltletrd |
|- ( ( N e. RR /\ 2 <_ N ) -> 1 < N ) |
54 |
|
1red |
|- ( N e. RR -> 1 e. RR ) |
55 |
|
id |
|- ( N e. RR -> N e. RR ) |
56 |
54 55
|
posdifd |
|- ( N e. RR -> ( 1 < N <-> 0 < ( N - 1 ) ) ) |
57 |
56
|
adantr |
|- ( ( N e. RR /\ 2 <_ N ) -> ( 1 < N <-> 0 < ( N - 1 ) ) ) |
58 |
53 57
|
mpbid |
|- ( ( N e. RR /\ 2 <_ N ) -> 0 < ( N - 1 ) ) |
59 |
58
|
ex |
|- ( N e. RR -> ( 2 <_ N -> 0 < ( N - 1 ) ) ) |
60 |
45 59
|
syl |
|- ( N e. ZZ -> ( 2 <_ N -> 0 < ( N - 1 ) ) ) |
61 |
60
|
a1i |
|- ( 2 e. ZZ -> ( N e. ZZ -> ( 2 <_ N -> 0 < ( N - 1 ) ) ) ) |
62 |
61
|
3imp |
|- ( ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) -> 0 < ( N - 1 ) ) |
63 |
44 62
|
sylbi |
|- ( N e. ( ZZ>= ` 2 ) -> 0 < ( N - 1 ) ) |
64 |
63
|
ad2antlr |
|- ( ( ( W e. Word V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` W ) = ( N - 1 ) ) -> 0 < ( N - 1 ) ) |
65 |
|
breq2 |
|- ( ( # ` W ) = ( N - 1 ) -> ( 0 < ( # ` W ) <-> 0 < ( N - 1 ) ) ) |
66 |
65
|
adantl |
|- ( ( ( W e. Word V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` W ) = ( N - 1 ) ) -> ( 0 < ( # ` W ) <-> 0 < ( N - 1 ) ) ) |
67 |
64 66
|
mpbird |
|- ( ( ( W e. Word V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` W ) = ( N - 1 ) ) -> 0 < ( # ` W ) ) |
68 |
|
hashneq0 |
|- ( W e. Word V -> ( 0 < ( # ` W ) <-> W =/= (/) ) ) |
69 |
68
|
adantr |
|- ( ( W e. Word V /\ N e. ( ZZ>= ` 2 ) ) -> ( 0 < ( # ` W ) <-> W =/= (/) ) ) |
70 |
69
|
adantr |
|- ( ( ( W e. Word V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` W ) = ( N - 1 ) ) -> ( 0 < ( # ` W ) <-> W =/= (/) ) ) |
71 |
67 70
|
mpbid |
|- ( ( ( W e. Word V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` W ) = ( N - 1 ) ) -> W =/= (/) ) |
72 |
71
|
3adantl2 |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` W ) = ( N - 1 ) ) -> W =/= (/) ) |
73 |
40 43 72
|
3jca |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` W ) = ( N - 1 ) ) -> ( W e. Word V /\ <" Z "> e. Word V /\ W =/= (/) ) ) |
74 |
73
|
ex |
|- ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( ( # ` W ) = ( N - 1 ) -> ( W e. Word V /\ <" Z "> e. Word V /\ W =/= (/) ) ) ) |
75 |
24 74
|
syld |
|- ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( ( # ` ( W ++ <" Z "> ) ) = N -> ( W e. Word V /\ <" Z "> e. Word V /\ W =/= (/) ) ) ) |
76 |
75
|
imp |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) -> ( W e. Word V /\ <" Z "> e. Word V /\ W =/= (/) ) ) |
77 |
|
ccatval1lsw |
|- ( ( W e. Word V /\ <" Z "> e. Word V /\ W =/= (/) ) -> ( ( W ++ <" Z "> ) ` ( ( # ` W ) - 1 ) ) = ( lastS ` W ) ) |
78 |
76 77
|
syl |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) -> ( ( W ++ <" Z "> ) ` ( ( # ` W ) - 1 ) ) = ( lastS ` W ) ) |
79 |
39 78
|
eqtrd |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) -> ( ( W ++ <" Z "> ) ` ( N - 2 ) ) = ( lastS ` W ) ) |
80 |
|
2m1e1 |
|- ( 2 - 1 ) = 1 |
81 |
80
|
a1i |
|- ( N e. ( ZZ>= ` 2 ) -> ( 2 - 1 ) = 1 ) |
82 |
81
|
eqcomd |
|- ( N e. ( ZZ>= ` 2 ) -> 1 = ( 2 - 1 ) ) |
83 |
82
|
oveq2d |
|- ( N e. ( ZZ>= ` 2 ) -> ( N - 1 ) = ( N - ( 2 - 1 ) ) ) |
84 |
|
2cnd |
|- ( N e. ( ZZ>= ` 2 ) -> 2 e. CC ) |
85 |
25 84 26
|
subsubd |
|- ( N e. ( ZZ>= ` 2 ) -> ( N - ( 2 - 1 ) ) = ( ( N - 2 ) + 1 ) ) |
86 |
83 85
|
eqtr2d |
|- ( N e. ( ZZ>= ` 2 ) -> ( ( N - 2 ) + 1 ) = ( N - 1 ) ) |
87 |
86
|
3ad2ant3 |
|- ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( ( N - 2 ) + 1 ) = ( N - 1 ) ) |
88 |
|
eqeq2 |
|- ( ( # ` W ) = ( N - 1 ) -> ( ( ( N - 2 ) + 1 ) = ( # ` W ) <-> ( ( N - 2 ) + 1 ) = ( N - 1 ) ) ) |
89 |
87 88
|
syl5ibrcom |
|- ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( ( # ` W ) = ( N - 1 ) -> ( ( N - 2 ) + 1 ) = ( # ` W ) ) ) |
90 |
24 89
|
syld |
|- ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( ( # ` ( W ++ <" Z "> ) ) = N -> ( ( N - 2 ) + 1 ) = ( # ` W ) ) ) |
91 |
90
|
imp |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) -> ( ( N - 2 ) + 1 ) = ( # ` W ) ) |
92 |
91
|
fveq2d |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) -> ( ( W ++ <" Z "> ) ` ( ( N - 2 ) + 1 ) ) = ( ( W ++ <" Z "> ) ` ( # ` W ) ) ) |
93 |
|
id |
|- ( ( W e. Word V /\ Z e. V ) -> ( W e. Word V /\ Z e. V ) ) |
94 |
93
|
3adant3 |
|- ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( W e. Word V /\ Z e. V ) ) |
95 |
94
|
adantr |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) -> ( W e. Word V /\ Z e. V ) ) |
96 |
|
ccatws1ls |
|- ( ( W e. Word V /\ Z e. V ) -> ( ( W ++ <" Z "> ) ` ( # ` W ) ) = Z ) |
97 |
95 96
|
syl |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) -> ( ( W ++ <" Z "> ) ` ( # ` W ) ) = Z ) |
98 |
92 97
|
eqtrd |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) -> ( ( W ++ <" Z "> ) ` ( ( N - 2 ) + 1 ) ) = Z ) |
99 |
79 98
|
preq12d |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) -> { ( ( W ++ <" Z "> ) ` ( N - 2 ) ) , ( ( W ++ <" Z "> ) ` ( ( N - 2 ) + 1 ) ) } = { ( lastS ` W ) , Z } ) |
100 |
99
|
eleq1d |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) -> ( { ( ( W ++ <" Z "> ) ` ( N - 2 ) ) , ( ( W ++ <" Z "> ) ` ( ( N - 2 ) + 1 ) ) } e. E <-> { ( lastS ` W ) , Z } e. E ) ) |
101 |
18 100
|
sylibd |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) -> ( A. i e. ( 0 ..^ ( ( # ` ( W ++ <" Z "> ) ) - 1 ) ) { ( ( W ++ <" Z "> ) ` i ) , ( ( W ++ <" Z "> ) ` ( i + 1 ) ) } e. E -> { ( lastS ` W ) , Z } e. E ) ) |
102 |
101
|
ex |
|- ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( ( # ` ( W ++ <" Z "> ) ) = N -> ( A. i e. ( 0 ..^ ( ( # ` ( W ++ <" Z "> ) ) - 1 ) ) { ( ( W ++ <" Z "> ) ` i ) , ( ( W ++ <" Z "> ) ` ( i + 1 ) ) } e. E -> { ( lastS ` W ) , Z } e. E ) ) ) |
103 |
102
|
com13 |
|- ( A. i e. ( 0 ..^ ( ( # ` ( W ++ <" Z "> ) ) - 1 ) ) { ( ( W ++ <" Z "> ) ` i ) , ( ( W ++ <" Z "> ) ` ( i + 1 ) ) } e. E -> ( ( # ` ( W ++ <" Z "> ) ) = N -> ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> { ( lastS ` W ) , Z } e. E ) ) ) |
104 |
103
|
3ad2ant2 |
|- ( ( ( W ++ <" Z "> ) e. Word V /\ A. i e. ( 0 ..^ ( ( # ` ( W ++ <" Z "> ) ) - 1 ) ) { ( ( W ++ <" Z "> ) ` i ) , ( ( W ++ <" Z "> ) ` ( i + 1 ) ) } e. E /\ { ( lastS ` ( W ++ <" Z "> ) ) , ( ( W ++ <" Z "> ) ` 0 ) } e. E ) -> ( ( # ` ( W ++ <" Z "> ) ) = N -> ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> { ( lastS ` W ) , Z } e. E ) ) ) |
105 |
104
|
imp31 |
|- ( ( ( ( ( W ++ <" Z "> ) e. Word V /\ A. i e. ( 0 ..^ ( ( # ` ( W ++ <" Z "> ) ) - 1 ) ) { ( ( W ++ <" Z "> ) ` i ) , ( ( W ++ <" Z "> ) ` ( i + 1 ) ) } e. E /\ { ( lastS ` ( W ++ <" Z "> ) ) , ( ( W ++ <" Z "> ) ` 0 ) } e. E ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) /\ ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) ) -> { ( lastS ` W ) , Z } e. E ) |
106 |
94
|
adantr |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` W ) = ( N - 1 ) ) -> ( W e. Word V /\ Z e. V ) ) |
107 |
|
lswccats1 |
|- ( ( W e. Word V /\ Z e. V ) -> ( lastS ` ( W ++ <" Z "> ) ) = Z ) |
108 |
106 107
|
syl |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` W ) = ( N - 1 ) ) -> ( lastS ` ( W ++ <" Z "> ) ) = Z ) |
109 |
63
|
3ad2ant3 |
|- ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> 0 < ( N - 1 ) ) |
110 |
109
|
adantr |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` W ) = ( N - 1 ) ) -> 0 < ( N - 1 ) ) |
111 |
65
|
adantl |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` W ) = ( N - 1 ) ) -> ( 0 < ( # ` W ) <-> 0 < ( N - 1 ) ) ) |
112 |
110 111
|
mpbird |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` W ) = ( N - 1 ) ) -> 0 < ( # ` W ) ) |
113 |
|
ccatfv0 |
|- ( ( W e. Word V /\ <" Z "> e. Word V /\ 0 < ( # ` W ) ) -> ( ( W ++ <" Z "> ) ` 0 ) = ( W ` 0 ) ) |
114 |
40 43 112 113
|
syl3anc |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` W ) = ( N - 1 ) ) -> ( ( W ++ <" Z "> ) ` 0 ) = ( W ` 0 ) ) |
115 |
108 114
|
preq12d |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( # ` W ) = ( N - 1 ) ) -> { ( lastS ` ( W ++ <" Z "> ) ) , ( ( W ++ <" Z "> ) ` 0 ) } = { Z , ( W ` 0 ) } ) |
116 |
115
|
ex |
|- ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( ( # ` W ) = ( N - 1 ) -> { ( lastS ` ( W ++ <" Z "> ) ) , ( ( W ++ <" Z "> ) ` 0 ) } = { Z , ( W ` 0 ) } ) ) |
117 |
24 116
|
syld |
|- ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( ( # ` ( W ++ <" Z "> ) ) = N -> { ( lastS ` ( W ++ <" Z "> ) ) , ( ( W ++ <" Z "> ) ` 0 ) } = { Z , ( W ` 0 ) } ) ) |
118 |
117
|
impcom |
|- ( ( ( # ` ( W ++ <" Z "> ) ) = N /\ ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) ) -> { ( lastS ` ( W ++ <" Z "> ) ) , ( ( W ++ <" Z "> ) ` 0 ) } = { Z , ( W ` 0 ) } ) |
119 |
118
|
eleq1d |
|- ( ( ( # ` ( W ++ <" Z "> ) ) = N /\ ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) ) -> ( { ( lastS ` ( W ++ <" Z "> ) ) , ( ( W ++ <" Z "> ) ` 0 ) } e. E <-> { Z , ( W ` 0 ) } e. E ) ) |
120 |
119
|
biimpcd |
|- ( { ( lastS ` ( W ++ <" Z "> ) ) , ( ( W ++ <" Z "> ) ` 0 ) } e. E -> ( ( ( # ` ( W ++ <" Z "> ) ) = N /\ ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) ) -> { Z , ( W ` 0 ) } e. E ) ) |
121 |
120
|
3ad2ant3 |
|- ( ( ( W ++ <" Z "> ) e. Word V /\ A. i e. ( 0 ..^ ( ( # ` ( W ++ <" Z "> ) ) - 1 ) ) { ( ( W ++ <" Z "> ) ` i ) , ( ( W ++ <" Z "> ) ` ( i + 1 ) ) } e. E /\ { ( lastS ` ( W ++ <" Z "> ) ) , ( ( W ++ <" Z "> ) ` 0 ) } e. E ) -> ( ( ( # ` ( W ++ <" Z "> ) ) = N /\ ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) ) -> { Z , ( W ` 0 ) } e. E ) ) |
122 |
121
|
impl |
|- ( ( ( ( ( W ++ <" Z "> ) e. Word V /\ A. i e. ( 0 ..^ ( ( # ` ( W ++ <" Z "> ) ) - 1 ) ) { ( ( W ++ <" Z "> ) ` i ) , ( ( W ++ <" Z "> ) ` ( i + 1 ) ) } e. E /\ { ( lastS ` ( W ++ <" Z "> ) ) , ( ( W ++ <" Z "> ) ` 0 ) } e. E ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) /\ ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) ) -> { Z , ( W ` 0 ) } e. E ) |
123 |
105 122
|
jca |
|- ( ( ( ( ( W ++ <" Z "> ) e. Word V /\ A. i e. ( 0 ..^ ( ( # ` ( W ++ <" Z "> ) ) - 1 ) ) { ( ( W ++ <" Z "> ) ` i ) , ( ( W ++ <" Z "> ) ` ( i + 1 ) ) } e. E /\ { ( lastS ` ( W ++ <" Z "> ) ) , ( ( W ++ <" Z "> ) ` 0 ) } e. E ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) /\ ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) ) -> ( { ( lastS ` W ) , Z } e. E /\ { Z , ( W ` 0 ) } e. E ) ) |
124 |
123
|
ex |
|- ( ( ( ( W ++ <" Z "> ) e. Word V /\ A. i e. ( 0 ..^ ( ( # ` ( W ++ <" Z "> ) ) - 1 ) ) { ( ( W ++ <" Z "> ) ` i ) , ( ( W ++ <" Z "> ) ` ( i + 1 ) ) } e. E /\ { ( lastS ` ( W ++ <" Z "> ) ) , ( ( W ++ <" Z "> ) ` 0 ) } e. E ) /\ ( # ` ( W ++ <" Z "> ) ) = N ) -> ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( { ( lastS ` W ) , Z } e. E /\ { Z , ( W ` 0 ) } e. E ) ) ) |
125 |
4 124
|
syl6bi |
|- ( N e. NN -> ( ( W ++ <" Z "> ) e. ( N ClWWalksN G ) -> ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( { ( lastS ` W ) , Z } e. E /\ { Z , ( W ` 0 ) } e. E ) ) ) ) |
126 |
3 125
|
mpcom |
|- ( ( W ++ <" Z "> ) e. ( N ClWWalksN G ) -> ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( { ( lastS ` W ) , Z } e. E /\ { Z , ( W ` 0 ) } e. E ) ) ) |
127 |
126
|
impcom |
|- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( W ++ <" Z "> ) e. ( N ClWWalksN G ) ) -> ( { ( lastS ` W ) , Z } e. E /\ { Z , ( W ` 0 ) } e. E ) ) |