Step |
Hyp |
Ref |
Expression |
1 |
|
wwlksnredwwlkn.e |
|- E = ( Edg ` G ) |
2 |
|
eqidd |
|- ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> ( W prefix ( N + 1 ) ) = ( W prefix ( N + 1 ) ) ) |
3 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
4 |
3 1
|
wwlknp |
|- ( W e. ( ( N + 1 ) WWalksN G ) -> ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) |
5 |
|
simprl |
|- ( ( N e. NN0 /\ ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) ) -> W e. Word ( Vtx ` G ) ) |
6 |
|
peano2nn0 |
|- ( N e. NN0 -> ( N + 1 ) e. NN0 ) |
7 |
|
peano2nn0 |
|- ( ( N + 1 ) e. NN0 -> ( ( N + 1 ) + 1 ) e. NN0 ) |
8 |
6 7
|
syl |
|- ( N e. NN0 -> ( ( N + 1 ) + 1 ) e. NN0 ) |
9 |
|
id |
|- ( N e. NN0 -> N e. NN0 ) |
10 |
|
nn0p1nn |
|- ( ( N + 1 ) e. NN0 -> ( ( N + 1 ) + 1 ) e. NN ) |
11 |
6 10
|
syl |
|- ( N e. NN0 -> ( ( N + 1 ) + 1 ) e. NN ) |
12 |
|
nn0re |
|- ( N e. NN0 -> N e. RR ) |
13 |
|
id |
|- ( N e. RR -> N e. RR ) |
14 |
|
peano2re |
|- ( N e. RR -> ( N + 1 ) e. RR ) |
15 |
|
peano2re |
|- ( ( N + 1 ) e. RR -> ( ( N + 1 ) + 1 ) e. RR ) |
16 |
14 15
|
syl |
|- ( N e. RR -> ( ( N + 1 ) + 1 ) e. RR ) |
17 |
13 14 16
|
3jca |
|- ( N e. RR -> ( N e. RR /\ ( N + 1 ) e. RR /\ ( ( N + 1 ) + 1 ) e. RR ) ) |
18 |
12 17
|
syl |
|- ( N e. NN0 -> ( N e. RR /\ ( N + 1 ) e. RR /\ ( ( N + 1 ) + 1 ) e. RR ) ) |
19 |
12
|
ltp1d |
|- ( N e. NN0 -> N < ( N + 1 ) ) |
20 |
|
nn0re |
|- ( ( N + 1 ) e. NN0 -> ( N + 1 ) e. RR ) |
21 |
6 20
|
syl |
|- ( N e. NN0 -> ( N + 1 ) e. RR ) |
22 |
21
|
ltp1d |
|- ( N e. NN0 -> ( N + 1 ) < ( ( N + 1 ) + 1 ) ) |
23 |
|
lttr |
|- ( ( N e. RR /\ ( N + 1 ) e. RR /\ ( ( N + 1 ) + 1 ) e. RR ) -> ( ( N < ( N + 1 ) /\ ( N + 1 ) < ( ( N + 1 ) + 1 ) ) -> N < ( ( N + 1 ) + 1 ) ) ) |
24 |
23
|
imp |
|- ( ( ( N e. RR /\ ( N + 1 ) e. RR /\ ( ( N + 1 ) + 1 ) e. RR ) /\ ( N < ( N + 1 ) /\ ( N + 1 ) < ( ( N + 1 ) + 1 ) ) ) -> N < ( ( N + 1 ) + 1 ) ) |
25 |
18 19 22 24
|
syl12anc |
|- ( N e. NN0 -> N < ( ( N + 1 ) + 1 ) ) |
26 |
|
elfzo0 |
|- ( N e. ( 0 ..^ ( ( N + 1 ) + 1 ) ) <-> ( N e. NN0 /\ ( ( N + 1 ) + 1 ) e. NN /\ N < ( ( N + 1 ) + 1 ) ) ) |
27 |
9 11 25 26
|
syl3anbrc |
|- ( N e. NN0 -> N e. ( 0 ..^ ( ( N + 1 ) + 1 ) ) ) |
28 |
|
fz0add1fz1 |
|- ( ( ( ( N + 1 ) + 1 ) e. NN0 /\ N e. ( 0 ..^ ( ( N + 1 ) + 1 ) ) ) -> ( N + 1 ) e. ( 1 ... ( ( N + 1 ) + 1 ) ) ) |
29 |
8 27 28
|
syl2anc |
|- ( N e. NN0 -> ( N + 1 ) e. ( 1 ... ( ( N + 1 ) + 1 ) ) ) |
30 |
29
|
adantr |
|- ( ( N e. NN0 /\ ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) ) -> ( N + 1 ) e. ( 1 ... ( ( N + 1 ) + 1 ) ) ) |
31 |
|
oveq2 |
|- ( ( # ` W ) = ( ( N + 1 ) + 1 ) -> ( 1 ... ( # ` W ) ) = ( 1 ... ( ( N + 1 ) + 1 ) ) ) |
32 |
31
|
eleq2d |
|- ( ( # ` W ) = ( ( N + 1 ) + 1 ) -> ( ( N + 1 ) e. ( 1 ... ( # ` W ) ) <-> ( N + 1 ) e. ( 1 ... ( ( N + 1 ) + 1 ) ) ) ) |
33 |
32
|
adantl |
|- ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) -> ( ( N + 1 ) e. ( 1 ... ( # ` W ) ) <-> ( N + 1 ) e. ( 1 ... ( ( N + 1 ) + 1 ) ) ) ) |
34 |
33
|
adantl |
|- ( ( N e. NN0 /\ ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) ) -> ( ( N + 1 ) e. ( 1 ... ( # ` W ) ) <-> ( N + 1 ) e. ( 1 ... ( ( N + 1 ) + 1 ) ) ) ) |
35 |
30 34
|
mpbird |
|- ( ( N e. NN0 /\ ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) ) -> ( N + 1 ) e. ( 1 ... ( # ` W ) ) ) |
36 |
5 35
|
jca |
|- ( ( N e. NN0 /\ ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) ) -> ( W e. Word ( Vtx ` G ) /\ ( N + 1 ) e. ( 1 ... ( # ` W ) ) ) ) |
37 |
36
|
3adantr3 |
|- ( ( N e. NN0 /\ ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> ( W e. Word ( Vtx ` G ) /\ ( N + 1 ) e. ( 1 ... ( # ` W ) ) ) ) |
38 |
|
pfxfvlsw |
|- ( ( W e. Word ( Vtx ` G ) /\ ( N + 1 ) e. ( 1 ... ( # ` W ) ) ) -> ( lastS ` ( W prefix ( N + 1 ) ) ) = ( W ` ( ( N + 1 ) - 1 ) ) ) |
39 |
37 38
|
syl |
|- ( ( N e. NN0 /\ ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> ( lastS ` ( W prefix ( N + 1 ) ) ) = ( W ` ( ( N + 1 ) - 1 ) ) ) |
40 |
|
lsw |
|- ( W e. Word ( Vtx ` G ) -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) ) |
41 |
40
|
3ad2ant1 |
|- ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) ) |
42 |
41
|
adantl |
|- ( ( N e. NN0 /\ ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) ) |
43 |
39 42
|
preq12d |
|- ( ( N e. NN0 /\ ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> { ( lastS ` ( W prefix ( N + 1 ) ) ) , ( lastS ` W ) } = { ( W ` ( ( N + 1 ) - 1 ) ) , ( W ` ( ( # ` W ) - 1 ) ) } ) |
44 |
|
oveq1 |
|- ( ( # ` W ) = ( ( N + 1 ) + 1 ) -> ( ( # ` W ) - 1 ) = ( ( ( N + 1 ) + 1 ) - 1 ) ) |
45 |
44
|
3ad2ant2 |
|- ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) -> ( ( # ` W ) - 1 ) = ( ( ( N + 1 ) + 1 ) - 1 ) ) |
46 |
45
|
adantl |
|- ( ( N e. NN0 /\ ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> ( ( # ` W ) - 1 ) = ( ( ( N + 1 ) + 1 ) - 1 ) ) |
47 |
46
|
fveq2d |
|- ( ( N e. NN0 /\ ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` ( ( ( N + 1 ) + 1 ) - 1 ) ) ) |
48 |
47
|
preq2d |
|- ( ( N e. NN0 /\ ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> { ( W ` ( ( N + 1 ) - 1 ) ) , ( W ` ( ( # ` W ) - 1 ) ) } = { ( W ` ( ( N + 1 ) - 1 ) ) , ( W ` ( ( ( N + 1 ) + 1 ) - 1 ) ) } ) |
49 |
|
nn0cn |
|- ( N e. NN0 -> N e. CC ) |
50 |
|
1cnd |
|- ( N e. NN0 -> 1 e. CC ) |
51 |
49 50
|
pncand |
|- ( N e. NN0 -> ( ( N + 1 ) - 1 ) = N ) |
52 |
51
|
fveq2d |
|- ( N e. NN0 -> ( W ` ( ( N + 1 ) - 1 ) ) = ( W ` N ) ) |
53 |
6
|
nn0cnd |
|- ( N e. NN0 -> ( N + 1 ) e. CC ) |
54 |
53 50
|
pncand |
|- ( N e. NN0 -> ( ( ( N + 1 ) + 1 ) - 1 ) = ( N + 1 ) ) |
55 |
54
|
fveq2d |
|- ( N e. NN0 -> ( W ` ( ( ( N + 1 ) + 1 ) - 1 ) ) = ( W ` ( N + 1 ) ) ) |
56 |
52 55
|
preq12d |
|- ( N e. NN0 -> { ( W ` ( ( N + 1 ) - 1 ) ) , ( W ` ( ( ( N + 1 ) + 1 ) - 1 ) ) } = { ( W ` N ) , ( W ` ( N + 1 ) ) } ) |
57 |
56
|
adantr |
|- ( ( N e. NN0 /\ ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> { ( W ` ( ( N + 1 ) - 1 ) ) , ( W ` ( ( ( N + 1 ) + 1 ) - 1 ) ) } = { ( W ` N ) , ( W ` ( N + 1 ) ) } ) |
58 |
48 57
|
eqtrd |
|- ( ( N e. NN0 /\ ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> { ( W ` ( ( N + 1 ) - 1 ) ) , ( W ` ( ( # ` W ) - 1 ) ) } = { ( W ` N ) , ( W ` ( N + 1 ) ) } ) |
59 |
|
fveq2 |
|- ( i = N -> ( W ` i ) = ( W ` N ) ) |
60 |
|
fvoveq1 |
|- ( i = N -> ( W ` ( i + 1 ) ) = ( W ` ( N + 1 ) ) ) |
61 |
59 60
|
preq12d |
|- ( i = N -> { ( W ` i ) , ( W ` ( i + 1 ) ) } = { ( W ` N ) , ( W ` ( N + 1 ) ) } ) |
62 |
61
|
eleq1d |
|- ( i = N -> ( { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E <-> { ( W ` N ) , ( W ` ( N + 1 ) ) } e. E ) ) |
63 |
62
|
rspcv |
|- ( N e. ( 0 ..^ ( N + 1 ) ) -> ( A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E -> { ( W ` N ) , ( W ` ( N + 1 ) ) } e. E ) ) |
64 |
|
fzonn0p1 |
|- ( N e. NN0 -> N e. ( 0 ..^ ( N + 1 ) ) ) |
65 |
63 64
|
syl11 |
|- ( A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E -> ( N e. NN0 -> { ( W ` N ) , ( W ` ( N + 1 ) ) } e. E ) ) |
66 |
65
|
3ad2ant3 |
|- ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) -> ( N e. NN0 -> { ( W ` N ) , ( W ` ( N + 1 ) ) } e. E ) ) |
67 |
66
|
impcom |
|- ( ( N e. NN0 /\ ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> { ( W ` N ) , ( W ` ( N + 1 ) ) } e. E ) |
68 |
58 67
|
eqeltrd |
|- ( ( N e. NN0 /\ ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> { ( W ` ( ( N + 1 ) - 1 ) ) , ( W ` ( ( # ` W ) - 1 ) ) } e. E ) |
69 |
43 68
|
eqeltrd |
|- ( ( N e. NN0 /\ ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> { ( lastS ` ( W prefix ( N + 1 ) ) ) , ( lastS ` W ) } e. E ) |
70 |
4 69
|
sylan2 |
|- ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> { ( lastS ` ( W prefix ( N + 1 ) ) ) , ( lastS ` W ) } e. E ) |
71 |
|
wwlksnred |
|- ( N e. NN0 -> ( W e. ( ( N + 1 ) WWalksN G ) -> ( W prefix ( N + 1 ) ) e. ( N WWalksN G ) ) ) |
72 |
71
|
imp |
|- ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> ( W prefix ( N + 1 ) ) e. ( N WWalksN G ) ) |
73 |
|
eqeq2 |
|- ( y = ( W prefix ( N + 1 ) ) -> ( ( W prefix ( N + 1 ) ) = y <-> ( W prefix ( N + 1 ) ) = ( W prefix ( N + 1 ) ) ) ) |
74 |
|
fveq2 |
|- ( y = ( W prefix ( N + 1 ) ) -> ( lastS ` y ) = ( lastS ` ( W prefix ( N + 1 ) ) ) ) |
75 |
74
|
preq1d |
|- ( y = ( W prefix ( N + 1 ) ) -> { ( lastS ` y ) , ( lastS ` W ) } = { ( lastS ` ( W prefix ( N + 1 ) ) ) , ( lastS ` W ) } ) |
76 |
75
|
eleq1d |
|- ( y = ( W prefix ( N + 1 ) ) -> ( { ( lastS ` y ) , ( lastS ` W ) } e. E <-> { ( lastS ` ( W prefix ( N + 1 ) ) ) , ( lastS ` W ) } e. E ) ) |
77 |
73 76
|
anbi12d |
|- ( y = ( W prefix ( N + 1 ) ) -> ( ( ( W prefix ( N + 1 ) ) = y /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) <-> ( ( W prefix ( N + 1 ) ) = ( W prefix ( N + 1 ) ) /\ { ( lastS ` ( W prefix ( N + 1 ) ) ) , ( lastS ` W ) } e. E ) ) ) |
78 |
77
|
adantl |
|- ( ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) /\ y = ( W prefix ( N + 1 ) ) ) -> ( ( ( W prefix ( N + 1 ) ) = y /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) <-> ( ( W prefix ( N + 1 ) ) = ( W prefix ( N + 1 ) ) /\ { ( lastS ` ( W prefix ( N + 1 ) ) ) , ( lastS ` W ) } e. E ) ) ) |
79 |
72 78
|
rspcedv |
|- ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> ( ( ( W prefix ( N + 1 ) ) = ( W prefix ( N + 1 ) ) /\ { ( lastS ` ( W prefix ( N + 1 ) ) ) , ( lastS ` W ) } e. E ) -> E. y e. ( N WWalksN G ) ( ( W prefix ( N + 1 ) ) = y /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) ) ) |
80 |
2 70 79
|
mp2and |
|- ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> E. y e. ( N WWalksN G ) ( ( W prefix ( N + 1 ) ) = y /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) ) |
81 |
80
|
ex |
|- ( N e. NN0 -> ( W e. ( ( N + 1 ) WWalksN G ) -> E. y e. ( N WWalksN G ) ( ( W prefix ( N + 1 ) ) = y /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) ) ) |