Step |
Hyp |
Ref |
Expression |
1 |
|
wwlksnextprop.x |
|- X = ( ( N + 1 ) WWalksN G ) |
2 |
|
wwlksnextprop.e |
|- E = ( Edg ` G ) |
3 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
4 |
3 2
|
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 |
|
fzonn0p1 |
|- ( N e. NN0 -> N e. ( 0 ..^ ( N + 1 ) ) ) |
6 |
5
|
adantl |
|- ( ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) /\ N e. NN0 ) -> N e. ( 0 ..^ ( N + 1 ) ) ) |
7 |
|
fveq2 |
|- ( i = N -> ( W ` i ) = ( W ` N ) ) |
8 |
|
fvoveq1 |
|- ( i = N -> ( W ` ( i + 1 ) ) = ( W ` ( N + 1 ) ) ) |
9 |
7 8
|
preq12d |
|- ( i = N -> { ( W ` i ) , ( W ` ( i + 1 ) ) } = { ( W ` N ) , ( W ` ( N + 1 ) ) } ) |
10 |
9
|
eleq1d |
|- ( i = N -> ( { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E <-> { ( W ` N ) , ( W ` ( N + 1 ) ) } e. E ) ) |
11 |
10
|
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 ) ) |
12 |
6 11
|
syl |
|- ( ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) /\ N e. NN0 ) -> ( A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E -> { ( W ` N ) , ( W ` ( N + 1 ) ) } e. E ) ) |
13 |
12
|
imp |
|- ( ( ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) /\ N e. NN0 ) /\ A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) -> { ( W ` N ) , ( W ` ( N + 1 ) ) } e. E ) |
14 |
|
simpll |
|- ( ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) /\ N e. NN0 ) -> W e. Word ( Vtx ` G ) ) |
15 |
|
1zzd |
|- ( ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) /\ N e. NN0 ) -> 1 e. ZZ ) |
16 |
|
lencl |
|- ( W e. Word ( Vtx ` G ) -> ( # ` W ) e. NN0 ) |
17 |
16
|
nn0zd |
|- ( W e. Word ( Vtx ` G ) -> ( # ` W ) e. ZZ ) |
18 |
17
|
ad2antrr |
|- ( ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) /\ N e. NN0 ) -> ( # ` W ) e. ZZ ) |
19 |
|
peano2nn0 |
|- ( N e. NN0 -> ( N + 1 ) e. NN0 ) |
20 |
19
|
nn0zd |
|- ( N e. NN0 -> ( N + 1 ) e. ZZ ) |
21 |
20
|
adantl |
|- ( ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) /\ N e. NN0 ) -> ( N + 1 ) e. ZZ ) |
22 |
|
nn0ge0 |
|- ( N e. NN0 -> 0 <_ N ) |
23 |
|
1red |
|- ( N e. NN0 -> 1 e. RR ) |
24 |
|
nn0re |
|- ( N e. NN0 -> N e. RR ) |
25 |
23 24
|
addge02d |
|- ( N e. NN0 -> ( 0 <_ N <-> 1 <_ ( N + 1 ) ) ) |
26 |
22 25
|
mpbid |
|- ( N e. NN0 -> 1 <_ ( N + 1 ) ) |
27 |
26
|
adantl |
|- ( ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) /\ N e. NN0 ) -> 1 <_ ( N + 1 ) ) |
28 |
19
|
nn0red |
|- ( N e. NN0 -> ( N + 1 ) e. RR ) |
29 |
28
|
lep1d |
|- ( N e. NN0 -> ( N + 1 ) <_ ( ( N + 1 ) + 1 ) ) |
30 |
|
breq2 |
|- ( ( # ` W ) = ( ( N + 1 ) + 1 ) -> ( ( N + 1 ) <_ ( # ` W ) <-> ( N + 1 ) <_ ( ( N + 1 ) + 1 ) ) ) |
31 |
29 30
|
syl5ibrcom |
|- ( N e. NN0 -> ( ( # ` W ) = ( ( N + 1 ) + 1 ) -> ( N + 1 ) <_ ( # ` W ) ) ) |
32 |
31
|
a1i |
|- ( ( # ` W ) e. NN0 -> ( N e. NN0 -> ( ( # ` W ) = ( ( N + 1 ) + 1 ) -> ( N + 1 ) <_ ( # ` W ) ) ) ) |
33 |
32
|
com23 |
|- ( ( # ` W ) e. NN0 -> ( ( # ` W ) = ( ( N + 1 ) + 1 ) -> ( N e. NN0 -> ( N + 1 ) <_ ( # ` W ) ) ) ) |
34 |
16 33
|
syl |
|- ( W e. Word ( Vtx ` G ) -> ( ( # ` W ) = ( ( N + 1 ) + 1 ) -> ( N e. NN0 -> ( N + 1 ) <_ ( # ` W ) ) ) ) |
35 |
34
|
imp31 |
|- ( ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) /\ N e. NN0 ) -> ( N + 1 ) <_ ( # ` W ) ) |
36 |
15 18 21 27 35
|
elfzd |
|- ( ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) /\ N e. NN0 ) -> ( N + 1 ) e. ( 1 ... ( # ` W ) ) ) |
37 |
|
pfxfvlsw |
|- ( ( W e. Word ( Vtx ` G ) /\ ( N + 1 ) e. ( 1 ... ( # ` W ) ) ) -> ( lastS ` ( W prefix ( N + 1 ) ) ) = ( W ` ( ( N + 1 ) - 1 ) ) ) |
38 |
14 36 37
|
syl2anc |
|- ( ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) /\ N e. NN0 ) -> ( lastS ` ( W prefix ( N + 1 ) ) ) = ( W ` ( ( N + 1 ) - 1 ) ) ) |
39 |
|
nn0cn |
|- ( N e. NN0 -> N e. CC ) |
40 |
|
1cnd |
|- ( N e. NN0 -> 1 e. CC ) |
41 |
39 40
|
pncand |
|- ( N e. NN0 -> ( ( N + 1 ) - 1 ) = N ) |
42 |
41
|
fveq2d |
|- ( N e. NN0 -> ( W ` ( ( N + 1 ) - 1 ) ) = ( W ` N ) ) |
43 |
42
|
adantl |
|- ( ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) /\ N e. NN0 ) -> ( W ` ( ( N + 1 ) - 1 ) ) = ( W ` N ) ) |
44 |
38 43
|
eqtrd |
|- ( ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) /\ N e. NN0 ) -> ( lastS ` ( W prefix ( N + 1 ) ) ) = ( W ` N ) ) |
45 |
|
lsw |
|- ( W e. Word ( Vtx ` G ) -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) ) |
46 |
45
|
ad2antrr |
|- ( ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) /\ N e. NN0 ) -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) ) |
47 |
|
fvoveq1 |
|- ( ( # ` W ) = ( ( N + 1 ) + 1 ) -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` ( ( ( N + 1 ) + 1 ) - 1 ) ) ) |
48 |
47
|
adantl |
|- ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` ( ( ( N + 1 ) + 1 ) - 1 ) ) ) |
49 |
19
|
nn0cnd |
|- ( N e. NN0 -> ( N + 1 ) e. CC ) |
50 |
49 40
|
pncand |
|- ( N e. NN0 -> ( ( ( N + 1 ) + 1 ) - 1 ) = ( N + 1 ) ) |
51 |
50
|
fveq2d |
|- ( N e. NN0 -> ( W ` ( ( ( N + 1 ) + 1 ) - 1 ) ) = ( W ` ( N + 1 ) ) ) |
52 |
48 51
|
sylan9eq |
|- ( ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) /\ N e. NN0 ) -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` ( N + 1 ) ) ) |
53 |
46 52
|
eqtrd |
|- ( ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) /\ N e. NN0 ) -> ( lastS ` W ) = ( W ` ( N + 1 ) ) ) |
54 |
44 53
|
preq12d |
|- ( ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) /\ N e. NN0 ) -> { ( lastS ` ( W prefix ( N + 1 ) ) ) , ( lastS ` W ) } = { ( W ` N ) , ( W ` ( N + 1 ) ) } ) |
55 |
54
|
eleq1d |
|- ( ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) /\ N e. NN0 ) -> ( { ( lastS ` ( W prefix ( N + 1 ) ) ) , ( lastS ` W ) } e. E <-> { ( W ` N ) , ( W ` ( N + 1 ) ) } e. E ) ) |
56 |
55
|
adantr |
|- ( ( ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) /\ N e. NN0 ) /\ A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) -> ( { ( lastS ` ( W prefix ( N + 1 ) ) ) , ( lastS ` W ) } e. E <-> { ( W ` N ) , ( W ` ( N + 1 ) ) } e. E ) ) |
57 |
13 56
|
mpbird |
|- ( ( ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) /\ N e. NN0 ) /\ A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) -> { ( lastS ` ( W prefix ( N + 1 ) ) ) , ( lastS ` W ) } e. E ) |
58 |
57
|
exp31 |
|- ( ( W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) -> ( N e. NN0 -> ( A. i e. ( 0 ..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E -> { ( lastS ` ( W prefix ( N + 1 ) ) ) , ( lastS ` W ) } e. E ) ) ) |
59 |
58
|
com23 |
|- ( ( 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 -> { ( lastS ` ( W prefix ( N + 1 ) ) ) , ( lastS ` W ) } e. E ) ) ) |
60 |
59
|
3impia |
|- ( ( 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 -> { ( lastS ` ( W prefix ( N + 1 ) ) ) , ( lastS ` W ) } e. E ) ) |
61 |
4 60
|
syl |
|- ( W e. ( ( N + 1 ) WWalksN G ) -> ( N e. NN0 -> { ( lastS ` ( W prefix ( N + 1 ) ) ) , ( lastS ` W ) } e. E ) ) |
62 |
61 1
|
eleq2s |
|- ( W e. X -> ( N e. NN0 -> { ( lastS ` ( W prefix ( N + 1 ) ) ) , ( lastS ` W ) } e. E ) ) |
63 |
62
|
imp |
|- ( ( W e. X /\ N e. NN0 ) -> { ( lastS ` ( W prefix ( N + 1 ) ) ) , ( lastS ` W ) } e. E ) |