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 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) → ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 + 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) ) |
5 |
|
fzonn0p1 |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ( 0 ..^ ( 𝑁 + 1 ) ) ) |
6 |
5
|
adantl |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) ∧ 𝑁 ∈ ℕ0 ) → 𝑁 ∈ ( 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 ↔ { ( 𝑊 ‘ 𝑁 ) , ( 𝑊 ‘ ( 𝑁 + 1 ) ) } ∈ 𝐸 ) ) |
11 |
10
|
rspcv |
⊢ ( 𝑁 ∈ ( 0 ..^ ( 𝑁 + 1 ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 + 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 → { ( 𝑊 ‘ 𝑁 ) , ( 𝑊 ‘ ( 𝑁 + 1 ) ) } ∈ 𝐸 ) ) |
12 |
6 11
|
syl |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) ∧ 𝑁 ∈ ℕ0 ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 + 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 → { ( 𝑊 ‘ 𝑁 ) , ( 𝑊 ‘ ( 𝑁 + 1 ) ) } ∈ 𝐸 ) ) |
13 |
12
|
imp |
⊢ ( ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) ∧ 𝑁 ∈ ℕ0 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 + 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) → { ( 𝑊 ‘ 𝑁 ) , ( 𝑊 ‘ ( 𝑁 + 1 ) ) } ∈ 𝐸 ) |
14 |
|
simpll |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) ∧ 𝑁 ∈ ℕ0 ) → 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ) |
15 |
|
1zzd |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) ∧ 𝑁 ∈ ℕ0 ) → 1 ∈ ℤ ) |
16 |
|
lencl |
⊢ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) → ( ♯ ‘ 𝑊 ) ∈ ℕ0 ) |
17 |
16
|
nn0zd |
⊢ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) → ( ♯ ‘ 𝑊 ) ∈ ℤ ) |
18 |
17
|
ad2antrr |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) ∧ 𝑁 ∈ ℕ0 ) → ( ♯ ‘ 𝑊 ) ∈ ℤ ) |
19 |
|
peano2nn0 |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 + 1 ) ∈ ℕ0 ) |
20 |
19
|
nn0zd |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 + 1 ) ∈ ℤ ) |
21 |
20
|
adantl |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) ∧ 𝑁 ∈ ℕ0 ) → ( 𝑁 + 1 ) ∈ ℤ ) |
22 |
|
nn0ge0 |
⊢ ( 𝑁 ∈ ℕ0 → 0 ≤ 𝑁 ) |
23 |
|
1red |
⊢ ( 𝑁 ∈ ℕ0 → 1 ∈ ℝ ) |
24 |
|
nn0re |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ ) |
25 |
23 24
|
addge02d |
⊢ ( 𝑁 ∈ ℕ0 → ( 0 ≤ 𝑁 ↔ 1 ≤ ( 𝑁 + 1 ) ) ) |
26 |
22 25
|
mpbid |
⊢ ( 𝑁 ∈ ℕ0 → 1 ≤ ( 𝑁 + 1 ) ) |
27 |
26
|
adantl |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) ∧ 𝑁 ∈ ℕ0 ) → 1 ≤ ( 𝑁 + 1 ) ) |
28 |
19
|
nn0red |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 + 1 ) ∈ ℝ ) |
29 |
28
|
lep1d |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 + 1 ) ≤ ( ( 𝑁 + 1 ) + 1 ) ) |
30 |
|
breq2 |
⊢ ( ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) → ( ( 𝑁 + 1 ) ≤ ( ♯ ‘ 𝑊 ) ↔ ( 𝑁 + 1 ) ≤ ( ( 𝑁 + 1 ) + 1 ) ) ) |
31 |
29 30
|
syl5ibrcom |
⊢ ( 𝑁 ∈ ℕ0 → ( ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) → ( 𝑁 + 1 ) ≤ ( ♯ ‘ 𝑊 ) ) ) |
32 |
31
|
a1i |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → ( 𝑁 ∈ ℕ0 → ( ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) → ( 𝑁 + 1 ) ≤ ( ♯ ‘ 𝑊 ) ) ) ) |
33 |
32
|
com23 |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → ( ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) → ( 𝑁 ∈ ℕ0 → ( 𝑁 + 1 ) ≤ ( ♯ ‘ 𝑊 ) ) ) ) |
34 |
16 33
|
syl |
⊢ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) → ( ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) → ( 𝑁 ∈ ℕ0 → ( 𝑁 + 1 ) ≤ ( ♯ ‘ 𝑊 ) ) ) ) |
35 |
34
|
imp31 |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) ∧ 𝑁 ∈ ℕ0 ) → ( 𝑁 + 1 ) ≤ ( ♯ ‘ 𝑊 ) ) |
36 |
15 18 21 27 35
|
elfzd |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) ∧ 𝑁 ∈ ℕ0 ) → ( 𝑁 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) |
37 |
|
pfxfvlsw |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( 𝑁 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( lastS ‘ ( 𝑊 prefix ( 𝑁 + 1 ) ) ) = ( 𝑊 ‘ ( ( 𝑁 + 1 ) − 1 ) ) ) |
38 |
14 36 37
|
syl2anc |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) ∧ 𝑁 ∈ ℕ0 ) → ( lastS ‘ ( 𝑊 prefix ( 𝑁 + 1 ) ) ) = ( 𝑊 ‘ ( ( 𝑁 + 1 ) − 1 ) ) ) |
39 |
|
nn0cn |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ ) |
40 |
|
1cnd |
⊢ ( 𝑁 ∈ ℕ0 → 1 ∈ ℂ ) |
41 |
39 40
|
pncand |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 𝑁 + 1 ) − 1 ) = 𝑁 ) |
42 |
41
|
fveq2d |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑊 ‘ ( ( 𝑁 + 1 ) − 1 ) ) = ( 𝑊 ‘ 𝑁 ) ) |
43 |
42
|
adantl |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) ∧ 𝑁 ∈ ℕ0 ) → ( 𝑊 ‘ ( ( 𝑁 + 1 ) − 1 ) ) = ( 𝑊 ‘ 𝑁 ) ) |
44 |
38 43
|
eqtrd |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) ∧ 𝑁 ∈ ℕ0 ) → ( lastS ‘ ( 𝑊 prefix ( 𝑁 + 1 ) ) ) = ( 𝑊 ‘ 𝑁 ) ) |
45 |
|
lsw |
⊢ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ) |
46 |
45
|
ad2antrr |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) ∧ 𝑁 ∈ ℕ0 ) → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ) |
47 |
|
fvoveq1 |
⊢ ( ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) → ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) = ( 𝑊 ‘ ( ( ( 𝑁 + 1 ) + 1 ) − 1 ) ) ) |
48 |
47
|
adantl |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) → ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) = ( 𝑊 ‘ ( ( ( 𝑁 + 1 ) + 1 ) − 1 ) ) ) |
49 |
19
|
nn0cnd |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 + 1 ) ∈ ℂ ) |
50 |
49 40
|
pncand |
⊢ ( 𝑁 ∈ ℕ0 → ( ( ( 𝑁 + 1 ) + 1 ) − 1 ) = ( 𝑁 + 1 ) ) |
51 |
50
|
fveq2d |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑊 ‘ ( ( ( 𝑁 + 1 ) + 1 ) − 1 ) ) = ( 𝑊 ‘ ( 𝑁 + 1 ) ) ) |
52 |
48 51
|
sylan9eq |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) ∧ 𝑁 ∈ ℕ0 ) → ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) = ( 𝑊 ‘ ( 𝑁 + 1 ) ) ) |
53 |
46 52
|
eqtrd |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) ∧ 𝑁 ∈ ℕ0 ) → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ ( 𝑁 + 1 ) ) ) |
54 |
44 53
|
preq12d |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) ∧ 𝑁 ∈ ℕ0 ) → { ( lastS ‘ ( 𝑊 prefix ( 𝑁 + 1 ) ) ) , ( lastS ‘ 𝑊 ) } = { ( 𝑊 ‘ 𝑁 ) , ( 𝑊 ‘ ( 𝑁 + 1 ) ) } ) |
55 |
54
|
eleq1d |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) ∧ 𝑁 ∈ ℕ0 ) → ( { ( lastS ‘ ( 𝑊 prefix ( 𝑁 + 1 ) ) ) , ( lastS ‘ 𝑊 ) } ∈ 𝐸 ↔ { ( 𝑊 ‘ 𝑁 ) , ( 𝑊 ‘ ( 𝑁 + 1 ) ) } ∈ 𝐸 ) ) |
56 |
55
|
adantr |
⊢ ( ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) ∧ 𝑁 ∈ ℕ0 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 + 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) → ( { ( lastS ‘ ( 𝑊 prefix ( 𝑁 + 1 ) ) ) , ( lastS ‘ 𝑊 ) } ∈ 𝐸 ↔ { ( 𝑊 ‘ 𝑁 ) , ( 𝑊 ‘ ( 𝑁 + 1 ) ) } ∈ 𝐸 ) ) |
57 |
13 56
|
mpbird |
⊢ ( ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) ∧ 𝑁 ∈ ℕ0 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 + 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) → { ( lastS ‘ ( 𝑊 prefix ( 𝑁 + 1 ) ) ) , ( lastS ‘ 𝑊 ) } ∈ 𝐸 ) |
58 |
57
|
exp31 |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) → ( 𝑁 ∈ ℕ0 → ( ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 + 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 → { ( lastS ‘ ( 𝑊 prefix ( 𝑁 + 1 ) ) ) , ( lastS ‘ 𝑊 ) } ∈ 𝐸 ) ) ) |
59 |
58
|
com23 |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 + 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 → ( 𝑁 ∈ ℕ0 → { ( lastS ‘ ( 𝑊 prefix ( 𝑁 + 1 ) ) ) , ( lastS ‘ 𝑊 ) } ∈ 𝐸 ) ) ) |
60 |
59
|
3impia |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 + 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) → ( 𝑁 ∈ ℕ0 → { ( lastS ‘ ( 𝑊 prefix ( 𝑁 + 1 ) ) ) , ( lastS ‘ 𝑊 ) } ∈ 𝐸 ) ) |
61 |
4 60
|
syl |
⊢ ( 𝑊 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) → ( 𝑁 ∈ ℕ0 → { ( lastS ‘ ( 𝑊 prefix ( 𝑁 + 1 ) ) ) , ( lastS ‘ 𝑊 ) } ∈ 𝐸 ) ) |
62 |
61 1
|
eleq2s |
⊢ ( 𝑊 ∈ 𝑋 → ( 𝑁 ∈ ℕ0 → { ( lastS ‘ ( 𝑊 prefix ( 𝑁 + 1 ) ) ) , ( lastS ‘ 𝑊 ) } ∈ 𝐸 ) ) |
63 |
62
|
imp |
⊢ ( ( 𝑊 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) → { ( lastS ‘ ( 𝑊 prefix ( 𝑁 + 1 ) ) ) , ( lastS ‘ 𝑊 ) } ∈ 𝐸 ) |