Step |
Hyp |
Ref |
Expression |
1 |
|
wwlksnext.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
wwlksnext.e |
⊢ 𝐸 = ( Edg ‘ 𝐺 ) |
3 |
1 2
|
wwlknp |
⊢ ( 𝑊 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) → ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 + 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) ) |
4 |
|
wwlksnred |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑊 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) → ( 𝑊 prefix ( 𝑁 + 1 ) ) ∈ ( 𝑁 WWalksN 𝐺 ) ) ) |
5 |
4
|
ad2antrr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) ) → ( 𝑊 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) → ( 𝑊 prefix ( 𝑁 + 1 ) ) ∈ ( 𝑁 WWalksN 𝐺 ) ) ) |
6 |
|
fveqeq2 |
⊢ ( 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) → ( ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ↔ ( ♯ ‘ ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ) = ( ( 𝑁 + 1 ) + 1 ) ) ) |
7 |
6
|
3ad2ant2 |
⊢ ( ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) → ( ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ↔ ( ♯ ‘ ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ) = ( ( 𝑁 + 1 ) + 1 ) ) ) |
8 |
7
|
adantl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) ) → ( ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ↔ ( ♯ ‘ ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ) = ( ( 𝑁 + 1 ) + 1 ) ) ) |
9 |
|
s1cl |
⊢ ( 𝑆 ∈ 𝑉 → 〈“ 𝑆 ”〉 ∈ Word 𝑉 ) |
10 |
9
|
adantl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) → 〈“ 𝑆 ”〉 ∈ Word 𝑉 ) |
11 |
10
|
anim1ci |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ 𝑇 ∈ Word 𝑉 ) → ( 𝑇 ∈ Word 𝑉 ∧ 〈“ 𝑆 ”〉 ∈ Word 𝑉 ) ) |
12 |
|
ccatlen |
⊢ ( ( 𝑇 ∈ Word 𝑉 ∧ 〈“ 𝑆 ”〉 ∈ Word 𝑉 ) → ( ♯ ‘ ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ) = ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 〈“ 𝑆 ”〉 ) ) ) |
13 |
11 12
|
syl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ 𝑇 ∈ Word 𝑉 ) → ( ♯ ‘ ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ) = ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 〈“ 𝑆 ”〉 ) ) ) |
14 |
13
|
eqeq1d |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ 𝑇 ∈ Word 𝑉 ) → ( ( ♯ ‘ ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ) = ( ( 𝑁 + 1 ) + 1 ) ↔ ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 〈“ 𝑆 ”〉 ) ) = ( ( 𝑁 + 1 ) + 1 ) ) ) |
15 |
|
s1len |
⊢ ( ♯ ‘ 〈“ 𝑆 ”〉 ) = 1 |
16 |
15
|
a1i |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ 𝑇 ∈ Word 𝑉 ) → ( ♯ ‘ 〈“ 𝑆 ”〉 ) = 1 ) |
17 |
16
|
oveq2d |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ 𝑇 ∈ Word 𝑉 ) → ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 〈“ 𝑆 ”〉 ) ) = ( ( ♯ ‘ 𝑇 ) + 1 ) ) |
18 |
17
|
eqeq1d |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ 𝑇 ∈ Word 𝑉 ) → ( ( ( ♯ ‘ 𝑇 ) + ( ♯ ‘ 〈“ 𝑆 ”〉 ) ) = ( ( 𝑁 + 1 ) + 1 ) ↔ ( ( ♯ ‘ 𝑇 ) + 1 ) = ( ( 𝑁 + 1 ) + 1 ) ) ) |
19 |
|
lencl |
⊢ ( 𝑇 ∈ Word 𝑉 → ( ♯ ‘ 𝑇 ) ∈ ℕ0 ) |
20 |
19
|
nn0cnd |
⊢ ( 𝑇 ∈ Word 𝑉 → ( ♯ ‘ 𝑇 ) ∈ ℂ ) |
21 |
20
|
adantl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ 𝑇 ∈ Word 𝑉 ) → ( ♯ ‘ 𝑇 ) ∈ ℂ ) |
22 |
|
peano2nn0 |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 + 1 ) ∈ ℕ0 ) |
23 |
22
|
nn0cnd |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 + 1 ) ∈ ℂ ) |
24 |
23
|
ad2antrr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ 𝑇 ∈ Word 𝑉 ) → ( 𝑁 + 1 ) ∈ ℂ ) |
25 |
|
1cnd |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ 𝑇 ∈ Word 𝑉 ) → 1 ∈ ℂ ) |
26 |
21 24 25
|
addcan2d |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ 𝑇 ∈ Word 𝑉 ) → ( ( ( ♯ ‘ 𝑇 ) + 1 ) = ( ( 𝑁 + 1 ) + 1 ) ↔ ( ♯ ‘ 𝑇 ) = ( 𝑁 + 1 ) ) ) |
27 |
14 18 26
|
3bitrd |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ 𝑇 ∈ Word 𝑉 ) → ( ( ♯ ‘ ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ) = ( ( 𝑁 + 1 ) + 1 ) ↔ ( ♯ ‘ 𝑇 ) = ( 𝑁 + 1 ) ) ) |
28 |
|
oveq2 |
⊢ ( ( 𝑁 + 1 ) = ( ♯ ‘ 𝑇 ) → ( ( 𝑇 ++ 〈“ 𝑆 ”〉 ) prefix ( 𝑁 + 1 ) ) = ( ( 𝑇 ++ 〈“ 𝑆 ”〉 ) prefix ( ♯ ‘ 𝑇 ) ) ) |
29 |
28
|
eqcoms |
⊢ ( ( ♯ ‘ 𝑇 ) = ( 𝑁 + 1 ) → ( ( 𝑇 ++ 〈“ 𝑆 ”〉 ) prefix ( 𝑁 + 1 ) ) = ( ( 𝑇 ++ 〈“ 𝑆 ”〉 ) prefix ( ♯ ‘ 𝑇 ) ) ) |
30 |
|
pfxccat1 |
⊢ ( ( 𝑇 ∈ Word 𝑉 ∧ 〈“ 𝑆 ”〉 ∈ Word 𝑉 ) → ( ( 𝑇 ++ 〈“ 𝑆 ”〉 ) prefix ( ♯ ‘ 𝑇 ) ) = 𝑇 ) |
31 |
11 30
|
syl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ 𝑇 ∈ Word 𝑉 ) → ( ( 𝑇 ++ 〈“ 𝑆 ”〉 ) prefix ( ♯ ‘ 𝑇 ) ) = 𝑇 ) |
32 |
29 31
|
sylan9eqr |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ 𝑇 ∈ Word 𝑉 ) ∧ ( ♯ ‘ 𝑇 ) = ( 𝑁 + 1 ) ) → ( ( 𝑇 ++ 〈“ 𝑆 ”〉 ) prefix ( 𝑁 + 1 ) ) = 𝑇 ) |
33 |
32
|
ex |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ 𝑇 ∈ Word 𝑉 ) → ( ( ♯ ‘ 𝑇 ) = ( 𝑁 + 1 ) → ( ( 𝑇 ++ 〈“ 𝑆 ”〉 ) prefix ( 𝑁 + 1 ) ) = 𝑇 ) ) |
34 |
27 33
|
sylbid |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ 𝑇 ∈ Word 𝑉 ) → ( ( ♯ ‘ ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ) = ( ( 𝑁 + 1 ) + 1 ) → ( ( 𝑇 ++ 〈“ 𝑆 ”〉 ) prefix ( 𝑁 + 1 ) ) = 𝑇 ) ) |
35 |
34
|
3ad2antr1 |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) ) → ( ( ♯ ‘ ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ) = ( ( 𝑁 + 1 ) + 1 ) → ( ( 𝑇 ++ 〈“ 𝑆 ”〉 ) prefix ( 𝑁 + 1 ) ) = 𝑇 ) ) |
36 |
8 35
|
sylbid |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) ) → ( ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) → ( ( 𝑇 ++ 〈“ 𝑆 ”〉 ) prefix ( 𝑁 + 1 ) ) = 𝑇 ) ) |
37 |
36
|
imp |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) → ( ( 𝑇 ++ 〈“ 𝑆 ”〉 ) prefix ( 𝑁 + 1 ) ) = 𝑇 ) |
38 |
|
oveq1 |
⊢ ( 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) → ( 𝑊 prefix ( 𝑁 + 1 ) ) = ( ( 𝑇 ++ 〈“ 𝑆 ”〉 ) prefix ( 𝑁 + 1 ) ) ) |
39 |
38
|
eqeq1d |
⊢ ( 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) → ( ( 𝑊 prefix ( 𝑁 + 1 ) ) = 𝑇 ↔ ( ( 𝑇 ++ 〈“ 𝑆 ”〉 ) prefix ( 𝑁 + 1 ) ) = 𝑇 ) ) |
40 |
39
|
3ad2ant2 |
⊢ ( ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) → ( ( 𝑊 prefix ( 𝑁 + 1 ) ) = 𝑇 ↔ ( ( 𝑇 ++ 〈“ 𝑆 ”〉 ) prefix ( 𝑁 + 1 ) ) = 𝑇 ) ) |
41 |
40
|
ad2antlr |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) → ( ( 𝑊 prefix ( 𝑁 + 1 ) ) = 𝑇 ↔ ( ( 𝑇 ++ 〈“ 𝑆 ”〉 ) prefix ( 𝑁 + 1 ) ) = 𝑇 ) ) |
42 |
37 41
|
mpbird |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) → ( 𝑊 prefix ( 𝑁 + 1 ) ) = 𝑇 ) |
43 |
42
|
eleq1d |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) → ( ( 𝑊 prefix ( 𝑁 + 1 ) ) ∈ ( 𝑁 WWalksN 𝐺 ) ↔ 𝑇 ∈ ( 𝑁 WWalksN 𝐺 ) ) ) |
44 |
43
|
biimpd |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ) → ( ( 𝑊 prefix ( 𝑁 + 1 ) ) ∈ ( 𝑁 WWalksN 𝐺 ) → 𝑇 ∈ ( 𝑁 WWalksN 𝐺 ) ) ) |
45 |
44
|
ex |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) ) → ( ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) → ( ( 𝑊 prefix ( 𝑁 + 1 ) ) ∈ ( 𝑁 WWalksN 𝐺 ) → 𝑇 ∈ ( 𝑁 WWalksN 𝐺 ) ) ) ) |
46 |
45
|
com23 |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) ) → ( ( 𝑊 prefix ( 𝑁 + 1 ) ) ∈ ( 𝑁 WWalksN 𝐺 ) → ( ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) → 𝑇 ∈ ( 𝑁 WWalksN 𝐺 ) ) ) ) |
47 |
5 46
|
syld |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) ) → ( 𝑊 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) → ( ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) → 𝑇 ∈ ( 𝑁 WWalksN 𝐺 ) ) ) ) |
48 |
47
|
com13 |
⊢ ( ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) → ( 𝑊 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) → ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) ) → 𝑇 ∈ ( 𝑁 WWalksN 𝐺 ) ) ) ) |
49 |
48
|
3ad2ant2 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = ( ( 𝑁 + 1 ) + 1 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 + 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) → ( 𝑊 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) → ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) ) → 𝑇 ∈ ( 𝑁 WWalksN 𝐺 ) ) ) ) |
50 |
3 49
|
mpcom |
⊢ ( 𝑊 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) → ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) ) → 𝑇 ∈ ( 𝑁 WWalksN 𝐺 ) ) ) |
51 |
50
|
com12 |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) ) → ( 𝑊 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) → 𝑇 ∈ ( 𝑁 WWalksN 𝐺 ) ) ) |
52 |
1 2
|
wwlksnext |
⊢ ( ( 𝑇 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ 𝑆 ∈ 𝑉 ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) → ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ) |
53 |
|
eleq1 |
⊢ ( 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) → ( 𝑊 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ↔ ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ) ) |
54 |
52 53
|
syl5ibrcom |
⊢ ( ( 𝑇 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ 𝑆 ∈ 𝑉 ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) → ( 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) → 𝑊 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ) ) |
55 |
54
|
3exp |
⊢ ( 𝑇 ∈ ( 𝑁 WWalksN 𝐺 ) → ( 𝑆 ∈ 𝑉 → ( { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 → ( 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) → 𝑊 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ) ) ) ) |
56 |
55
|
com23 |
⊢ ( 𝑇 ∈ ( 𝑁 WWalksN 𝐺 ) → ( { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 → ( 𝑆 ∈ 𝑉 → ( 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) → 𝑊 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ) ) ) ) |
57 |
56
|
com14 |
⊢ ( 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) → ( { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 → ( 𝑆 ∈ 𝑉 → ( 𝑇 ∈ ( 𝑁 WWalksN 𝐺 ) → 𝑊 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ) ) ) ) |
58 |
57
|
imp |
⊢ ( ( 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) → ( 𝑆 ∈ 𝑉 → ( 𝑇 ∈ ( 𝑁 WWalksN 𝐺 ) → 𝑊 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ) ) ) |
59 |
58
|
3adant1 |
⊢ ( ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) → ( 𝑆 ∈ 𝑉 → ( 𝑇 ∈ ( 𝑁 WWalksN 𝐺 ) → 𝑊 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ) ) ) |
60 |
59
|
com12 |
⊢ ( 𝑆 ∈ 𝑉 → ( ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) → ( 𝑇 ∈ ( 𝑁 WWalksN 𝐺 ) → 𝑊 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ) ) ) |
61 |
60
|
adantl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) → ( ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) → ( 𝑇 ∈ ( 𝑁 WWalksN 𝐺 ) → 𝑊 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ) ) ) |
62 |
61
|
imp |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) ) → ( 𝑇 ∈ ( 𝑁 WWalksN 𝐺 ) → 𝑊 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ) ) |
63 |
51 62
|
impbid |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉 ) ∧ ( 𝑇 ∈ Word 𝑉 ∧ 𝑊 = ( 𝑇 ++ 〈“ 𝑆 ”〉 ) ∧ { ( lastS ‘ 𝑇 ) , 𝑆 } ∈ 𝐸 ) ) → ( 𝑊 ∈ ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ↔ 𝑇 ∈ ( 𝑁 WWalksN 𝐺 ) ) ) |