Step |
Hyp |
Ref |
Expression |
1 |
|
lencl |
⊢ ( 𝑊 ∈ Word 𝑉 → ( ♯ ‘ 𝑊 ) ∈ ℕ0 ) |
2 |
|
1z |
⊢ 1 ∈ ℤ |
3 |
|
nn0z |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → ( ♯ ‘ 𝑊 ) ∈ ℤ ) |
4 |
|
zltp1le |
⊢ ( ( 1 ∈ ℤ ∧ ( ♯ ‘ 𝑊 ) ∈ ℤ ) → ( 1 < ( ♯ ‘ 𝑊 ) ↔ ( 1 + 1 ) ≤ ( ♯ ‘ 𝑊 ) ) ) |
5 |
2 3 4
|
sylancr |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → ( 1 < ( ♯ ‘ 𝑊 ) ↔ ( 1 + 1 ) ≤ ( ♯ ‘ 𝑊 ) ) ) |
6 |
|
1p1e2 |
⊢ ( 1 + 1 ) = 2 |
7 |
6
|
a1i |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → ( 1 + 1 ) = 2 ) |
8 |
7
|
breq1d |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → ( ( 1 + 1 ) ≤ ( ♯ ‘ 𝑊 ) ↔ 2 ≤ ( ♯ ‘ 𝑊 ) ) ) |
9 |
8
|
biimpd |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → ( ( 1 + 1 ) ≤ ( ♯ ‘ 𝑊 ) → 2 ≤ ( ♯ ‘ 𝑊 ) ) ) |
10 |
5 9
|
sylbid |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → ( 1 < ( ♯ ‘ 𝑊 ) → 2 ≤ ( ♯ ‘ 𝑊 ) ) ) |
11 |
10
|
imp |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ 1 < ( ♯ ‘ 𝑊 ) ) → 2 ≤ ( ♯ ‘ 𝑊 ) ) |
12 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
13 |
|
simpl |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ 1 < ( ♯ ‘ 𝑊 ) ) → ( ♯ ‘ 𝑊 ) ∈ ℕ0 ) |
14 |
|
nn0sub |
⊢ ( ( 2 ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ0 ) → ( 2 ≤ ( ♯ ‘ 𝑊 ) ↔ ( ( ♯ ‘ 𝑊 ) − 2 ) ∈ ℕ0 ) ) |
15 |
12 13 14
|
sylancr |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ 1 < ( ♯ ‘ 𝑊 ) ) → ( 2 ≤ ( ♯ ‘ 𝑊 ) ↔ ( ( ♯ ‘ 𝑊 ) − 2 ) ∈ ℕ0 ) ) |
16 |
11 15
|
mpbid |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ 1 < ( ♯ ‘ 𝑊 ) ) → ( ( ♯ ‘ 𝑊 ) − 2 ) ∈ ℕ0 ) |
17 |
3
|
adantr |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ 1 < ( ♯ ‘ 𝑊 ) ) → ( ♯ ‘ 𝑊 ) ∈ ℤ ) |
18 |
|
0red |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → 0 ∈ ℝ ) |
19 |
|
1red |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → 1 ∈ ℝ ) |
20 |
|
nn0re |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → ( ♯ ‘ 𝑊 ) ∈ ℝ ) |
21 |
18 19 20
|
3jca |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → ( 0 ∈ ℝ ∧ 1 ∈ ℝ ∧ ( ♯ ‘ 𝑊 ) ∈ ℝ ) ) |
22 |
|
0lt1 |
⊢ 0 < 1 |
23 |
|
lttr |
⊢ ( ( 0 ∈ ℝ ∧ 1 ∈ ℝ ∧ ( ♯ ‘ 𝑊 ) ∈ ℝ ) → ( ( 0 < 1 ∧ 1 < ( ♯ ‘ 𝑊 ) ) → 0 < ( ♯ ‘ 𝑊 ) ) ) |
24 |
23
|
expd |
⊢ ( ( 0 ∈ ℝ ∧ 1 ∈ ℝ ∧ ( ♯ ‘ 𝑊 ) ∈ ℝ ) → ( 0 < 1 → ( 1 < ( ♯ ‘ 𝑊 ) → 0 < ( ♯ ‘ 𝑊 ) ) ) ) |
25 |
21 22 24
|
mpisyl |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → ( 1 < ( ♯ ‘ 𝑊 ) → 0 < ( ♯ ‘ 𝑊 ) ) ) |
26 |
25
|
imp |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ 1 < ( ♯ ‘ 𝑊 ) ) → 0 < ( ♯ ‘ 𝑊 ) ) |
27 |
|
elnnz |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ↔ ( ( ♯ ‘ 𝑊 ) ∈ ℤ ∧ 0 < ( ♯ ‘ 𝑊 ) ) ) |
28 |
17 26 27
|
sylanbrc |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ 1 < ( ♯ ‘ 𝑊 ) ) → ( ♯ ‘ 𝑊 ) ∈ ℕ ) |
29 |
|
2rp |
⊢ 2 ∈ ℝ+ |
30 |
29
|
a1i |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → 2 ∈ ℝ+ ) |
31 |
20 30
|
ltsubrpd |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → ( ( ♯ ‘ 𝑊 ) − 2 ) < ( ♯ ‘ 𝑊 ) ) |
32 |
31
|
adantr |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ 1 < ( ♯ ‘ 𝑊 ) ) → ( ( ♯ ‘ 𝑊 ) − 2 ) < ( ♯ ‘ 𝑊 ) ) |
33 |
|
elfzo0 |
⊢ ( ( ( ♯ ‘ 𝑊 ) − 2 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ↔ ( ( ( ♯ ‘ 𝑊 ) − 2 ) ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ ( ( ♯ ‘ 𝑊 ) − 2 ) < ( ♯ ‘ 𝑊 ) ) ) |
34 |
16 28 32 33
|
syl3anbrc |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ 1 < ( ♯ ‘ 𝑊 ) ) → ( ( ♯ ‘ 𝑊 ) − 2 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
35 |
1 34
|
sylan |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) → ( ( ♯ ‘ 𝑊 ) − 2 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
36 |
35
|
3adant2 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) → ( ( ♯ ‘ 𝑊 ) − 2 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
37 |
|
pfxsuffeqwrdeq |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ ( ( ♯ ‘ 𝑊 ) − 2 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 = 𝑈 ↔ ( ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ∧ ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∧ ( 𝑊 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) = ( 𝑈 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) ) ) ) ) |
38 |
36 37
|
syld3an3 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) → ( 𝑊 = 𝑈 ↔ ( ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ∧ ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∧ ( 𝑊 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) = ( 𝑈 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) ) ) ) ) |
39 |
|
swrd2lsw |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) → ( 𝑊 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) = 〈“ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ( lastS ‘ 𝑊 ) ”〉 ) |
40 |
39
|
3adant2 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) → ( 𝑊 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) = 〈“ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ( lastS ‘ 𝑊 ) ”〉 ) |
41 |
40
|
adantr |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ) → ( 𝑊 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) = 〈“ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ( lastS ‘ 𝑊 ) ”〉 ) |
42 |
|
breq2 |
⊢ ( ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) → ( 1 < ( ♯ ‘ 𝑊 ) ↔ 1 < ( ♯ ‘ 𝑈 ) ) ) |
43 |
42
|
3anbi3d |
⊢ ( ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) → ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) ↔ ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑈 ) ) ) ) |
44 |
|
swrd2lsw |
⊢ ( ( 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑈 ) ) → ( 𝑈 substr 〈 ( ( ♯ ‘ 𝑈 ) − 2 ) , ( ♯ ‘ 𝑈 ) 〉 ) = 〈“ ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ( lastS ‘ 𝑈 ) ”〉 ) |
45 |
44
|
3adant1 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑈 ) ) → ( 𝑈 substr 〈 ( ( ♯ ‘ 𝑈 ) − 2 ) , ( ♯ ‘ 𝑈 ) 〉 ) = 〈“ ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ( lastS ‘ 𝑈 ) ”〉 ) |
46 |
43 45
|
syl6bi |
⊢ ( ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) → ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) → ( 𝑈 substr 〈 ( ( ♯ ‘ 𝑈 ) − 2 ) , ( ♯ ‘ 𝑈 ) 〉 ) = 〈“ ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ( lastS ‘ 𝑈 ) ”〉 ) ) |
47 |
46
|
impcom |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ) → ( 𝑈 substr 〈 ( ( ♯ ‘ 𝑈 ) − 2 ) , ( ♯ ‘ 𝑈 ) 〉 ) = 〈“ ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ( lastS ‘ 𝑈 ) ”〉 ) |
48 |
|
oveq1 |
⊢ ( ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) → ( ( ♯ ‘ 𝑊 ) − 2 ) = ( ( ♯ ‘ 𝑈 ) − 2 ) ) |
49 |
|
id |
⊢ ( ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) → ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ) |
50 |
48 49
|
opeq12d |
⊢ ( ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) → 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 = 〈 ( ( ♯ ‘ 𝑈 ) − 2 ) , ( ♯ ‘ 𝑈 ) 〉 ) |
51 |
50
|
oveq2d |
⊢ ( ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) → ( 𝑈 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) = ( 𝑈 substr 〈 ( ( ♯ ‘ 𝑈 ) − 2 ) , ( ♯ ‘ 𝑈 ) 〉 ) ) |
52 |
51
|
eqeq1d |
⊢ ( ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) → ( ( 𝑈 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) = 〈“ ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ( lastS ‘ 𝑈 ) ”〉 ↔ ( 𝑈 substr 〈 ( ( ♯ ‘ 𝑈 ) − 2 ) , ( ♯ ‘ 𝑈 ) 〉 ) = 〈“ ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ( lastS ‘ 𝑈 ) ”〉 ) ) |
53 |
52
|
adantl |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ) → ( ( 𝑈 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) = 〈“ ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ( lastS ‘ 𝑈 ) ”〉 ↔ ( 𝑈 substr 〈 ( ( ♯ ‘ 𝑈 ) − 2 ) , ( ♯ ‘ 𝑈 ) 〉 ) = 〈“ ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ( lastS ‘ 𝑈 ) ”〉 ) ) |
54 |
47 53
|
mpbird |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ) → ( 𝑈 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) = 〈“ ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ( lastS ‘ 𝑈 ) ”〉 ) |
55 |
41 54
|
eqeq12d |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ) → ( ( 𝑊 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) = ( 𝑈 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) ↔ 〈“ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ( lastS ‘ 𝑊 ) ”〉 = 〈“ ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ( lastS ‘ 𝑈 ) ”〉 ) ) |
56 |
|
fvexd |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ) → ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∈ V ) |
57 |
|
fvexd |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ) → ( lastS ‘ 𝑊 ) ∈ V ) |
58 |
|
fvexd |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ) → ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ∈ V ) |
59 |
|
fvexd |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ) → ( lastS ‘ 𝑈 ) ∈ V ) |
60 |
|
s2eq2s1eq |
⊢ ( ( ( ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∈ V ∧ ( lastS ‘ 𝑊 ) ∈ V ) ∧ ( ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ∈ V ∧ ( lastS ‘ 𝑈 ) ∈ V ) ) → ( 〈“ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ( lastS ‘ 𝑊 ) ”〉 = 〈“ ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ( lastS ‘ 𝑈 ) ”〉 ↔ ( 〈“ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ”〉 = 〈“ ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ”〉 ∧ 〈“ ( lastS ‘ 𝑊 ) ”〉 = 〈“ ( lastS ‘ 𝑈 ) ”〉 ) ) ) |
61 |
56 57 58 59 60
|
syl22anc |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ) → ( 〈“ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ( lastS ‘ 𝑊 ) ”〉 = 〈“ ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ( lastS ‘ 𝑈 ) ”〉 ↔ ( 〈“ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ”〉 = 〈“ ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ”〉 ∧ 〈“ ( lastS ‘ 𝑊 ) ”〉 = 〈“ ( lastS ‘ 𝑈 ) ”〉 ) ) ) |
62 |
|
fvex |
⊢ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∈ V |
63 |
|
s111 |
⊢ ( ( ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∈ V ∧ ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ∈ V ) → ( 〈“ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ”〉 = 〈“ ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ”〉 ↔ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ) ) |
64 |
62 58 63
|
sylancr |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ) → ( 〈“ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ”〉 = 〈“ ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ”〉 ↔ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ) ) |
65 |
|
fvoveq1 |
⊢ ( ( ♯ ‘ 𝑈 ) = ( ♯ ‘ 𝑊 ) → ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) = ( 𝑈 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ) |
66 |
65
|
eqcoms |
⊢ ( ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) → ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) = ( 𝑈 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ) |
67 |
66
|
adantl |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ) → ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) = ( 𝑈 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ) |
68 |
67
|
eqeq2d |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ) → ( ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ↔ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ) ) |
69 |
64 68
|
bitrd |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ) → ( 〈“ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ”〉 = 〈“ ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ”〉 ↔ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ) ) |
70 |
|
fvex |
⊢ ( lastS ‘ 𝑊 ) ∈ V |
71 |
|
s111 |
⊢ ( ( ( lastS ‘ 𝑊 ) ∈ V ∧ ( lastS ‘ 𝑈 ) ∈ V ) → ( 〈“ ( lastS ‘ 𝑊 ) ”〉 = 〈“ ( lastS ‘ 𝑈 ) ”〉 ↔ ( lastS ‘ 𝑊 ) = ( lastS ‘ 𝑈 ) ) ) |
72 |
70 59 71
|
sylancr |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ) → ( 〈“ ( lastS ‘ 𝑊 ) ”〉 = 〈“ ( lastS ‘ 𝑈 ) ”〉 ↔ ( lastS ‘ 𝑊 ) = ( lastS ‘ 𝑈 ) ) ) |
73 |
69 72
|
anbi12d |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ) → ( ( 〈“ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ”〉 = 〈“ ( 𝑈 ‘ ( ( ♯ ‘ 𝑈 ) − 2 ) ) ”〉 ∧ 〈“ ( lastS ‘ 𝑊 ) ”〉 = 〈“ ( lastS ‘ 𝑈 ) ”〉 ) ↔ ( ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∧ ( lastS ‘ 𝑊 ) = ( lastS ‘ 𝑈 ) ) ) ) |
74 |
55 61 73
|
3bitrd |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ) → ( ( 𝑊 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) = ( 𝑈 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) ↔ ( ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∧ ( lastS ‘ 𝑊 ) = ( lastS ‘ 𝑈 ) ) ) ) |
75 |
74
|
anbi2d |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ) → ( ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∧ ( 𝑊 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) = ( 𝑈 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) ) ↔ ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∧ ( ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∧ ( lastS ‘ 𝑊 ) = ( lastS ‘ 𝑈 ) ) ) ) ) |
76 |
|
3anass |
⊢ ( ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∧ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∧ ( lastS ‘ 𝑊 ) = ( lastS ‘ 𝑈 ) ) ↔ ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∧ ( ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∧ ( lastS ‘ 𝑊 ) = ( lastS ‘ 𝑈 ) ) ) ) |
77 |
75 76
|
bitr4di |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) ∧ ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ) → ( ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∧ ( 𝑊 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) = ( 𝑈 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) ) ↔ ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∧ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∧ ( lastS ‘ 𝑊 ) = ( lastS ‘ 𝑈 ) ) ) ) |
78 |
77
|
pm5.32da |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) → ( ( ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ∧ ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∧ ( 𝑊 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) = ( 𝑈 substr 〈 ( ( ♯ ‘ 𝑊 ) − 2 ) , ( ♯ ‘ 𝑊 ) 〉 ) ) ) ↔ ( ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ∧ ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∧ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∧ ( lastS ‘ 𝑊 ) = ( lastS ‘ 𝑈 ) ) ) ) ) |
79 |
38 78
|
bitrd |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < ( ♯ ‘ 𝑊 ) ) → ( 𝑊 = 𝑈 ↔ ( ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑈 ) ∧ ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 prefix ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∧ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) = ( 𝑈 ‘ ( ( ♯ ‘ 𝑊 ) − 2 ) ) ∧ ( lastS ‘ 𝑊 ) = ( lastS ‘ 𝑈 ) ) ) ) ) |