Step |
Hyp |
Ref |
Expression |
1 |
|
lsw |
⊢ ( 𝑊 ∈ Word 𝑉 → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ) |
2 |
1
|
3ad2ant1 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ∅ ∉ 𝑉 ∧ ( ♯ ‘ 𝑊 ) ≠ 0 ) → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ) |
3 |
|
wrdf |
⊢ ( 𝑊 ∈ Word 𝑉 → 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑉 ) |
4 |
|
lencl |
⊢ ( 𝑊 ∈ Word 𝑉 → ( ♯ ‘ 𝑊 ) ∈ ℕ0 ) |
5 |
|
simpll |
⊢ ( ( ( 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑉 ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ0 ) ∧ ( ♯ ‘ 𝑊 ) ≠ 0 ) → 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑉 ) |
6 |
|
elnnne0 |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ↔ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) ≠ 0 ) ) |
7 |
6
|
biimpri |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) ≠ 0 ) → ( ♯ ‘ 𝑊 ) ∈ ℕ ) |
8 |
|
nnm1nn0 |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ → ( ( ♯ ‘ 𝑊 ) − 1 ) ∈ ℕ0 ) |
9 |
7 8
|
syl |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) ≠ 0 ) → ( ( ♯ ‘ 𝑊 ) − 1 ) ∈ ℕ0 ) |
10 |
|
nn0re |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → ( ♯ ‘ 𝑊 ) ∈ ℝ ) |
11 |
10
|
ltm1d |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → ( ( ♯ ‘ 𝑊 ) − 1 ) < ( ♯ ‘ 𝑊 ) ) |
12 |
11
|
adantr |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) ≠ 0 ) → ( ( ♯ ‘ 𝑊 ) − 1 ) < ( ♯ ‘ 𝑊 ) ) |
13 |
|
elfzo0 |
⊢ ( ( ( ♯ ‘ 𝑊 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ↔ ( ( ( ♯ ‘ 𝑊 ) − 1 ) ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ ( ( ♯ ‘ 𝑊 ) − 1 ) < ( ♯ ‘ 𝑊 ) ) ) |
14 |
9 7 12 13
|
syl3anbrc |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) ≠ 0 ) → ( ( ♯ ‘ 𝑊 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
15 |
14
|
adantll |
⊢ ( ( ( 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑉 ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ0 ) ∧ ( ♯ ‘ 𝑊 ) ≠ 0 ) → ( ( ♯ ‘ 𝑊 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
16 |
5 15
|
ffvelrnd |
⊢ ( ( ( 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑉 ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ0 ) ∧ ( ♯ ‘ 𝑊 ) ≠ 0 ) → ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ∈ 𝑉 ) |
17 |
16
|
ex |
⊢ ( ( 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑉 ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ0 ) → ( ( ♯ ‘ 𝑊 ) ≠ 0 → ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ∈ 𝑉 ) ) |
18 |
3 4 17
|
syl2anc |
⊢ ( 𝑊 ∈ Word 𝑉 → ( ( ♯ ‘ 𝑊 ) ≠ 0 → ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ∈ 𝑉 ) ) |
19 |
|
eleq1a |
⊢ ( ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ∈ 𝑉 → ( ∅ = ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) → ∅ ∈ 𝑉 ) ) |
20 |
19
|
com12 |
⊢ ( ∅ = ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) → ( ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ∈ 𝑉 → ∅ ∈ 𝑉 ) ) |
21 |
20
|
eqcoms |
⊢ ( ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) = ∅ → ( ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ∈ 𝑉 → ∅ ∈ 𝑉 ) ) |
22 |
21
|
com12 |
⊢ ( ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ∈ 𝑉 → ( ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) = ∅ → ∅ ∈ 𝑉 ) ) |
23 |
|
nnel |
⊢ ( ¬ ∅ ∉ 𝑉 ↔ ∅ ∈ 𝑉 ) |
24 |
22 23
|
syl6ibr |
⊢ ( ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ∈ 𝑉 → ( ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) = ∅ → ¬ ∅ ∉ 𝑉 ) ) |
25 |
24
|
necon2ad |
⊢ ( ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ∈ 𝑉 → ( ∅ ∉ 𝑉 → ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ≠ ∅ ) ) |
26 |
18 25
|
syl6 |
⊢ ( 𝑊 ∈ Word 𝑉 → ( ( ♯ ‘ 𝑊 ) ≠ 0 → ( ∅ ∉ 𝑉 → ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ≠ ∅ ) ) ) |
27 |
26
|
com23 |
⊢ ( 𝑊 ∈ Word 𝑉 → ( ∅ ∉ 𝑉 → ( ( ♯ ‘ 𝑊 ) ≠ 0 → ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ≠ ∅ ) ) ) |
28 |
27
|
3imp |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ∅ ∉ 𝑉 ∧ ( ♯ ‘ 𝑊 ) ≠ 0 ) → ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ≠ ∅ ) |
29 |
2 28
|
eqnetrd |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ∅ ∉ 𝑉 ∧ ( ♯ ‘ 𝑊 ) ≠ 0 ) → ( lastS ‘ 𝑊 ) ≠ ∅ ) |