Proof of Theorem pfxnd0
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | df-nel 3046 | . . . . 5
⊢ (𝐿 ∉
(0...(♯‘𝑊))
↔ ¬ 𝐿 ∈
(0...(♯‘𝑊))) | 
| 2 | 1 | a1i 11 | . . . 4
⊢ (𝑊 ∈ Word 𝑉 → (𝐿 ∉ (0...(♯‘𝑊)) ↔ ¬ 𝐿 ∈
(0...(♯‘𝑊)))) | 
| 3 |  | elfz2nn0 13659 | . . . . . 6
⊢ (𝐿 ∈
(0...(♯‘𝑊))
↔ (𝐿 ∈
ℕ0 ∧ (♯‘𝑊) ∈ ℕ0 ∧ 𝐿 ≤ (♯‘𝑊))) | 
| 4 | 3 | a1i 11 | . . . . 5
⊢ (𝑊 ∈ Word 𝑉 → (𝐿 ∈ (0...(♯‘𝑊)) ↔ (𝐿 ∈ ℕ0 ∧
(♯‘𝑊) ∈
ℕ0 ∧ 𝐿
≤ (♯‘𝑊)))) | 
| 5 | 4 | notbid 318 | . . . 4
⊢ (𝑊 ∈ Word 𝑉 → (¬ 𝐿 ∈ (0...(♯‘𝑊)) ↔ ¬ (𝐿 ∈ ℕ0
∧ (♯‘𝑊)
∈ ℕ0 ∧ 𝐿 ≤ (♯‘𝑊)))) | 
| 6 |  | 3ianor 1106 | . . . . 5
⊢ (¬
(𝐿 ∈
ℕ0 ∧ (♯‘𝑊) ∈ ℕ0 ∧ 𝐿 ≤ (♯‘𝑊)) ↔ (¬ 𝐿 ∈ ℕ0 ∨
¬ (♯‘𝑊)
∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑊))) | 
| 7 | 6 | a1i 11 | . . . 4
⊢ (𝑊 ∈ Word 𝑉 → (¬ (𝐿 ∈ ℕ0 ∧
(♯‘𝑊) ∈
ℕ0 ∧ 𝐿
≤ (♯‘𝑊))
↔ (¬ 𝐿 ∈
ℕ0 ∨ ¬ (♯‘𝑊) ∈ ℕ0 ∨ ¬
𝐿 ≤ (♯‘𝑊)))) | 
| 8 | 2, 5, 7 | 3bitrd 305 | . . 3
⊢ (𝑊 ∈ Word 𝑉 → (𝐿 ∉ (0...(♯‘𝑊)) ↔ (¬ 𝐿 ∈ ℕ0 ∨
¬ (♯‘𝑊)
∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑊)))) | 
| 9 |  | 3orrot 1091 | . . . . 5
⊢ ((¬
𝐿 ∈
ℕ0 ∨ ¬ (♯‘𝑊) ∈ ℕ0 ∨ ¬
𝐿 ≤ (♯‘𝑊)) ↔ (¬
(♯‘𝑊) ∈
ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑊) ∨ ¬ 𝐿 ∈
ℕ0)) | 
| 10 |  | 3orass 1089 | . . . . . 6
⊢ ((¬
(♯‘𝑊) ∈
ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑊) ∨ ¬ 𝐿 ∈ ℕ0) ↔ (¬
(♯‘𝑊) ∈
ℕ0 ∨ (¬ 𝐿 ≤ (♯‘𝑊) ∨ ¬ 𝐿 ∈
ℕ0))) | 
| 11 |  | lencl 14572 | . . . . . . . . 9
⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈
ℕ0) | 
| 12 | 11 | pm2.24d 151 | . . . . . . . 8
⊢ (𝑊 ∈ Word 𝑉 → (¬ (♯‘𝑊) ∈ ℕ0
→ (𝑊 prefix 𝐿) = ∅)) | 
| 13 | 12 | com12 32 | . . . . . . 7
⊢ (¬
(♯‘𝑊) ∈
ℕ0 → (𝑊 ∈ Word 𝑉 → (𝑊 prefix 𝐿) = ∅)) | 
| 14 |  | simpr 484 | . . . . . . . . . . 11
⊢ ((𝑊 ∈ V ∧ 𝐿 ∈ ℕ0)
→ 𝐿 ∈
ℕ0) | 
| 15 |  | pfxnndmnd 14711 | . . . . . . . . . . 11
⊢ (¬
(𝑊 ∈ V ∧ 𝐿 ∈ ℕ0)
→ (𝑊 prefix 𝐿) = ∅) | 
| 16 | 14, 15 | nsyl5 159 | . . . . . . . . . 10
⊢ (¬
𝐿 ∈
ℕ0 → (𝑊 prefix 𝐿) = ∅) | 
| 17 | 16 | a1d 25 | . . . . . . . . 9
⊢ (¬
𝐿 ∈
ℕ0 → (𝑊 ∈ Word 𝑉 → (𝑊 prefix 𝐿) = ∅)) | 
| 18 |  | notnotb 315 | . . . . . . . . . . 11
⊢ (𝐿 ∈ ℕ0
↔ ¬ ¬ 𝐿 ∈
ℕ0) | 
| 19 | 11 | nn0red 12590 | . . . . . . . . . . . . . . 15
⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℝ) | 
| 20 |  | nn0re 12537 | . . . . . . . . . . . . . . 15
⊢ (𝐿 ∈ ℕ0
→ 𝐿 ∈
ℝ) | 
| 21 |  | ltnle 11341 | . . . . . . . . . . . . . . 15
⊢
(((♯‘𝑊)
∈ ℝ ∧ 𝐿
∈ ℝ) → ((♯‘𝑊) < 𝐿 ↔ ¬ 𝐿 ≤ (♯‘𝑊))) | 
| 22 | 19, 20, 21 | syl2an 596 | . . . . . . . . . . . . . 14
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℕ0) →
((♯‘𝑊) <
𝐿 ↔ ¬ 𝐿 ≤ (♯‘𝑊))) | 
| 23 |  | pfxnd 14726 | . . . . . . . . . . . . . . 15
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℕ0 ∧
(♯‘𝑊) <
𝐿) → (𝑊 prefix 𝐿) = ∅) | 
| 24 | 23 | 3expia 1121 | . . . . . . . . . . . . . 14
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℕ0) →
((♯‘𝑊) <
𝐿 → (𝑊 prefix 𝐿) = ∅)) | 
| 25 | 22, 24 | sylbird 260 | . . . . . . . . . . . . 13
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℕ0) → (¬
𝐿 ≤ (♯‘𝑊) → (𝑊 prefix 𝐿) = ∅)) | 
| 26 | 25 | expcom 413 | . . . . . . . . . . . 12
⊢ (𝐿 ∈ ℕ0
→ (𝑊 ∈ Word 𝑉 → (¬ 𝐿 ≤ (♯‘𝑊) → (𝑊 prefix 𝐿) = ∅))) | 
| 27 | 26 | com23 86 | . . . . . . . . . . 11
⊢ (𝐿 ∈ ℕ0
→ (¬ 𝐿 ≤
(♯‘𝑊) →
(𝑊 ∈ Word 𝑉 → (𝑊 prefix 𝐿) = ∅))) | 
| 28 | 18, 27 | sylbir 235 | . . . . . . . . . 10
⊢ (¬
¬ 𝐿 ∈
ℕ0 → (¬ 𝐿 ≤ (♯‘𝑊) → (𝑊 ∈ Word 𝑉 → (𝑊 prefix 𝐿) = ∅))) | 
| 29 | 28 | imp 406 | . . . . . . . . 9
⊢ ((¬
¬ 𝐿 ∈
ℕ0 ∧ ¬ 𝐿 ≤ (♯‘𝑊)) → (𝑊 ∈ Word 𝑉 → (𝑊 prefix 𝐿) = ∅)) | 
| 30 | 17, 29 | jaoi3 1060 | . . . . . . . 8
⊢ ((¬
𝐿 ∈
ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑊)) → (𝑊 ∈ Word 𝑉 → (𝑊 prefix 𝐿) = ∅)) | 
| 31 | 30 | orcoms 872 | . . . . . . 7
⊢ ((¬
𝐿 ≤ (♯‘𝑊) ∨ ¬ 𝐿 ∈ ℕ0) → (𝑊 ∈ Word 𝑉 → (𝑊 prefix 𝐿) = ∅)) | 
| 32 | 13, 31 | jaoi 857 | . . . . . 6
⊢ ((¬
(♯‘𝑊) ∈
ℕ0 ∨ (¬ 𝐿 ≤ (♯‘𝑊) ∨ ¬ 𝐿 ∈ ℕ0)) → (𝑊 ∈ Word 𝑉 → (𝑊 prefix 𝐿) = ∅)) | 
| 33 | 10, 32 | sylbi 217 | . . . . 5
⊢ ((¬
(♯‘𝑊) ∈
ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑊) ∨ ¬ 𝐿 ∈ ℕ0) → (𝑊 ∈ Word 𝑉 → (𝑊 prefix 𝐿) = ∅)) | 
| 34 | 9, 33 | sylbi 217 | . . . 4
⊢ ((¬
𝐿 ∈
ℕ0 ∨ ¬ (♯‘𝑊) ∈ ℕ0 ∨ ¬
𝐿 ≤ (♯‘𝑊)) → (𝑊 ∈ Word 𝑉 → (𝑊 prefix 𝐿) = ∅)) | 
| 35 | 34 | com12 32 | . . 3
⊢ (𝑊 ∈ Word 𝑉 → ((¬ 𝐿 ∈ ℕ0 ∨ ¬
(♯‘𝑊) ∈
ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑊)) → (𝑊 prefix 𝐿) = ∅)) | 
| 36 | 8, 35 | sylbid 240 | . 2
⊢ (𝑊 ∈ Word 𝑉 → (𝐿 ∉ (0...(♯‘𝑊)) → (𝑊 prefix 𝐿) = ∅)) | 
| 37 | 36 | imp 406 | 1
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∉ (0...(♯‘𝑊))) → (𝑊 prefix 𝐿) = ∅) |