Proof of Theorem pfxccat3
| Step | Hyp | Ref
| Expression |
| 1 | | simpll 767 |
. . . 4
⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ 𝑁 ≤ 𝐿) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) |
| 2 | | simplrl 777 |
. . . . 5
⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ 𝑁 ≤ 𝐿) → 𝑀 ∈ (0...𝑁)) |
| 3 | | lencl 14571 |
. . . . . . . . 9
⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈
ℕ0) |
| 4 | | elfznn0 13660 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) → 𝑁 ∈
ℕ0) |
| 5 | 4 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) → 𝑁 ∈
ℕ0) |
| 6 | 5 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 ≤ 𝐿) → 𝑁 ∈
ℕ0) |
| 7 | | simplr 769 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 ≤ 𝐿) → (♯‘𝐴) ∈
ℕ0) |
| 8 | | swrdccatin2.l |
. . . . . . . . . . . . . . 15
⊢ 𝐿 = (♯‘𝐴) |
| 9 | 8 | breq2i 5151 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ≤ 𝐿 ↔ 𝑁 ≤ (♯‘𝐴)) |
| 10 | 9 | biimpi 216 |
. . . . . . . . . . . . 13
⊢ (𝑁 ≤ 𝐿 → 𝑁 ≤ (♯‘𝐴)) |
| 11 | 10 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 ≤ 𝐿) → 𝑁 ≤ (♯‘𝐴)) |
| 12 | | elfz2nn0 13658 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈
(0...(♯‘𝐴))
↔ (𝑁 ∈
ℕ0 ∧ (♯‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝐴))) |
| 13 | 6, 7, 11, 12 | syl3anbrc 1344 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 ≤ 𝐿) → 𝑁 ∈ (0...(♯‘𝐴))) |
| 14 | 13 | exp31 419 |
. . . . . . . . . 10
⊢ (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) → ((♯‘𝐴) ∈ ℕ0 → (𝑁 ≤ 𝐿 → 𝑁 ∈ (0...(♯‘𝐴))))) |
| 15 | 14 | adantl 481 |
. . . . . . . . 9
⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((♯‘𝐴) ∈ ℕ0
→ (𝑁 ≤ 𝐿 → 𝑁 ∈ (0...(♯‘𝐴))))) |
| 16 | 3, 15 | syl5com 31 |
. . . . . . . 8
⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝑁 ≤ 𝐿 → 𝑁 ∈ (0...(♯‘𝐴))))) |
| 17 | 16 | adantr 480 |
. . . . . . 7
⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝑁 ≤ 𝐿 → 𝑁 ∈ (0...(♯‘𝐴))))) |
| 18 | 17 | imp 406 |
. . . . . 6
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (𝑁 ≤ 𝐿 → 𝑁 ∈ (0...(♯‘𝐴)))) |
| 19 | 18 | imp 406 |
. . . . 5
⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ 𝑁 ≤ 𝐿) → 𝑁 ∈ (0...(♯‘𝐴))) |
| 20 | 2, 19 | jca 511 |
. . . 4
⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ 𝑁 ≤ 𝐿) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) |
| 21 | | swrdccatin1 14763 |
. . . 4
⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = (𝐴 substr 〈𝑀, 𝑁〉))) |
| 22 | 1, 20, 21 | sylc 65 |
. . 3
⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ 𝑁 ≤ 𝐿) → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = (𝐴 substr 〈𝑀, 𝑁〉)) |
| 23 | | simp1l 1198 |
. . . 4
⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) |
| 24 | 8 | eleq1i 2832 |
. . . . . . . . . . 11
⊢ (𝐿 ∈ ℕ0
↔ (♯‘𝐴)
∈ ℕ0) |
| 25 | | elfz2nn0 13658 |
. . . . . . . . . . . . . 14
⊢ (𝑀 ∈ (0...𝑁) ↔ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0
∧ 𝑀 ≤ 𝑁)) |
| 26 | | nn0z 12638 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐿 ∈ ℕ0
→ 𝐿 ∈
ℤ) |
| 27 | 26 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℕ0 ∧ 𝑀
≤ 𝑁) ∧ 𝐿 ∈ ℕ0)
→ 𝐿 ∈
ℤ) |
| 28 | | nn0z 12638 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℤ) |
| 29 | 28 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℕ0 ∧ 𝑀
≤ 𝑁) → 𝑁 ∈
ℤ) |
| 30 | 29 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℕ0 ∧ 𝑀
≤ 𝑁) ∧ 𝐿 ∈ ℕ0)
→ 𝑁 ∈
ℤ) |
| 31 | | nn0z 12638 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑀 ∈ ℕ0
→ 𝑀 ∈
ℤ) |
| 32 | 31 | 3ad2ant1 1134 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℕ0 ∧ 𝑀
≤ 𝑁) → 𝑀 ∈
ℤ) |
| 33 | 32 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℕ0 ∧ 𝑀
≤ 𝑁) ∧ 𝐿 ∈ ℕ0)
→ 𝑀 ∈
ℤ) |
| 34 | 27, 30, 33 | 3jca 1129 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℕ0 ∧ 𝑀
≤ 𝑁) ∧ 𝐿 ∈ ℕ0)
→ (𝐿 ∈ ℤ
∧ 𝑁 ∈ ℤ
∧ 𝑀 ∈
ℤ)) |
| 35 | 34 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℕ0 ∧ 𝑀
≤ 𝑁) ∧ 𝐿 ∈ ℕ0)
∧ 𝐿 ≤ 𝑀) → (𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) |
| 36 | | simpl3 1194 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℕ0 ∧ 𝑀
≤ 𝑁) ∧ 𝐿 ∈ ℕ0)
→ 𝑀 ≤ 𝑁) |
| 37 | 36 | anim1ci 616 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℕ0 ∧ 𝑀
≤ 𝑁) ∧ 𝐿 ∈ ℕ0)
∧ 𝐿 ≤ 𝑀) → (𝐿 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)) |
| 38 | | elfz2 13554 |
. . . . . . . . . . . . . . . 16
⊢ (𝑀 ∈ (𝐿...𝑁) ↔ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁))) |
| 39 | 35, 37, 38 | sylanbrc 583 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℕ0 ∧ 𝑀
≤ 𝑁) ∧ 𝐿 ∈ ℕ0)
∧ 𝐿 ≤ 𝑀) → 𝑀 ∈ (𝐿...𝑁)) |
| 40 | 39 | exp31 419 |
. . . . . . . . . . . . . 14
⊢ ((𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℕ0 ∧ 𝑀
≤ 𝑁) → (𝐿 ∈ ℕ0
→ (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁)))) |
| 41 | 25, 40 | sylbi 217 |
. . . . . . . . . . . . 13
⊢ (𝑀 ∈ (0...𝑁) → (𝐿 ∈ ℕ0 → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁)))) |
| 42 | 41 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝐿 ∈ ℕ0 → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁)))) |
| 43 | 42 | com12 32 |
. . . . . . . . . . 11
⊢ (𝐿 ∈ ℕ0
→ ((𝑀 ∈
(0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁)))) |
| 44 | 24, 43 | sylbir 235 |
. . . . . . . . . 10
⊢
((♯‘𝐴)
∈ ℕ0 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁)))) |
| 45 | 3, 44 | syl 17 |
. . . . . . . . 9
⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁)))) |
| 46 | 45 | adantr 480 |
. . . . . . . 8
⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁)))) |
| 47 | 46 | imp 406 |
. . . . . . 7
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁))) |
| 48 | 47 | a1d 25 |
. . . . . 6
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝑁 ≤ 𝐿 → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁)))) |
| 49 | 48 | 3imp 1111 |
. . . . 5
⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → 𝑀 ∈ (𝐿...𝑁)) |
| 50 | | elfz2nn0 13658 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ↔ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0
∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) |
| 51 | | nn0z 12638 |
. . . . . . . . . . . . . . . . . 18
⊢
((♯‘𝐴)
∈ ℕ0 → (♯‘𝐴) ∈ ℤ) |
| 52 | 8, 51 | eqeltrid 2845 |
. . . . . . . . . . . . . . . . 17
⊢
((♯‘𝐴)
∈ ℕ0 → 𝐿 ∈ ℤ) |
| 53 | 52 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢
(((♯‘𝐴)
∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿) → 𝐿 ∈ ℤ) |
| 54 | 53 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ0
∧ (𝐿 +
(♯‘𝐵)) ∈
ℕ0 ∧ 𝑁
≤ (𝐿 +
(♯‘𝐵))) ∧
((♯‘𝐴) ∈
ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → 𝐿 ∈ ℤ) |
| 55 | | nn0z 12638 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐿 + (♯‘𝐵)) ∈ ℕ0
→ (𝐿 +
(♯‘𝐵)) ∈
ℤ) |
| 56 | 55 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℕ0
∧ (𝐿 +
(♯‘𝐵)) ∈
ℕ0 ∧ 𝑁
≤ (𝐿 +
(♯‘𝐵))) →
(𝐿 + (♯‘𝐵)) ∈
ℤ) |
| 57 | 56 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ0
∧ (𝐿 +
(♯‘𝐵)) ∈
ℕ0 ∧ 𝑁
≤ (𝐿 +
(♯‘𝐵))) ∧
((♯‘𝐴) ∈
ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → (𝐿 + (♯‘𝐵)) ∈ ℤ) |
| 58 | 28 | 3ad2ant1 1134 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℕ0
∧ (𝐿 +
(♯‘𝐵)) ∈
ℕ0 ∧ 𝑁
≤ (𝐿 +
(♯‘𝐵))) →
𝑁 ∈
ℤ) |
| 59 | 58 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ0
∧ (𝐿 +
(♯‘𝐵)) ∈
ℕ0 ∧ 𝑁
≤ (𝐿 +
(♯‘𝐵))) ∧
((♯‘𝐴) ∈
ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → 𝑁 ∈ ℤ) |
| 60 | 54, 57, 59 | 3jca 1129 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ0
∧ (𝐿 +
(♯‘𝐵)) ∈
ℕ0 ∧ 𝑁
≤ (𝐿 +
(♯‘𝐵))) ∧
((♯‘𝐴) ∈
ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → (𝐿 ∈ ℤ ∧ (𝐿 + (♯‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
| 61 | 8 | eqcomi 2746 |
. . . . . . . . . . . . . . . . . . 19
⊢
(♯‘𝐴) =
𝐿 |
| 62 | 61 | eleq1i 2832 |
. . . . . . . . . . . . . . . . . 18
⊢
((♯‘𝐴)
∈ ℕ0 ↔ 𝐿 ∈
ℕ0) |
| 63 | | nn0re 12535 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐿 ∈ ℕ0
→ 𝐿 ∈
ℝ) |
| 64 | | nn0re 12535 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℝ) |
| 65 | | ltnle 11340 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁 ↔ ¬ 𝑁 ≤ 𝐿)) |
| 66 | 63, 64, 65 | syl2anr 597 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑁 ∈ ℕ0
∧ 𝐿 ∈
ℕ0) → (𝐿 < 𝑁 ↔ ¬ 𝑁 ≤ 𝐿)) |
| 67 | 66 | bicomd 223 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑁 ∈ ℕ0
∧ 𝐿 ∈
ℕ0) → (¬ 𝑁 ≤ 𝐿 ↔ 𝐿 < 𝑁)) |
| 68 | | ltle 11349 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁 → 𝐿 ≤ 𝑁)) |
| 69 | 63, 64, 68 | syl2anr 597 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑁 ∈ ℕ0
∧ 𝐿 ∈
ℕ0) → (𝐿 < 𝑁 → 𝐿 ≤ 𝑁)) |
| 70 | 67, 69 | sylbid 240 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈ ℕ0
∧ 𝐿 ∈
ℕ0) → (¬ 𝑁 ≤ 𝐿 → 𝐿 ≤ 𝑁)) |
| 71 | 70 | ex 412 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈ ℕ0
→ (𝐿 ∈
ℕ0 → (¬ 𝑁 ≤ 𝐿 → 𝐿 ≤ 𝑁))) |
| 72 | 62, 71 | biimtrid 242 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℕ0
→ ((♯‘𝐴)
∈ ℕ0 → (¬ 𝑁 ≤ 𝐿 → 𝐿 ≤ 𝑁))) |
| 73 | 72 | 3ad2ant1 1134 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℕ0
∧ (𝐿 +
(♯‘𝐵)) ∈
ℕ0 ∧ 𝑁
≤ (𝐿 +
(♯‘𝐵))) →
((♯‘𝐴) ∈
ℕ0 → (¬ 𝑁 ≤ 𝐿 → 𝐿 ≤ 𝑁))) |
| 74 | 73 | imp32 418 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ0
∧ (𝐿 +
(♯‘𝐵)) ∈
ℕ0 ∧ 𝑁
≤ (𝐿 +
(♯‘𝐵))) ∧
((♯‘𝐴) ∈
ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → 𝐿 ≤ 𝑁) |
| 75 | | simpl3 1194 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ0
∧ (𝐿 +
(♯‘𝐵)) ∈
ℕ0 ∧ 𝑁
≤ (𝐿 +
(♯‘𝐵))) ∧
((♯‘𝐴) ∈
ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → 𝑁 ≤ (𝐿 + (♯‘𝐵))) |
| 76 | 74, 75 | jca 511 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ0
∧ (𝐿 +
(♯‘𝐵)) ∈
ℕ0 ∧ 𝑁
≤ (𝐿 +
(♯‘𝐵))) ∧
((♯‘𝐴) ∈
ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → (𝐿 ≤ 𝑁 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) |
| 77 | | elfz2 13554 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) ↔ ((𝐿 ∈ ℤ ∧ (𝐿 + (♯‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐿 ≤ 𝑁 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))))) |
| 78 | 60, 76, 77 | sylanbrc 583 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ0
∧ (𝐿 +
(♯‘𝐵)) ∈
ℕ0 ∧ 𝑁
≤ (𝐿 +
(♯‘𝐵))) ∧
((♯‘𝐴) ∈
ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) |
| 79 | 78 | exp32 420 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ0
∧ (𝐿 +
(♯‘𝐵)) ∈
ℕ0 ∧ 𝑁
≤ (𝐿 +
(♯‘𝐵))) →
((♯‘𝐴) ∈
ℕ0 → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))) |
| 80 | 50, 79 | sylbi 217 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) → ((♯‘𝐴) ∈ ℕ0 → (¬
𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))) |
| 81 | 80 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((♯‘𝐴) ∈ ℕ0
→ (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))) |
| 82 | 3, 81 | syl5com 31 |
. . . . . . . . 9
⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))) |
| 83 | 82 | adantr 480 |
. . . . . . . 8
⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))) |
| 84 | 83 | imp 406 |
. . . . . . 7
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) |
| 85 | 84 | a1dd 50 |
. . . . . 6
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝑁 ≤ 𝐿 → (𝐿 ≤ 𝑀 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))) |
| 86 | 85 | 3imp 1111 |
. . . . 5
⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) |
| 87 | 49, 86 | jca 511 |
. . . 4
⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) |
| 88 | 8 | swrdccatin2 14767 |
. . . 4
⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = (𝐵 substr 〈(𝑀 − 𝐿), (𝑁 − 𝐿)〉))) |
| 89 | 23, 87, 88 | sylc 65 |
. . 3
⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = (𝐵 substr 〈(𝑀 − 𝐿), (𝑁 − 𝐿)〉)) |
| 90 | | simp1l 1198 |
. . . 4
⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) |
| 91 | | nn0re 12535 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑀 ∈ ℕ0
→ 𝑀 ∈
ℝ) |
| 92 | 91 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℕ0) → 𝑀 ∈ ℝ) |
| 93 | | ltnle 11340 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 < 𝐿 ↔ ¬ 𝐿 ≤ 𝑀)) |
| 94 | 92, 63, 93 | syl2anr 597 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐿 ∈ ℕ0
∧ (𝑀 ∈
ℕ0 ∧ 𝑁
∈ ℕ0)) → (𝑀 < 𝐿 ↔ ¬ 𝐿 ≤ 𝑀)) |
| 95 | 94 | bicomd 223 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐿 ∈ ℕ0
∧ (𝑀 ∈
ℕ0 ∧ 𝑁
∈ ℕ0)) → (¬ 𝐿 ≤ 𝑀 ↔ 𝑀 < 𝐿)) |
| 96 | | simpll 767 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑀 ∈ ℕ0
∧ 𝐿 ∈
ℕ0) ∧ 𝑀 < 𝐿) → 𝑀 ∈
ℕ0) |
| 97 | | simplr 769 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑀 ∈ ℕ0
∧ 𝐿 ∈
ℕ0) ∧ 𝑀 < 𝐿) → 𝐿 ∈
ℕ0) |
| 98 | | ltle 11349 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 < 𝐿 → 𝑀 ≤ 𝐿)) |
| 99 | 91, 63, 98 | syl2an 596 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑀 ∈ ℕ0
∧ 𝐿 ∈
ℕ0) → (𝑀 < 𝐿 → 𝑀 ≤ 𝐿)) |
| 100 | 99 | imp 406 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑀 ∈ ℕ0
∧ 𝐿 ∈
ℕ0) ∧ 𝑀 < 𝐿) → 𝑀 ≤ 𝐿) |
| 101 | | elfz2nn0 13658 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑀 ∈ (0...𝐿) ↔ (𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0
∧ 𝑀 ≤ 𝐿)) |
| 102 | 96, 97, 100, 101 | syl3anbrc 1344 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑀 ∈ ℕ0
∧ 𝐿 ∈
ℕ0) ∧ 𝑀 < 𝐿) → 𝑀 ∈ (0...𝐿)) |
| 103 | 102 | exp31 419 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑀 ∈ ℕ0
→ (𝐿 ∈
ℕ0 → (𝑀 < 𝐿 → 𝑀 ∈ (0...𝐿)))) |
| 104 | 103 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℕ0) → (𝐿 ∈ ℕ0 → (𝑀 < 𝐿 → 𝑀 ∈ (0...𝐿)))) |
| 105 | 104 | impcom 407 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐿 ∈ ℕ0
∧ (𝑀 ∈
ℕ0 ∧ 𝑁
∈ ℕ0)) → (𝑀 < 𝐿 → 𝑀 ∈ (0...𝐿))) |
| 106 | 95, 105 | sylbid 240 |
. . . . . . . . . . . . . . 15
⊢ ((𝐿 ∈ ℕ0
∧ (𝑀 ∈
ℕ0 ∧ 𝑁
∈ ℕ0)) → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿))) |
| 107 | 106 | expcom 413 |
. . . . . . . . . . . . . 14
⊢ ((𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℕ0) → (𝐿 ∈ ℕ0 → (¬
𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿)))) |
| 108 | 107 | 3adant3 1133 |
. . . . . . . . . . . . 13
⊢ ((𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℕ0 ∧ 𝑀
≤ 𝑁) → (𝐿 ∈ ℕ0
→ (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿)))) |
| 109 | 25, 108 | sylbi 217 |
. . . . . . . . . . . 12
⊢ (𝑀 ∈ (0...𝑁) → (𝐿 ∈ ℕ0 → (¬
𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿)))) |
| 110 | 62, 109 | biimtrid 242 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ (0...𝑁) → ((♯‘𝐴) ∈ ℕ0 → (¬
𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿)))) |
| 111 | 110 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((♯‘𝐴) ∈ ℕ0
→ (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿)))) |
| 112 | 3, 111 | syl5com 31 |
. . . . . . . . 9
⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿)))) |
| 113 | 112 | adantr 480 |
. . . . . . . 8
⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿)))) |
| 114 | 113 | imp 406 |
. . . . . . 7
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿))) |
| 115 | 114 | a1d 25 |
. . . . . 6
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝑁 ≤ 𝐿 → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿)))) |
| 116 | 115 | 3imp 1111 |
. . . . 5
⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → 𝑀 ∈ (0...𝐿)) |
| 117 | 64 | 3ad2ant1 1134 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ ℕ0
∧ (𝐿 +
(♯‘𝐵)) ∈
ℕ0 ∧ 𝑁
≤ (𝐿 +
(♯‘𝐵))) →
𝑁 ∈
ℝ) |
| 118 | 65 | bicomd 223 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (¬
𝑁 ≤ 𝐿 ↔ 𝐿 < 𝑁)) |
| 119 | 63, 117, 118 | syl2an 596 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐿 ∈ ℕ0
∧ (𝑁 ∈
ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (¬ 𝑁 ≤ 𝐿 ↔ 𝐿 < 𝑁)) |
| 120 | 26 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐿 ∈ ℕ0
∧ (𝑁 ∈
ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → 𝐿 ∈ ℤ) |
| 121 | 56 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐿 ∈ ℕ0
∧ (𝑁 ∈
ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (𝐿 + (♯‘𝐵)) ∈ ℤ) |
| 122 | 58 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐿 ∈ ℕ0
∧ (𝑁 ∈
ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → 𝑁 ∈ ℤ) |
| 123 | 120, 121,
122 | 3jca 1129 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐿 ∈ ℕ0
∧ (𝑁 ∈
ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (𝐿 ∈ ℤ ∧ (𝐿 + (♯‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
| 124 | 123 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐿 ∈ ℕ0
∧ (𝑁 ∈
ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → (𝐿 ∈ ℤ ∧ (𝐿 + (♯‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
| 125 | 63, 117, 68 | syl2an 596 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐿 ∈ ℕ0
∧ (𝑁 ∈
ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (𝐿 < 𝑁 → 𝐿 ≤ 𝑁)) |
| 126 | 125 | imp 406 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐿 ∈ ℕ0
∧ (𝑁 ∈
ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝐿 ≤ 𝑁) |
| 127 | | simplr3 1218 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐿 ∈ ℕ0
∧ (𝑁 ∈
ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝑁 ≤ (𝐿 + (♯‘𝐵))) |
| 128 | 126, 127 | jca 511 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐿 ∈ ℕ0
∧ (𝑁 ∈
ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → (𝐿 ≤ 𝑁 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) |
| 129 | 124, 128,
77 | sylanbrc 583 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐿 ∈ ℕ0
∧ (𝑁 ∈
ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) |
| 130 | 129 | ex 412 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐿 ∈ ℕ0
∧ (𝑁 ∈
ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (𝐿 < 𝑁 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) |
| 131 | 119, 130 | sylbid 240 |
. . . . . . . . . . . . . . 15
⊢ ((𝐿 ∈ ℕ0
∧ (𝑁 ∈
ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) |
| 132 | 131 | ex 412 |
. . . . . . . . . . . . . 14
⊢ (𝐿 ∈ ℕ0
→ ((𝑁 ∈
ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))) |
| 133 | 62, 132 | sylbi 217 |
. . . . . . . . . . . . 13
⊢
((♯‘𝐴)
∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0
∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))) |
| 134 | 3, 133 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ Word 𝑉 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0
∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))) |
| 135 | 134 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0
∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))) |
| 136 | 135 | com12 32 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ0
∧ (𝐿 +
(♯‘𝐵)) ∈
ℕ0 ∧ 𝑁
≤ (𝐿 +
(♯‘𝐵))) →
((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))) |
| 137 | 50, 136 | sylbi 217 |
. . . . . . . . 9
⊢ (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))) |
| 138 | 137 | adantl 481 |
. . . . . . . 8
⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))) |
| 139 | 138 | impcom 407 |
. . . . . . 7
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) |
| 140 | 139 | a1dd 50 |
. . . . . 6
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝑁 ≤ 𝐿 → (¬ 𝐿 ≤ 𝑀 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))) |
| 141 | 140 | 3imp 1111 |
. . . . 5
⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) |
| 142 | 116, 141 | jca 511 |
. . . 4
⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) |
| 143 | 8 | pfxccatin12 14771 |
. . . 4
⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = ((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 prefix (𝑁 − 𝐿))))) |
| 144 | 90, 142, 143 | sylc 65 |
. . 3
⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = ((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 prefix (𝑁 − 𝐿)))) |
| 145 | 22, 89, 144 | 2if2 4581 |
. 2
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = if(𝑁 ≤ 𝐿, (𝐴 substr 〈𝑀, 𝑁〉), if(𝐿 ≤ 𝑀, (𝐵 substr 〈(𝑀 − 𝐿), (𝑁 − 𝐿)〉), ((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 prefix (𝑁 − 𝐿)))))) |
| 146 | 145 | ex 412 |
1
⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = if(𝑁 ≤ 𝐿, (𝐴 substr 〈𝑀, 𝑁〉), if(𝐿 ≤ 𝑀, (𝐵 substr 〈(𝑀 − 𝐿), (𝑁 − 𝐿)〉), ((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 prefix (𝑁 − 𝐿))))))) |