Step | Hyp | Ref
| Expression |
1 | | swrdval2 14543 |
. . 3
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑊‘(𝑥 + 𝑀)))) |
2 | 1 | rneqd 5897 |
. 2
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝑀, 𝑁⟩) = ran (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑊‘(𝑥 + 𝑀)))) |
3 | | eqidd 2734 |
. . . . . . 7
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → (♯‘𝑊) = (♯‘𝑊)) |
4 | | simpl1 1192 |
. . . . . . 7
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → 𝑊 ∈ Word 𝑉) |
5 | 3, 4 | wrdfd 31848 |
. . . . . 6
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → 𝑊:(0..^(♯‘𝑊))⟶𝑉) |
6 | 5 | ffund 6676 |
. . . . 5
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → Fun 𝑊) |
7 | | elfzuz3 13447 |
. . . . . . . . . 10
⊢ (𝑁 ∈
(0...(♯‘𝑊))
→ (♯‘𝑊)
∈ (ℤ≥‘𝑁)) |
8 | 7 | 3ad2ant3 1136 |
. . . . . . . . 9
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (♯‘𝑊) ∈
(ℤ≥‘𝑁)) |
9 | 8 | adantr 482 |
. . . . . . . 8
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → (♯‘𝑊) ∈ (ℤ≥‘𝑁)) |
10 | | fzoss2 13609 |
. . . . . . . 8
⊢
((♯‘𝑊)
∈ (ℤ≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝑊))) |
11 | 9, 10 | syl 17 |
. . . . . . 7
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → (0..^𝑁) ⊆ (0..^(♯‘𝑊))) |
12 | | elfzuz 13446 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈
(ℤ≥‘0)) |
13 | 12 | 3ad2ant2 1135 |
. . . . . . . . . 10
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → 𝑀 ∈
(ℤ≥‘0)) |
14 | 13 | adantr 482 |
. . . . . . . . 9
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → 𝑀 ∈
(ℤ≥‘0)) |
15 | | fzoss1 13608 |
. . . . . . . . 9
⊢ (𝑀 ∈
(ℤ≥‘0) → (𝑀..^𝑁) ⊆ (0..^𝑁)) |
16 | 14, 15 | syl 17 |
. . . . . . . 8
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → (𝑀..^𝑁) ⊆ (0..^𝑁)) |
17 | | simpr 486 |
. . . . . . . . 9
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → 𝑥 ∈ (0..^(𝑁 − 𝑀))) |
18 | | simpl3 1194 |
. . . . . . . . . 10
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → 𝑁 ∈ (0...(♯‘𝑊))) |
19 | 18 | elfzelzd 13451 |
. . . . . . . . 9
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → 𝑁 ∈ ℤ) |
20 | | simpl2 1193 |
. . . . . . . . . 10
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → 𝑀 ∈ (0...𝑁)) |
21 | 20 | elfzelzd 13451 |
. . . . . . . . 9
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → 𝑀 ∈ ℤ) |
22 | | fzoaddel2 13637 |
. . . . . . . . 9
⊢ ((𝑥 ∈ (0..^(𝑁 − 𝑀)) ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑥 + 𝑀) ∈ (𝑀..^𝑁)) |
23 | 17, 19, 21, 22 | syl3anc 1372 |
. . . . . . . 8
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → (𝑥 + 𝑀) ∈ (𝑀..^𝑁)) |
24 | 16, 23 | sseldd 3949 |
. . . . . . 7
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → (𝑥 + 𝑀) ∈ (0..^𝑁)) |
25 | 11, 24 | sseldd 3949 |
. . . . . 6
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → (𝑥 + 𝑀) ∈ (0..^(♯‘𝑊))) |
26 | | wrddm 14418 |
. . . . . . . 8
⊢ (𝑊 ∈ Word 𝑉 → dom 𝑊 = (0..^(♯‘𝑊))) |
27 | 26 | 3ad2ant1 1134 |
. . . . . . 7
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → dom 𝑊 = (0..^(♯‘𝑊))) |
28 | 27 | adantr 482 |
. . . . . 6
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → dom 𝑊 = (0..^(♯‘𝑊))) |
29 | 25, 28 | eleqtrrd 2837 |
. . . . 5
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → (𝑥 + 𝑀) ∈ dom 𝑊) |
30 | | fvelrn 7031 |
. . . . 5
⊢ ((Fun
𝑊 ∧ (𝑥 + 𝑀) ∈ dom 𝑊) → (𝑊‘(𝑥 + 𝑀)) ∈ ran 𝑊) |
31 | 6, 29, 30 | syl2anc 585 |
. . . 4
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → (𝑊‘(𝑥 + 𝑀)) ∈ ran 𝑊) |
32 | 31 | ralrimiva 3140 |
. . 3
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ∀𝑥 ∈ (0..^(𝑁 − 𝑀))(𝑊‘(𝑥 + 𝑀)) ∈ ran 𝑊) |
33 | | eqid 2733 |
. . . 4
⊢ (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑊‘(𝑥 + 𝑀))) = (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑊‘(𝑥 + 𝑀))) |
34 | 33 | rnmptss 7074 |
. . 3
⊢
(∀𝑥 ∈
(0..^(𝑁 − 𝑀))(𝑊‘(𝑥 + 𝑀)) ∈ ran 𝑊 → ran (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑊‘(𝑥 + 𝑀))) ⊆ ran 𝑊) |
35 | 32, 34 | syl 17 |
. 2
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ran (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑊‘(𝑥 + 𝑀))) ⊆ ran 𝑊) |
36 | 2, 35 | eqsstrd 3986 |
1
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝑀, 𝑁⟩) ⊆ ran 𝑊) |