Step | Hyp | Ref
| Expression |
1 | | resubcl 10804 |
. . . 4
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 − 𝑦) ∈ ℝ) |
2 | 1 | adantl 482 |
. . 3
⊢ (((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
∧ (𝑥 ∈ ℝ
∧ 𝑦 ∈ ℝ))
→ (𝑥 − 𝑦) ∈
ℝ) |
3 | | 0red 10497 |
. . . . . . 7
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ 0 ∈ ℝ) |
4 | 3 | s1cld 13805 |
. . . . . 6
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ 〈“0”〉 ∈ Word ℝ) |
5 | | simpl 483 |
. . . . . 6
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ 𝐹 ∈ Word
ℝ) |
6 | | ccatcl 13776 |
. . . . . 6
⊢
((〈“0”〉 ∈ Word ℝ ∧ 𝐹 ∈ Word ℝ) →
(〈“0”〉 ++ 𝐹) ∈ Word ℝ) |
7 | 4, 5, 6 | syl2anc 584 |
. . . . 5
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ (〈“0”〉 ++ 𝐹) ∈ Word ℝ) |
8 | | wrdf 13716 |
. . . . 5
⊢
((〈“0”〉 ++ 𝐹) ∈ Word ℝ →
(〈“0”〉 ++ 𝐹):(0..^(♯‘(〈“0”〉
++ 𝐹)))⟶ℝ) |
9 | 7, 8 | syl 17 |
. . . 4
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ (〈“0”〉 ++ 𝐹):(0..^(♯‘(〈“0”〉
++ 𝐹)))⟶ℝ) |
10 | | ccatlen 13777 |
. . . . . . . . 9
⊢
((〈“0”〉 ∈ Word ℝ ∧ 𝐹 ∈ Word ℝ) →
(♯‘(〈“0”〉 ++ 𝐹)) =
((♯‘〈“0”〉) + (♯‘𝐹))) |
11 | 4, 5, 10 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ (♯‘(〈“0”〉 ++ 𝐹)) =
((♯‘〈“0”〉) + (♯‘𝐹))) |
12 | | s1len 13808 |
. . . . . . . . 9
⊢
(♯‘〈“0”〉) = 1 |
13 | 12 | oveq1i 7033 |
. . . . . . . 8
⊢
((♯‘〈“0”〉) + (♯‘𝐹)) = (1 + (♯‘𝐹)) |
14 | 11, 13 | syl6eq 2849 |
. . . . . . 7
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ (♯‘(〈“0”〉 ++ 𝐹)) = (1 + (♯‘𝐹))) |
15 | | 1cnd 10489 |
. . . . . . . 8
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ 1 ∈ ℂ) |
16 | | wrdfin 13732 |
. . . . . . . . . 10
⊢ (𝐹 ∈ Word ℝ →
𝐹 ∈
Fin) |
17 | | hashcl 13571 |
. . . . . . . . . 10
⊢ (𝐹 ∈ Fin →
(♯‘𝐹) ∈
ℕ0) |
18 | 5, 16, 17 | 3syl 18 |
. . . . . . . . 9
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ (♯‘𝐹)
∈ ℕ0) |
19 | 18 | nn0cnd 11811 |
. . . . . . . 8
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ (♯‘𝐹)
∈ ℂ) |
20 | 15, 19 | addcomd 10695 |
. . . . . . 7
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ (1 + (♯‘𝐹)) = ((♯‘𝐹) + 1)) |
21 | 14, 20 | eqtrd 2833 |
. . . . . 6
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ (♯‘(〈“0”〉 ++ 𝐹)) = ((♯‘𝐹) + 1)) |
22 | 21 | oveq2d 7039 |
. . . . 5
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ (0..^(♯‘(〈“0”〉 ++ 𝐹))) = (0..^((♯‘𝐹) + 1))) |
23 | 22 | feq2d 6375 |
. . . 4
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ ((〈“0”〉 ++ 𝐹):(0..^(♯‘(〈“0”〉
++ 𝐹)))⟶ℝ ↔
(〈“0”〉 ++ 𝐹):(0..^((♯‘𝐹) + 1))⟶ℝ)) |
24 | 9, 23 | mpbid 233 |
. . 3
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ (〈“0”〉 ++ 𝐹):(0..^((♯‘𝐹) + 1))⟶ℝ) |
25 | | remulcl 10475 |
. . . . 5
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ) |
26 | 25 | adantl 482 |
. . . 4
⊢ (((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
∧ (𝑥 ∈ ℝ
∧ 𝑦 ∈ ℝ))
→ (𝑥 · 𝑦) ∈
ℝ) |
27 | | ccatcl 13776 |
. . . . . . 7
⊢ ((𝐹 ∈ Word ℝ ∧
〈“0”〉 ∈ Word ℝ) → (𝐹 ++ 〈“0”〉) ∈ Word
ℝ) |
28 | 4, 27 | syldan 591 |
. . . . . 6
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ (𝐹 ++
〈“0”〉) ∈ Word ℝ) |
29 | | wrdf 13716 |
. . . . . 6
⊢ ((𝐹 ++ 〈“0”〉)
∈ Word ℝ → (𝐹 ++
〈“0”〉):(0..^(♯‘(𝐹 ++
〈“0”〉)))⟶ℝ) |
30 | 28, 29 | syl 17 |
. . . . 5
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ (𝐹 ++
〈“0”〉):(0..^(♯‘(𝐹 ++
〈“0”〉)))⟶ℝ) |
31 | | ccatlen 13777 |
. . . . . . . . 9
⊢ ((𝐹 ∈ Word ℝ ∧
〈“0”〉 ∈ Word ℝ) → (♯‘(𝐹 ++
〈“0”〉)) = ((♯‘𝐹) +
(♯‘〈“0”〉))) |
32 | 4, 31 | syldan 591 |
. . . . . . . 8
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ (♯‘(𝐹 ++
〈“0”〉)) = ((♯‘𝐹) +
(♯‘〈“0”〉))) |
33 | 12 | oveq2i 7034 |
. . . . . . . 8
⊢
((♯‘𝐹) +
(♯‘〈“0”〉)) = ((♯‘𝐹) + 1) |
34 | 32, 33 | syl6eq 2849 |
. . . . . . 7
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ (♯‘(𝐹 ++
〈“0”〉)) = ((♯‘𝐹) + 1)) |
35 | 34 | oveq2d 7039 |
. . . . . 6
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ (0..^(♯‘(𝐹 ++ 〈“0”〉))) =
(0..^((♯‘𝐹) +
1))) |
36 | 35 | feq2d 6375 |
. . . . 5
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ ((𝐹 ++
〈“0”〉):(0..^(♯‘(𝐹 ++
〈“0”〉)))⟶ℝ ↔ (𝐹 ++
〈“0”〉):(0..^((♯‘𝐹) + 1))⟶ℝ)) |
37 | 30, 36 | mpbid 233 |
. . . 4
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ (𝐹 ++
〈“0”〉):(0..^((♯‘𝐹) + 1))⟶ℝ) |
38 | | ovexd 7057 |
. . . 4
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ (0..^((♯‘𝐹) + 1)) ∈ V) |
39 | | simpr 485 |
. . . . 5
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ 𝐶 ∈
ℝ+) |
40 | 39 | rpred 12285 |
. . . 4
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ 𝐶 ∈
ℝ) |
41 | 26, 37, 38, 40 | ofcf 30975 |
. . 3
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ ((𝐹 ++
〈“0”〉)∘𝑓/𝑐 ·
𝐶):(0..^((♯‘𝐹) + 1))⟶ℝ) |
42 | | inidm 4121 |
. . 3
⊢
((0..^((♯‘𝐹) + 1)) ∩ (0..^((♯‘𝐹) + 1))) =
(0..^((♯‘𝐹) +
1)) |
43 | 2, 24, 41, 38, 38, 42 | off 7289 |
. 2
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ ((〈“0”〉 ++ 𝐹) ∘𝑓 −
((𝐹 ++
〈“0”〉)∘𝑓/𝑐 ·
𝐶)):(0..^((♯‘𝐹) + 1))⟶ℝ) |
44 | | signs.h |
. . 3
⊢ 𝐻 = ((〈“0”〉
++ 𝐹)
∘𝑓 − ((𝐹 ++
〈“0”〉)∘𝑓/𝑐 ·
𝐶)) |
45 | 44 | feq1i 6380 |
. 2
⊢ (𝐻:(0..^((♯‘𝐹) + 1))⟶ℝ ↔
((〈“0”〉 ++ 𝐹) ∘𝑓 −
((𝐹 ++
〈“0”〉)∘𝑓/𝑐 ·
𝐶)):(0..^((♯‘𝐹) + 1))⟶ℝ) |
46 | 43, 45 | sylibr 235 |
1
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+)
→ 𝐻:(0..^((♯‘𝐹) + 1))⟶ℝ) |