Proof of Theorem signsvfn
Step | Hyp | Ref
| Expression |
1 | | eldifi 4062 |
. . . . . 6
⊢ (𝐹 ∈ (Word ℝ ∖
{∅}) → 𝐹 ∈
Word ℝ) |
2 | | s1cl 14316 |
. . . . . 6
⊢ (𝐾 ∈ ℝ →
〈“𝐾”〉
∈ Word ℝ) |
3 | | ccatcl 14286 |
. . . . . 6
⊢ ((𝐹 ∈ Word ℝ ∧
〈“𝐾”〉
∈ Word ℝ) → (𝐹 ++ 〈“𝐾”〉) ∈ Word
ℝ) |
4 | 1, 2, 3 | syl2an 596 |
. . . . 5
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → (𝐹 ++
〈“𝐾”〉) ∈ Word
ℝ) |
5 | | signsv.p |
. . . . . 6
⊢ ⨣ =
(𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦
if(𝑏 = 0, 𝑎, 𝑏)) |
6 | | signsv.w |
. . . . . 6
⊢ 𝑊 = {〈(Base‘ndx), {-1,
0, 1}〉, 〈(+g‘ndx), ⨣
〉} |
7 | | signsv.t |
. . . . . 6
⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈
(0..^(♯‘𝑓))
↦ (𝑊
Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖)))))) |
8 | | signsv.v |
. . . . . 6
⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈
(1..^(♯‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0)) |
9 | 5, 6, 7, 8 | signsvvfval 32566 |
. . . . 5
⊢ ((𝐹 ++ 〈“𝐾”〉) ∈ Word
ℝ → (𝑉‘(𝐹 ++ 〈“𝐾”〉)) = Σ𝑗 ∈ (1..^(♯‘(𝐹 ++ 〈“𝐾”〉)))if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)), 1, 0)) |
10 | 4, 9 | syl 17 |
. . . 4
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → (𝑉‘(𝐹 ++ 〈“𝐾”〉)) = Σ𝑗 ∈ (1..^(♯‘(𝐹 ++ 〈“𝐾”〉)))if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)), 1, 0)) |
11 | | ccatlen 14287 |
. . . . . . . 8
⊢ ((𝐹 ∈ Word ℝ ∧
〈“𝐾”〉
∈ Word ℝ) → (♯‘(𝐹 ++ 〈“𝐾”〉)) = ((♯‘𝐹) +
(♯‘〈“𝐾”〉))) |
12 | 1, 2, 11 | syl2an 596 |
. . . . . . 7
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → (♯‘(𝐹 ++ 〈“𝐾”〉)) = ((♯‘𝐹) +
(♯‘〈“𝐾”〉))) |
13 | | s1len 14320 |
. . . . . . . 8
⊢
(♯‘〈“𝐾”〉) = 1 |
14 | 13 | oveq2i 7295 |
. . . . . . 7
⊢
((♯‘𝐹) +
(♯‘〈“𝐾”〉)) = ((♯‘𝐹) + 1) |
15 | 12, 14 | eqtrdi 2795 |
. . . . . 6
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → (♯‘(𝐹 ++ 〈“𝐾”〉)) = ((♯‘𝐹) + 1)) |
16 | 15 | oveq2d 7300 |
. . . . 5
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → (1..^(♯‘(𝐹 ++ 〈“𝐾”〉))) =
(1..^((♯‘𝐹) +
1))) |
17 | 16 | sumeq1d 15422 |
. . . 4
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → Σ𝑗
∈ (1..^(♯‘(𝐹 ++ 〈“𝐾”〉)))if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)), 1, 0) = Σ𝑗 ∈
(1..^((♯‘𝐹) +
1))if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)), 1, 0)) |
18 | | eldifsn 4721 |
. . . . . . . 8
⊢ (𝐹 ∈ (Word ℝ ∖
{∅}) ↔ (𝐹 ∈
Word ℝ ∧ 𝐹 ≠
∅)) |
19 | | lennncl 14246 |
. . . . . . . 8
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅) →
(♯‘𝐹) ∈
ℕ) |
20 | 18, 19 | sylbi 216 |
. . . . . . 7
⊢ (𝐹 ∈ (Word ℝ ∖
{∅}) → (♯‘𝐹) ∈ ℕ) |
21 | | nnuz 12630 |
. . . . . . 7
⊢ ℕ =
(ℤ≥‘1) |
22 | 20, 21 | eleqtrdi 2850 |
. . . . . 6
⊢ (𝐹 ∈ (Word ℝ ∖
{∅}) → (♯‘𝐹) ∈
(ℤ≥‘1)) |
23 | 22 | adantr 481 |
. . . . 5
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → (♯‘𝐹) ∈
(ℤ≥‘1)) |
24 | | 1cnd 10979 |
. . . . . 6
⊢ ((((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1...(♯‘𝐹)))
∧ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1))) → 1 ∈
ℂ) |
25 | | 0cnd 10977 |
. . . . . 6
⊢ ((((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1...(♯‘𝐹)))
∧ ¬ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1))) → 0 ∈
ℂ) |
26 | 24, 25 | ifclda 4495 |
. . . . 5
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1...(♯‘𝐹)))
→ if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)), 1, 0) ∈
ℂ) |
27 | | fveq2 6783 |
. . . . . . 7
⊢ (𝑗 = (♯‘𝐹) → ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) = ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(♯‘𝐹))) |
28 | | fvoveq1 7307 |
. . . . . . 7
⊢ (𝑗 = (♯‘𝐹) → ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)) = ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘((♯‘𝐹) − 1))) |
29 | 27, 28 | neeq12d 3006 |
. . . . . 6
⊢ (𝑗 = (♯‘𝐹) → (((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)) ↔ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘((♯‘𝐹) − 1)))) |
30 | 29 | ifbid 4483 |
. . . . 5
⊢ (𝑗 = (♯‘𝐹) → if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)), 1, 0) = if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘((♯‘𝐹) − 1)), 1,
0)) |
31 | 23, 26, 30 | fzosump1 15473 |
. . . 4
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → Σ𝑗
∈ (1..^((♯‘𝐹) + 1))if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)), 1, 0) = (Σ𝑗 ∈
(1..^(♯‘𝐹))if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)), 1, 0) + if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘((♯‘𝐹) − 1)), 1,
0))) |
32 | 10, 17, 31 | 3eqtrd 2783 |
. . 3
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → (𝑉‘(𝐹 ++ 〈“𝐾”〉)) = (Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)), 1, 0) + if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘((♯‘𝐹) − 1)), 1,
0))) |
33 | 32 | adantlr 712 |
. 2
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (𝑉‘(𝐹 ++ 〈“𝐾”〉)) = (Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)), 1, 0) + if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘((♯‘𝐹) − 1)), 1,
0))) |
34 | | simpl 483 |
. . . . . . . . . . 11
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → 𝐹 ∈
(Word ℝ ∖ {∅})) |
35 | 34 | eldifad 3900 |
. . . . . . . . . 10
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → 𝐹 ∈
Word ℝ) |
36 | 35 | adantr 481 |
. . . . . . . . 9
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ 𝐹 ∈ Word
ℝ) |
37 | | simplr 766 |
. . . . . . . . 9
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ 𝐾 ∈
ℝ) |
38 | | fzo0ss1 13426 |
. . . . . . . . . . 11
⊢
(1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹)) |
39 | 38 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))) |
40 | 39 | sselda 3922 |
. . . . . . . . 9
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ 𝑗 ∈
(0..^(♯‘𝐹))) |
41 | 5, 6, 7, 8 | signstfvp 32559 |
. . . . . . . . 9
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑗 ∈
(0..^(♯‘𝐹)))
→ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) = ((𝑇‘𝐹)‘𝑗)) |
42 | 36, 37, 40, 41 | syl3anc 1370 |
. . . . . . . 8
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) = ((𝑇‘𝐹)‘𝑗)) |
43 | | elfzoel2 13395 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈
(1..^(♯‘𝐹))
→ (♯‘𝐹)
∈ ℤ) |
44 | 43 | adantl 482 |
. . . . . . . . . . . 12
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ (♯‘𝐹)
∈ ℤ) |
45 | | 1nn0 12258 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℕ0 |
46 | | eluzmn 12598 |
. . . . . . . . . . . 12
⊢
(((♯‘𝐹)
∈ ℤ ∧ 1 ∈ ℕ0) → (♯‘𝐹) ∈
(ℤ≥‘((♯‘𝐹) − 1))) |
47 | 44, 45, 46 | sylancl 586 |
. . . . . . . . . . 11
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ (♯‘𝐹)
∈ (ℤ≥‘((♯‘𝐹) − 1))) |
48 | | fzoss2 13424 |
. . . . . . . . . . 11
⊢
((♯‘𝐹)
∈ (ℤ≥‘((♯‘𝐹) − 1)) →
(0..^((♯‘𝐹)
− 1)) ⊆ (0..^(♯‘𝐹))) |
49 | 47, 48 | syl 17 |
. . . . . . . . . 10
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ (0..^((♯‘𝐹) − 1)) ⊆
(0..^(♯‘𝐹))) |
50 | | elfzo1elm1fzo0 13497 |
. . . . . . . . . . 11
⊢ (𝑗 ∈
(1..^(♯‘𝐹))
→ (𝑗 − 1) ∈
(0..^((♯‘𝐹)
− 1))) |
51 | 50 | adantl 482 |
. . . . . . . . . 10
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ (𝑗 − 1) ∈
(0..^((♯‘𝐹)
− 1))) |
52 | 49, 51 | sseldd 3923 |
. . . . . . . . 9
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ (𝑗 − 1) ∈
(0..^(♯‘𝐹))) |
53 | 5, 6, 7, 8 | signstfvp 32559 |
. . . . . . . . 9
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ (𝑗 − 1) ∈
(0..^(♯‘𝐹)))
→ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)) = ((𝑇‘𝐹)‘(𝑗 − 1))) |
54 | 36, 37, 52, 53 | syl3anc 1370 |
. . . . . . . 8
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)) = ((𝑇‘𝐹)‘(𝑗 − 1))) |
55 | 42, 54 | neeq12d 3006 |
. . . . . . 7
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ (((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)) ↔ ((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)))) |
56 | 55 | ifbid 4483 |
. . . . . 6
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)), 1, 0) = if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0)) |
57 | 56 | sumeq2dv 15424 |
. . . . 5
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → Σ𝑗
∈ (1..^(♯‘𝐹))if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)), 1, 0) = Σ𝑗 ∈
(1..^(♯‘𝐹))if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0)) |
58 | 5, 6, 7, 8 | signsvvfval 32566 |
. . . . . 6
⊢ (𝐹 ∈ Word ℝ →
(𝑉‘𝐹) = Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0)) |
59 | 35, 58 | syl 17 |
. . . . 5
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → (𝑉‘𝐹) = Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0)) |
60 | 57, 59 | eqtr4d 2782 |
. . . 4
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → Σ𝑗
∈ (1..^(♯‘𝐹))if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)), 1, 0) = (𝑉‘𝐹)) |
61 | 60 | adantlr 712 |
. . 3
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → Σ𝑗 ∈
(1..^(♯‘𝐹))if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)), 1, 0) = (𝑉‘𝐹)) |
62 | 5, 6, 7, 8 | signstfvn 32557 |
. . . . . . 7
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(♯‘𝐹)) = (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾))) |
63 | 62 | adantlr 712 |
. . . . . 6
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(♯‘𝐹)) = (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾))) |
64 | 35 | adantlr 712 |
. . . . . . 7
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → 𝐹 ∈ Word ℝ) |
65 | | simpr 485 |
. . . . . . 7
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → 𝐾 ∈ ℝ) |
66 | | fzo0end 13488 |
. . . . . . . . 9
⊢
((♯‘𝐹)
∈ ℕ → ((♯‘𝐹) − 1) ∈
(0..^(♯‘𝐹))) |
67 | 20, 66 | syl 17 |
. . . . . . . 8
⊢ (𝐹 ∈ (Word ℝ ∖
{∅}) → ((♯‘𝐹) − 1) ∈
(0..^(♯‘𝐹))) |
68 | 67 | ad2antrr 723 |
. . . . . . 7
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) →
((♯‘𝐹) −
1) ∈ (0..^(♯‘𝐹))) |
69 | 5, 6, 7, 8 | signstfvp 32559 |
. . . . . . 7
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧
((♯‘𝐹) −
1) ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘((♯‘𝐹) − 1)) = ((𝑇‘𝐹)‘((♯‘𝐹) − 1))) |
70 | 64, 65, 68, 69 | syl3anc 1370 |
. . . . . 6
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘((♯‘𝐹) − 1)) = ((𝑇‘𝐹)‘((♯‘𝐹) − 1))) |
71 | 63, 70 | neeq12d 3006 |
. . . . 5
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘((♯‘𝐹) − 1)) ↔ (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾)) ≠ ((𝑇‘𝐹)‘((♯‘𝐹) − 1)))) |
72 | 5, 6, 7, 8 | signstfvcl 32561 |
. . . . . . 7
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧
((♯‘𝐹) −
1) ∈ (0..^(♯‘𝐹))) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ {-1,
1}) |
73 | 68, 72 | syldan 591 |
. . . . . 6
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ {-1,
1}) |
74 | | rexr 11030 |
. . . . . . . 8
⊢ (𝐾 ∈ ℝ → 𝐾 ∈
ℝ*) |
75 | | sgncl 32514 |
. . . . . . . 8
⊢ (𝐾 ∈ ℝ*
→ (sgn‘𝐾) ∈
{-1, 0, 1}) |
76 | 74, 75 | syl 17 |
. . . . . . 7
⊢ (𝐾 ∈ ℝ →
(sgn‘𝐾) ∈ {-1,
0, 1}) |
77 | 76 | adantl 482 |
. . . . . 6
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (sgn‘𝐾) ∈ {-1, 0,
1}) |
78 | 5, 6 | signswch 32549 |
. . . . . 6
⊢ ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ {-1, 1} ∧
(sgn‘𝐾) ∈ {-1,
0, 1}) → ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾)) ≠ ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ↔ (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · (sgn‘𝐾)) < 0)) |
79 | 73, 77, 78 | syl2anc 584 |
. . . . 5
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾)) ≠ ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ↔ (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · (sgn‘𝐾)) < 0)) |
80 | 65 | rexrd 11034 |
. . . . . . . . 9
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → 𝐾 ∈
ℝ*) |
81 | | sgnsgn 32524 |
. . . . . . . . 9
⊢ (𝐾 ∈ ℝ*
→ (sgn‘(sgn‘𝐾)) = (sgn‘𝐾)) |
82 | 80, 81 | syl 17 |
. . . . . . . 8
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) →
(sgn‘(sgn‘𝐾)) =
(sgn‘𝐾)) |
83 | 82 | oveq2d 7300 |
. . . . . . 7
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) ·
(sgn‘(sgn‘𝐾)))
= ((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) · (sgn‘𝐾))) |
84 | 83 | breq1d 5085 |
. . . . . 6
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) ·
(sgn‘(sgn‘𝐾)))
< 0 ↔ ((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) · (sgn‘𝐾)) < 0)) |
85 | | neg1rr 12097 |
. . . . . . . . 9
⊢ -1 ∈
ℝ |
86 | | 1re 10984 |
. . . . . . . . 9
⊢ 1 ∈
ℝ |
87 | | prssi 4755 |
. . . . . . . . 9
⊢ ((-1
∈ ℝ ∧ 1 ∈ ℝ) → {-1, 1} ⊆
ℝ) |
88 | 85, 86, 87 | mp2an 689 |
. . . . . . . 8
⊢ {-1, 1}
⊆ ℝ |
89 | 88, 73 | sselid 3920 |
. . . . . . 7
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈
ℝ) |
90 | | sgnclre 32515 |
. . . . . . . 8
⊢ (𝐾 ∈ ℝ →
(sgn‘𝐾) ∈
ℝ) |
91 | 90 | adantl 482 |
. . . . . . 7
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (sgn‘𝐾) ∈
ℝ) |
92 | | sgnmulsgn 32525 |
. . . . . . 7
⊢ ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ ℝ ∧
(sgn‘𝐾) ∈
ℝ) → ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · (sgn‘𝐾)) < 0 ↔
((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) ·
(sgn‘(sgn‘𝐾)))
< 0)) |
93 | 89, 91, 92 | syl2anc 584 |
. . . . . 6
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · (sgn‘𝐾)) < 0 ↔
((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) ·
(sgn‘(sgn‘𝐾)))
< 0)) |
94 | | sgnmulsgn 32525 |
. . . . . . 7
⊢ ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ ℝ ∧ 𝐾 ∈ ℝ) →
((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0 ↔
((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) ·
(sgn‘𝐾)) <
0)) |
95 | 89, 94 | sylancom 588 |
. . . . . 6
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0 ↔ ((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) · (sgn‘𝐾)) < 0)) |
96 | 84, 93, 95 | 3bitr4d 311 |
. . . . 5
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · (sgn‘𝐾)) < 0 ↔ (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0)) |
97 | 71, 79, 96 | 3bitrd 305 |
. . . 4
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘((♯‘𝐹) − 1)) ↔ (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0)) |
98 | 97 | ifbid 4483 |
. . 3
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘((♯‘𝐹) − 1)), 1, 0) =
if((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0, 1,
0)) |
99 | 61, 98 | oveq12d 7302 |
. 2
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (Σ𝑗 ∈
(1..^(♯‘𝐹))if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)), 1, 0) + if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘((♯‘𝐹) − 1)), 1, 0)) = ((𝑉‘𝐹) + if((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0, 1, 0))) |
100 | 33, 99 | eqtrd 2779 |
1
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (𝑉‘(𝐹 ++ 〈“𝐾”〉)) = ((𝑉‘𝐹) + if((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0, 1, 0))) |