Proof of Theorem signsvfn
Step | Hyp | Ref
| Expression |
1 | | simpl 474 |
. . . . . . 7
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → 𝐹 ∈
(Word ℝ ∖ {∅})) |
2 | 1 | eldifad 3727 |
. . . . . 6
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → 𝐹 ∈
Word ℝ) |
3 | | simpr 479 |
. . . . . . 7
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → 𝐾 ∈
ℝ) |
4 | 3 | s1cld 13573 |
. . . . . 6
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → 〈“𝐾”〉 ∈ Word
ℝ) |
5 | | ccatcl 13546 |
. . . . . 6
⊢ ((𝐹 ∈ Word ℝ ∧
〈“𝐾”〉
∈ Word ℝ) → (𝐹 ++ 〈“𝐾”〉) ∈ Word
ℝ) |
6 | 2, 4, 5 | syl2anc 696 |
. . . . 5
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → (𝐹 ++
〈“𝐾”〉) ∈ Word
ℝ) |
7 | | signsv.p |
. . . . . 6
⊢ ⨣ =
(𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦
if(𝑏 = 0, 𝑎, 𝑏)) |
8 | | signsv.w |
. . . . . 6
⊢ 𝑊 = {〈(Base‘ndx), {-1,
0, 1}〉, 〈(+g‘ndx), ⨣
〉} |
9 | | signsv.t |
. . . . . 6
⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈
(0..^(♯‘𝑓))
↦ (𝑊
Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖)))))) |
10 | | signsv.v |
. . . . . 6
⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈
(1..^(♯‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0)) |
11 | 7, 8, 9, 10 | signsvvfval 30964 |
. . . . 5
⊢ ((𝐹 ++ 〈“𝐾”〉) ∈ Word
ℝ → (𝑉‘(𝐹 ++ 〈“𝐾”〉)) = Σ𝑗 ∈ (1..^(♯‘(𝐹 ++ 〈“𝐾”〉)))if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)), 1, 0)) |
12 | 6, 11 | 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)) |
13 | | ccatlen 13547 |
. . . . . . . 8
⊢ ((𝐹 ∈ Word ℝ ∧
〈“𝐾”〉
∈ Word ℝ) → (♯‘(𝐹 ++ 〈“𝐾”〉)) = ((♯‘𝐹) +
(♯‘〈“𝐾”〉))) |
14 | 2, 4, 13 | syl2anc 696 |
. . . . . . 7
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → (♯‘(𝐹 ++ 〈“𝐾”〉)) = ((♯‘𝐹) +
(♯‘〈“𝐾”〉))) |
15 | | s1len 13576 |
. . . . . . . 8
⊢
(♯‘〈“𝐾”〉) = 1 |
16 | 15 | oveq2i 6824 |
. . . . . . 7
⊢
((♯‘𝐹) +
(♯‘〈“𝐾”〉)) = ((♯‘𝐹) + 1) |
17 | 14, 16 | syl6eq 2810 |
. . . . . 6
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → (♯‘(𝐹 ++ 〈“𝐾”〉)) = ((♯‘𝐹) + 1)) |
18 | 17 | oveq2d 6829 |
. . . . 5
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → (1..^(♯‘(𝐹 ++ 〈“𝐾”〉))) =
(1..^((♯‘𝐹) +
1))) |
19 | 18 | sumeq1d 14630 |
. . . 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)) |
20 | | eldifsn 4462 |
. . . . . . . 8
⊢ (𝐹 ∈ (Word ℝ ∖
{∅}) ↔ (𝐹 ∈
Word ℝ ∧ 𝐹 ≠
∅)) |
21 | | lennncl 13511 |
. . . . . . . 8
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅) →
(♯‘𝐹) ∈
ℕ) |
22 | 20, 21 | sylbi 207 |
. . . . . . 7
⊢ (𝐹 ∈ (Word ℝ ∖
{∅}) → (♯‘𝐹) ∈ ℕ) |
23 | | nnuz 11916 |
. . . . . . 7
⊢ ℕ =
(ℤ≥‘1) |
24 | 22, 23 | syl6eleq 2849 |
. . . . . 6
⊢ (𝐹 ∈ (Word ℝ ∖
{∅}) → (♯‘𝐹) ∈
(ℤ≥‘1)) |
25 | 24 | adantr 472 |
. . . . 5
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → (♯‘𝐹) ∈
(ℤ≥‘1)) |
26 | | 1cnd 10248 |
. . . . . 6
⊢ ((((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1...(♯‘𝐹)))
∧ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1))) → 1 ∈
ℂ) |
27 | | 0cnd 10225 |
. . . . . 6
⊢ ((((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1...(♯‘𝐹)))
∧ ¬ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1))) → 0 ∈
ℂ) |
28 | 26, 27 | ifclda 4264 |
. . . . 5
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1...(♯‘𝐹)))
→ if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)), 1, 0) ∈
ℂ) |
29 | | fveq2 6352 |
. . . . . . 7
⊢ (𝑗 = (♯‘𝐹) → ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) = ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(♯‘𝐹))) |
30 | | oveq1 6820 |
. . . . . . . 8
⊢ (𝑗 = (♯‘𝐹) → (𝑗 − 1) = ((♯‘𝐹) − 1)) |
31 | 30 | fveq2d 6356 |
. . . . . . 7
⊢ (𝑗 = (♯‘𝐹) → ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)) = ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘((♯‘𝐹) − 1))) |
32 | 29, 31 | neeq12d 2993 |
. . . . . 6
⊢ (𝑗 = (♯‘𝐹) → (((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)) ↔ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘((♯‘𝐹) − 1)))) |
33 | 32 | ifbid 4252 |
. . . . 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)) |
34 | 25, 28, 33 | fzosump1 14680 |
. . . 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))) |
35 | 12, 19, 34 | 3eqtrd 2798 |
. . 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))) |
36 | 35 | adantlr 753 |
. 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))) |
37 | 2 | adantr 472 |
. . . . . . . . 9
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ 𝐹 ∈ Word
ℝ) |
38 | 3 | adantr 472 |
. . . . . . . . 9
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ 𝐾 ∈
ℝ) |
39 | | fzo0ss1 12692 |
. . . . . . . . . . 11
⊢
(1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹)) |
40 | 39 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))) |
41 | 40 | sselda 3744 |
. . . . . . . . 9
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ 𝑗 ∈
(0..^(♯‘𝐹))) |
42 | 7, 8, 9, 10 | signstfvp 30957 |
. . . . . . . . 9
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑗 ∈
(0..^(♯‘𝐹)))
→ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) = ((𝑇‘𝐹)‘𝑗)) |
43 | 37, 38, 41, 42 | syl3anc 1477 |
. . . . . . . 8
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) = ((𝑇‘𝐹)‘𝑗)) |
44 | | elfzoel2 12663 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈
(1..^(♯‘𝐹))
→ (♯‘𝐹)
∈ ℤ) |
45 | 44 | adantl 473 |
. . . . . . . . . . . 12
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ (♯‘𝐹)
∈ ℤ) |
46 | | 1nn0 11500 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℕ0 |
47 | | eluzmn 11886 |
. . . . . . . . . . . 12
⊢
(((♯‘𝐹)
∈ ℤ ∧ 1 ∈ ℕ0) → (♯‘𝐹) ∈
(ℤ≥‘((♯‘𝐹) − 1))) |
48 | 45, 46, 47 | sylancl 697 |
. . . . . . . . . . 11
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ (♯‘𝐹)
∈ (ℤ≥‘((♯‘𝐹) − 1))) |
49 | | fzoss2 12690 |
. . . . . . . . . . 11
⊢
((♯‘𝐹)
∈ (ℤ≥‘((♯‘𝐹) − 1)) →
(0..^((♯‘𝐹)
− 1)) ⊆ (0..^(♯‘𝐹))) |
50 | 48, 49 | syl 17 |
. . . . . . . . . 10
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ (0..^((♯‘𝐹) − 1)) ⊆
(0..^(♯‘𝐹))) |
51 | | simpr 479 |
. . . . . . . . . . 11
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ 𝑗 ∈
(1..^(♯‘𝐹))) |
52 | | elfzoelz 12664 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈
(1..^(♯‘𝐹))
→ 𝑗 ∈
ℤ) |
53 | 52 | adantl 473 |
. . . . . . . . . . . 12
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ 𝑗 ∈
ℤ) |
54 | | elfzom1b 12761 |
. . . . . . . . . . . 12
⊢ ((𝑗 ∈ ℤ ∧
(♯‘𝐹) ∈
ℤ) → (𝑗 ∈
(1..^(♯‘𝐹))
↔ (𝑗 − 1) ∈
(0..^((♯‘𝐹)
− 1)))) |
55 | 53, 45, 54 | syl2anc 696 |
. . . . . . . . . . 11
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ (𝑗 ∈
(1..^(♯‘𝐹))
↔ (𝑗 − 1) ∈
(0..^((♯‘𝐹)
− 1)))) |
56 | 51, 55 | mpbid 222 |
. . . . . . . . . 10
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ (𝑗 − 1) ∈
(0..^((♯‘𝐹)
− 1))) |
57 | 50, 56 | sseldd 3745 |
. . . . . . . . 9
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ (𝑗 − 1) ∈
(0..^(♯‘𝐹))) |
58 | 7, 8, 9, 10 | signstfvp 30957 |
. . . . . . . . 9
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ (𝑗 − 1) ∈
(0..^(♯‘𝐹)))
→ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)) = ((𝑇‘𝐹)‘(𝑗 − 1))) |
59 | 37, 38, 57, 58 | syl3anc 1477 |
. . . . . . . 8
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) ∧ 𝑗 ∈
(1..^(♯‘𝐹)))
→ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)) = ((𝑇‘𝐹)‘(𝑗 − 1))) |
60 | 43, 59 | neeq12d 2993 |
. . . . . . 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)))) |
61 | 60 | ifbid 4252 |
. . . . . 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)) |
62 | 61 | sumeq2dv 14632 |
. . . . 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)) |
63 | 7, 8, 9, 10 | signsvvfval 30964 |
. . . . . 6
⊢ (𝐹 ∈ Word ℝ →
(𝑉‘𝐹) = Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0)) |
64 | 2, 63 | syl 17 |
. . . . 5
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → (𝑉‘𝐹) = Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0)) |
65 | 62, 64 | eqtr4d 2797 |
. . . 4
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → Σ𝑗
∈ (1..^(♯‘𝐹))if(((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘𝑗) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(𝑗 − 1)), 1, 0) = (𝑉‘𝐹)) |
66 | 65 | adantlr 753 |
. . 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)) |
67 | 7, 8, 9, 10 | signstfvn 30955 |
. . . . . . 7
⊢ ((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ 𝐾 ∈
ℝ) → ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(♯‘𝐹)) = (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾))) |
68 | 67 | adantlr 753 |
. . . . . 6
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(♯‘𝐹)) = (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾))) |
69 | 2 | adantlr 753 |
. . . . . . 7
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → 𝐹 ∈ Word ℝ) |
70 | | simpr 479 |
. . . . . . 7
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → 𝐾 ∈ ℝ) |
71 | 22 | ad2antrr 764 |
. . . . . . . 8
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈
ℕ) |
72 | | fzo0end 12754 |
. . . . . . . 8
⊢
((♯‘𝐹)
∈ ℕ → ((♯‘𝐹) − 1) ∈
(0..^(♯‘𝐹))) |
73 | 71, 72 | syl 17 |
. . . . . . 7
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) →
((♯‘𝐹) −
1) ∈ (0..^(♯‘𝐹))) |
74 | 7, 8, 9, 10 | signstfvp 30957 |
. . . . . . 7
⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧
((♯‘𝐹) −
1) ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘((♯‘𝐹) − 1)) = ((𝑇‘𝐹)‘((♯‘𝐹) − 1))) |
75 | 69, 70, 73, 74 | syl3anc 1477 |
. . . . . 6
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘((♯‘𝐹) − 1)) = ((𝑇‘𝐹)‘((♯‘𝐹) − 1))) |
76 | 68, 75 | neeq12d 2993 |
. . . . 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)))) |
77 | 7, 8, 9, 10 | signstfvcl 30959 |
. . . . . . 7
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧
((♯‘𝐹) −
1) ∈ (0..^(♯‘𝐹))) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ {-1,
1}) |
78 | 73, 77 | syldan 488 |
. . . . . 6
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ {-1,
1}) |
79 | 70 | rexrd 10281 |
. . . . . . 7
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → 𝐾 ∈
ℝ*) |
80 | | sgncl 30909 |
. . . . . . 7
⊢ (𝐾 ∈ ℝ*
→ (sgn‘𝐾) ∈
{-1, 0, 1}) |
81 | 79, 80 | syl 17 |
. . . . . 6
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (sgn‘𝐾) ∈ {-1, 0,
1}) |
82 | 7, 8 | signswch 30947 |
. . . . . 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)) |
83 | 78, 81, 82 | syl2anc 696 |
. . . . 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)) |
84 | | sgnsgn 30919 |
. . . . . . . . 9
⊢ (𝐾 ∈ ℝ*
→ (sgn‘(sgn‘𝐾)) = (sgn‘𝐾)) |
85 | 79, 84 | syl 17 |
. . . . . . . 8
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) →
(sgn‘(sgn‘𝐾)) =
(sgn‘𝐾)) |
86 | 85 | oveq2d 6829 |
. . . . . . 7
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) ·
(sgn‘(sgn‘𝐾)))
= ((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) · (sgn‘𝐾))) |
87 | 86 | breq1d 4814 |
. . . . . 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)) |
88 | | neg1rr 11317 |
. . . . . . . . 9
⊢ -1 ∈
ℝ |
89 | | 1re 10231 |
. . . . . . . . 9
⊢ 1 ∈
ℝ |
90 | | prssi 4498 |
. . . . . . . . 9
⊢ ((-1
∈ ℝ ∧ 1 ∈ ℝ) → {-1, 1} ⊆
ℝ) |
91 | 88, 89, 90 | mp2an 710 |
. . . . . . . 8
⊢ {-1, 1}
⊆ ℝ |
92 | 91, 78 | sseldi 3742 |
. . . . . . 7
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈
ℝ) |
93 | | sgnclre 30910 |
. . . . . . . 8
⊢ (𝐾 ∈ ℝ →
(sgn‘𝐾) ∈
ℝ) |
94 | 93 | adantl 473 |
. . . . . . 7
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (sgn‘𝐾) ∈
ℝ) |
95 | | sgnmulsgn 30920 |
. . . . . . 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)) |
96 | 92, 94, 95 | syl2anc 696 |
. . . . . 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)) |
97 | | sgnmulsgn 30920 |
. . . . . . 7
⊢ ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ ℝ ∧ 𝐾 ∈ ℝ) →
((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0 ↔
((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) ·
(sgn‘𝐾)) <
0)) |
98 | 92, 70, 97 | syl2anc 696 |
. . . . . 6
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0 ↔ ((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) · (sgn‘𝐾)) < 0)) |
99 | 87, 96, 98 | 3bitr4d 300 |
. . . . 5
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · (sgn‘𝐾)) < 0 ↔ (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0)) |
100 | 76, 83, 99 | 3bitrd 294 |
. . . 4
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ 〈“𝐾”〉))‘((♯‘𝐹) − 1)) ↔ (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0)) |
101 | 100 | ifbid 4252 |
. . 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)) |
102 | 66, 101 | oveq12d 6831 |
. 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))) |
103 | 36, 102 | eqtrd 2794 |
1
⊢ (((𝐹 ∈ (Word ℝ ∖
{∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (𝑉‘(𝐹 ++ 〈“𝐾”〉)) = ((𝑉‘𝐹) + if((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0, 1, 0))) |