| Step | Hyp | Ref
| Expression |
| 1 | | signsv.p |
. . . . . . . 8
⊢ ⨣ =
(𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦
if(𝑏 = 0, 𝑎, 𝑏)) |
| 2 | | signsv.w |
. . . . . . . 8
⊢ 𝑊 = {〈(Base‘ndx), {-1,
0, 1}〉, 〈(+g‘ndx), ⨣
〉} |
| 3 | | signsv.t |
. . . . . . . 8
⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈
(0..^(♯‘𝑓))
↦ (𝑊
Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖)))))) |
| 4 | | signsv.v |
. . . . . . . 8
⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈
(1..^(♯‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0)) |
| 5 | 1, 2, 3, 4 | signstf 34581 |
. . . . . . 7
⊢ (𝐹 ∈ Word ℝ →
(𝑇‘𝐹) ∈ Word ℝ) |
| 6 | | wrdf 14557 |
. . . . . . 7
⊢ ((𝑇‘𝐹) ∈ Word ℝ → (𝑇‘𝐹):(0..^(♯‘(𝑇‘𝐹)))⟶ℝ) |
| 7 | | ffn 6736 |
. . . . . . 7
⊢ ((𝑇‘𝐹):(0..^(♯‘(𝑇‘𝐹)))⟶ℝ → (𝑇‘𝐹) Fn (0..^(♯‘(𝑇‘𝐹)))) |
| 8 | 5, 6, 7 | 3syl 18 |
. . . . . 6
⊢ (𝐹 ∈ Word ℝ →
(𝑇‘𝐹) Fn (0..^(♯‘(𝑇‘𝐹)))) |
| 9 | 1, 2, 3, 4 | signstlen 34582 |
. . . . . . . 8
⊢ (𝐹 ∈ Word ℝ →
(♯‘(𝑇‘𝐹)) = (♯‘𝐹)) |
| 10 | 9 | oveq2d 7447 |
. . . . . . 7
⊢ (𝐹 ∈ Word ℝ →
(0..^(♯‘(𝑇‘𝐹))) = (0..^(♯‘𝐹))) |
| 11 | 10 | fneq2d 6662 |
. . . . . 6
⊢ (𝐹 ∈ Word ℝ →
((𝑇‘𝐹) Fn (0..^(♯‘(𝑇‘𝐹))) ↔ (𝑇‘𝐹) Fn (0..^(♯‘𝐹)))) |
| 12 | 8, 11 | mpbid 232 |
. . . . 5
⊢ (𝐹 ∈ Word ℝ →
(𝑇‘𝐹) Fn (0..^(♯‘𝐹))) |
| 13 | | fnresin 32636 |
. . . . 5
⊢ ((𝑇‘𝐹) Fn (0..^(♯‘𝐹)) → ((𝑇‘𝐹) ↾ (0..^𝑁)) Fn ((0..^(♯‘𝐹)) ∩ (0..^𝑁))) |
| 14 | 12, 13 | syl 17 |
. . . 4
⊢ (𝐹 ∈ Word ℝ →
((𝑇‘𝐹) ↾ (0..^𝑁)) Fn ((0..^(♯‘𝐹)) ∩ (0..^𝑁))) |
| 15 | 14 | adantr 480 |
. . 3
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
→ ((𝑇‘𝐹) ↾ (0..^𝑁)) Fn ((0..^(♯‘𝐹)) ∩ (0..^𝑁))) |
| 16 | | elfzuz3 13561 |
. . . . . 6
⊢ (𝑁 ∈
(0...(♯‘𝐹))
→ (♯‘𝐹)
∈ (ℤ≥‘𝑁)) |
| 17 | | fzoss2 13727 |
. . . . . 6
⊢
((♯‘𝐹)
∈ (ℤ≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹))) |
| 18 | 16, 17 | syl 17 |
. . . . 5
⊢ (𝑁 ∈
(0...(♯‘𝐹))
→ (0..^𝑁) ⊆
(0..^(♯‘𝐹))) |
| 19 | 18 | adantl 481 |
. . . 4
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
→ (0..^𝑁) ⊆
(0..^(♯‘𝐹))) |
| 20 | | incom 4209 |
. . . . . 6
⊢
((0..^𝑁) ∩
(0..^(♯‘𝐹))) =
((0..^(♯‘𝐹))
∩ (0..^𝑁)) |
| 21 | | dfss2 3969 |
. . . . . . 7
⊢
((0..^𝑁) ⊆
(0..^(♯‘𝐹))
↔ ((0..^𝑁) ∩
(0..^(♯‘𝐹))) =
(0..^𝑁)) |
| 22 | 21 | biimpi 216 |
. . . . . 6
⊢
((0..^𝑁) ⊆
(0..^(♯‘𝐹))
→ ((0..^𝑁) ∩
(0..^(♯‘𝐹))) =
(0..^𝑁)) |
| 23 | 20, 22 | eqtr3id 2791 |
. . . . 5
⊢
((0..^𝑁) ⊆
(0..^(♯‘𝐹))
→ ((0..^(♯‘𝐹)) ∩ (0..^𝑁)) = (0..^𝑁)) |
| 24 | 23 | fneq2d 6662 |
. . . 4
⊢
((0..^𝑁) ⊆
(0..^(♯‘𝐹))
→ (((𝑇‘𝐹) ↾ (0..^𝑁)) Fn ((0..^(♯‘𝐹)) ∩ (0..^𝑁)) ↔ ((𝑇‘𝐹) ↾ (0..^𝑁)) Fn (0..^𝑁))) |
| 25 | 19, 24 | syl 17 |
. . 3
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
→ (((𝑇‘𝐹) ↾ (0..^𝑁)) Fn ((0..^(♯‘𝐹)) ∩ (0..^𝑁)) ↔ ((𝑇‘𝐹) ↾ (0..^𝑁)) Fn (0..^𝑁))) |
| 26 | 15, 25 | mpbid 232 |
. 2
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
→ ((𝑇‘𝐹) ↾ (0..^𝑁)) Fn (0..^𝑁)) |
| 27 | | wrdres 32919 |
. . . 4
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
→ (𝐹 ↾
(0..^𝑁)) ∈ Word
ℝ) |
| 28 | 1, 2, 3, 4 | signstf 34581 |
. . . 4
⊢ ((𝐹 ↾ (0..^𝑁)) ∈ Word ℝ → (𝑇‘(𝐹 ↾ (0..^𝑁))) ∈ Word ℝ) |
| 29 | | wrdf 14557 |
. . . 4
⊢ ((𝑇‘(𝐹 ↾ (0..^𝑁))) ∈ Word ℝ → (𝑇‘(𝐹 ↾ (0..^𝑁))):(0..^(♯‘(𝑇‘(𝐹 ↾ (0..^𝑁)))))⟶ℝ) |
| 30 | | ffn 6736 |
. . . 4
⊢ ((𝑇‘(𝐹 ↾ (0..^𝑁))):(0..^(♯‘(𝑇‘(𝐹 ↾ (0..^𝑁)))))⟶ℝ → (𝑇‘(𝐹 ↾ (0..^𝑁))) Fn (0..^(♯‘(𝑇‘(𝐹 ↾ (0..^𝑁)))))) |
| 31 | 27, 28, 29, 30 | 4syl 19 |
. . 3
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
→ (𝑇‘(𝐹 ↾ (0..^𝑁))) Fn (0..^(♯‘(𝑇‘(𝐹 ↾ (0..^𝑁)))))) |
| 32 | 1, 2, 3, 4 | signstlen 34582 |
. . . . . . 7
⊢ ((𝐹 ↾ (0..^𝑁)) ∈ Word ℝ →
(♯‘(𝑇‘(𝐹 ↾ (0..^𝑁)))) = (♯‘(𝐹 ↾ (0..^𝑁)))) |
| 33 | 27, 32 | syl 17 |
. . . . . 6
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
→ (♯‘(𝑇‘(𝐹 ↾ (0..^𝑁)))) = (♯‘(𝐹 ↾ (0..^𝑁)))) |
| 34 | | wrdfn 14566 |
. . . . . . . 8
⊢ (𝐹 ∈ Word ℝ →
𝐹 Fn
(0..^(♯‘𝐹))) |
| 35 | | fnssres 6691 |
. . . . . . . 8
⊢ ((𝐹 Fn (0..^(♯‘𝐹)) ∧ (0..^𝑁) ⊆ (0..^(♯‘𝐹))) → (𝐹 ↾ (0..^𝑁)) Fn (0..^𝑁)) |
| 36 | 34, 18, 35 | syl2an 596 |
. . . . . . 7
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
→ (𝐹 ↾
(0..^𝑁)) Fn (0..^𝑁)) |
| 37 | | hashfn 14414 |
. . . . . . 7
⊢ ((𝐹 ↾ (0..^𝑁)) Fn (0..^𝑁) → (♯‘(𝐹 ↾ (0..^𝑁))) = (♯‘(0..^𝑁))) |
| 38 | 36, 37 | syl 17 |
. . . . . 6
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
→ (♯‘(𝐹
↾ (0..^𝑁))) =
(♯‘(0..^𝑁))) |
| 39 | | elfznn0 13660 |
. . . . . . . 8
⊢ (𝑁 ∈
(0...(♯‘𝐹))
→ 𝑁 ∈
ℕ0) |
| 40 | | hashfzo0 14469 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ0
→ (♯‘(0..^𝑁)) = 𝑁) |
| 41 | 39, 40 | syl 17 |
. . . . . . 7
⊢ (𝑁 ∈
(0...(♯‘𝐹))
→ (♯‘(0..^𝑁)) = 𝑁) |
| 42 | 41 | adantl 481 |
. . . . . 6
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
→ (♯‘(0..^𝑁)) = 𝑁) |
| 43 | 33, 38, 42 | 3eqtrd 2781 |
. . . . 5
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
→ (♯‘(𝑇‘(𝐹 ↾ (0..^𝑁)))) = 𝑁) |
| 44 | 43 | oveq2d 7447 |
. . . 4
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
→ (0..^(♯‘(𝑇‘(𝐹 ↾ (0..^𝑁))))) = (0..^𝑁)) |
| 45 | 44 | fneq2d 6662 |
. . 3
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
→ ((𝑇‘(𝐹 ↾ (0..^𝑁))) Fn (0..^(♯‘(𝑇‘(𝐹 ↾ (0..^𝑁))))) ↔ (𝑇‘(𝐹 ↾ (0..^𝑁))) Fn (0..^𝑁))) |
| 46 | 31, 45 | mpbid 232 |
. 2
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
→ (𝑇‘(𝐹 ↾ (0..^𝑁))) Fn (0..^𝑁)) |
| 47 | | fvres 6925 |
. . . . 5
⊢ (𝑚 ∈ (0..^𝑁) → (((𝑇‘𝐹) ↾ (0..^𝑁))‘𝑚) = ((𝑇‘𝐹)‘𝑚)) |
| 48 | 47 | ad3antlr 731 |
. . . 4
⊢
(((((𝐹 ∈ Word
ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
∧ 𝑚 ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) → (((𝑇‘𝐹) ↾ (0..^𝑁))‘𝑚) = ((𝑇‘𝐹)‘𝑚)) |
| 49 | | simpr 484 |
. . . . . 6
⊢
(((((𝐹 ∈ Word
ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
∧ 𝑚 ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) → 𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) |
| 50 | 49 | fveq2d 6910 |
. . . . 5
⊢
(((((𝐹 ∈ Word
ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
∧ 𝑚 ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) → (𝑇‘𝐹) = (𝑇‘((𝐹 ↾ (0..^𝑁)) ++ 𝑔))) |
| 51 | 50 | fveq1d 6908 |
. . . 4
⊢
(((((𝐹 ∈ Word
ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
∧ 𝑚 ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) → ((𝑇‘𝐹)‘𝑚) = ((𝑇‘((𝐹 ↾ (0..^𝑁)) ++ 𝑔))‘𝑚)) |
| 52 | 27 | ad3antrrr 730 |
. . . . 5
⊢
(((((𝐹 ∈ Word
ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
∧ 𝑚 ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) → (𝐹 ↾ (0..^𝑁)) ∈ Word ℝ) |
| 53 | | simplr 769 |
. . . . 5
⊢
(((((𝐹 ∈ Word
ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
∧ 𝑚 ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) → 𝑔 ∈ Word ℝ) |
| 54 | 38, 42 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
→ (♯‘(𝐹
↾ (0..^𝑁))) = 𝑁) |
| 55 | 54 | oveq2d 7447 |
. . . . . . . 8
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
→ (0..^(♯‘(𝐹 ↾ (0..^𝑁)))) = (0..^𝑁)) |
| 56 | 55 | eleq2d 2827 |
. . . . . . 7
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
→ (𝑚 ∈
(0..^(♯‘(𝐹
↾ (0..^𝑁)))) ↔
𝑚 ∈ (0..^𝑁))) |
| 57 | 56 | biimpar 477 |
. . . . . 6
⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
∧ 𝑚 ∈ (0..^𝑁)) → 𝑚 ∈ (0..^(♯‘(𝐹 ↾ (0..^𝑁))))) |
| 58 | 57 | ad2antrr 726 |
. . . . 5
⊢
(((((𝐹 ∈ Word
ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
∧ 𝑚 ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) → 𝑚 ∈ (0..^(♯‘(𝐹 ↾ (0..^𝑁))))) |
| 59 | 1, 2, 3, 4 | signstfvc 34589 |
. . . . 5
⊢ (((𝐹 ↾ (0..^𝑁)) ∈ Word ℝ ∧ 𝑔 ∈ Word ℝ ∧ 𝑚 ∈
(0..^(♯‘(𝐹
↾ (0..^𝑁))))) →
((𝑇‘((𝐹 ↾ (0..^𝑁)) ++ 𝑔))‘𝑚) = ((𝑇‘(𝐹 ↾ (0..^𝑁)))‘𝑚)) |
| 60 | 52, 53, 58, 59 | syl3anc 1373 |
. . . 4
⊢
(((((𝐹 ∈ Word
ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
∧ 𝑚 ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) → ((𝑇‘((𝐹 ↾ (0..^𝑁)) ++ 𝑔))‘𝑚) = ((𝑇‘(𝐹 ↾ (0..^𝑁)))‘𝑚)) |
| 61 | 48, 51, 60 | 3eqtrd 2781 |
. . 3
⊢
(((((𝐹 ∈ Word
ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
∧ 𝑚 ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) → (((𝑇‘𝐹) ↾ (0..^𝑁))‘𝑚) = ((𝑇‘(𝐹 ↾ (0..^𝑁)))‘𝑚)) |
| 62 | | wrdsplex 32920 |
. . . 4
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
→ ∃𝑔 ∈ Word
ℝ𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) |
| 63 | 62 | adantr 480 |
. . 3
⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
∧ 𝑚 ∈ (0..^𝑁)) → ∃𝑔 ∈ Word ℝ𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) |
| 64 | 61, 63 | r19.29a 3162 |
. 2
⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
∧ 𝑚 ∈ (0..^𝑁)) → (((𝑇‘𝐹) ↾ (0..^𝑁))‘𝑚) = ((𝑇‘(𝐹 ↾ (0..^𝑁)))‘𝑚)) |
| 65 | 26, 46, 64 | eqfnfvd 7054 |
1
⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈
(0...(♯‘𝐹)))
→ ((𝑇‘𝐹) ↾ (0..^𝑁)) = (𝑇‘(𝐹 ↾ (0..^𝑁)))) |