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Mirrors > Home > MPE Home > Th. List > Mathboxes > signshf | Structured version Visualization version GIF version |
Description: 𝐻, corresponding to the word 𝐹 multiplied by (𝑥 − 𝐶), as a function. (Contributed by Thierry Arnoux, 29-Sep-2018.) |
Ref | Expression |
---|---|
signsv.p | ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏)) |
signsv.w | ⊢ 𝑊 = {〈(Base‘ndx), {-1, 0, 1}〉, 〈(+g‘ndx), ⨣ 〉} |
signsv.t | ⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(♯‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖)))))) |
signsv.v | ⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(♯‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0)) |
signs.h | ⊢ 𝐻 = ((〈“0”〉 ++ 𝐹) ∘f − ((𝐹 ++ 〈“0”〉) ∘f/c · 𝐶)) |
Ref | Expression |
---|---|
signshf | ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → 𝐻:(0..^((♯‘𝐹) + 1))⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resubcl 10953 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 − 𝑦) ∈ ℝ) | |
2 | 1 | adantl 484 | . . 3 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 − 𝑦) ∈ ℝ) |
3 | 0re 10646 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
4 | s1cl 13959 | . . . . . . . 8 ⊢ (0 ∈ ℝ → 〈“0”〉 ∈ Word ℝ) | |
5 | 3, 4 | ax-mp 5 | . . . . . . 7 ⊢ 〈“0”〉 ∈ Word ℝ |
6 | ccatcl 13929 | . . . . . . 7 ⊢ ((〈“0”〉 ∈ Word ℝ ∧ 𝐹 ∈ Word ℝ) → (〈“0”〉 ++ 𝐹) ∈ Word ℝ) | |
7 | 5, 6 | mpan 688 | . . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (〈“0”〉 ++ 𝐹) ∈ Word ℝ) |
8 | wrdf 13869 | . . . . . 6 ⊢ ((〈“0”〉 ++ 𝐹) ∈ Word ℝ → (〈“0”〉 ++ 𝐹):(0..^(♯‘(〈“0”〉 ++ 𝐹)))⟶ℝ) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ Word ℝ → (〈“0”〉 ++ 𝐹):(0..^(♯‘(〈“0”〉 ++ 𝐹)))⟶ℝ) |
10 | 1cnd 10639 | . . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → 1 ∈ ℂ) | |
11 | lencl 13886 | . . . . . . . . 9 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℕ0) | |
12 | 11 | nn0cnd 11960 | . . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℂ) |
13 | ccatlen 13930 | . . . . . . . . . 10 ⊢ ((〈“0”〉 ∈ Word ℝ ∧ 𝐹 ∈ Word ℝ) → (♯‘(〈“0”〉 ++ 𝐹)) = ((♯‘〈“0”〉) + (♯‘𝐹))) | |
14 | 5, 13 | mpan 688 | . . . . . . . . 9 ⊢ (𝐹 ∈ Word ℝ → (♯‘(〈“0”〉 ++ 𝐹)) = ((♯‘〈“0”〉) + (♯‘𝐹))) |
15 | s1len 13963 | . . . . . . . . . 10 ⊢ (♯‘〈“0”〉) = 1 | |
16 | 15 | oveq1i 7169 | . . . . . . . . 9 ⊢ ((♯‘〈“0”〉) + (♯‘𝐹)) = (1 + (♯‘𝐹)) |
17 | 14, 16 | syl6eq 2875 | . . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → (♯‘(〈“0”〉 ++ 𝐹)) = (1 + (♯‘𝐹))) |
18 | 10, 12, 17 | comraddd 10857 | . . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (♯‘(〈“0”〉 ++ 𝐹)) = ((♯‘𝐹) + 1)) |
19 | 18 | oveq2d 7175 | . . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (0..^(♯‘(〈“0”〉 ++ 𝐹))) = (0..^((♯‘𝐹) + 1))) |
20 | 19 | feq2d 6503 | . . . . 5 ⊢ (𝐹 ∈ Word ℝ → ((〈“0”〉 ++ 𝐹):(0..^(♯‘(〈“0”〉 ++ 𝐹)))⟶ℝ ↔ (〈“0”〉 ++ 𝐹):(0..^((♯‘𝐹) + 1))⟶ℝ)) |
21 | 9, 20 | mpbid 234 | . . . 4 ⊢ (𝐹 ∈ Word ℝ → (〈“0”〉 ++ 𝐹):(0..^((♯‘𝐹) + 1))⟶ℝ) |
22 | 21 | adantr 483 | . . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → (〈“0”〉 ++ 𝐹):(0..^((♯‘𝐹) + 1))⟶ℝ) |
23 | remulcl 10625 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ) | |
24 | 23 | adantl 484 | . . . 4 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ) |
25 | ccatcl 13929 | . . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 〈“0”〉 ∈ Word ℝ) → (𝐹 ++ 〈“0”〉) ∈ Word ℝ) | |
26 | 5, 25 | mpan2 689 | . . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (𝐹 ++ 〈“0”〉) ∈ Word ℝ) |
27 | wrdf 13869 | . . . . . . 7 ⊢ ((𝐹 ++ 〈“0”〉) ∈ Word ℝ → (𝐹 ++ 〈“0”〉):(0..^(♯‘(𝐹 ++ 〈“0”〉)))⟶ℝ) | |
28 | 26, 27 | syl 17 | . . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (𝐹 ++ 〈“0”〉):(0..^(♯‘(𝐹 ++ 〈“0”〉)))⟶ℝ) |
29 | ccatws1len 13977 | . . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → (♯‘(𝐹 ++ 〈“0”〉)) = ((♯‘𝐹) + 1)) | |
30 | 29 | oveq2d 7175 | . . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (0..^(♯‘(𝐹 ++ 〈“0”〉))) = (0..^((♯‘𝐹) + 1))) |
31 | 30 | feq2d 6503 | . . . . . 6 ⊢ (𝐹 ∈ Word ℝ → ((𝐹 ++ 〈“0”〉):(0..^(♯‘(𝐹 ++ 〈“0”〉)))⟶ℝ ↔ (𝐹 ++ 〈“0”〉):(0..^((♯‘𝐹) + 1))⟶ℝ)) |
32 | 28, 31 | mpbid 234 | . . . . 5 ⊢ (𝐹 ∈ Word ℝ → (𝐹 ++ 〈“0”〉):(0..^((♯‘𝐹) + 1))⟶ℝ) |
33 | 32 | adantr 483 | . . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → (𝐹 ++ 〈“0”〉):(0..^((♯‘𝐹) + 1))⟶ℝ) |
34 | ovexd 7194 | . . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → (0..^((♯‘𝐹) + 1)) ∈ V) | |
35 | rpre 12400 | . . . . 5 ⊢ (𝐶 ∈ ℝ+ → 𝐶 ∈ ℝ) | |
36 | 35 | adantl 484 | . . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → 𝐶 ∈ ℝ) |
37 | 24, 33, 34, 36 | ofcf 31366 | . . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → ((𝐹 ++ 〈“0”〉) ∘f/c · 𝐶):(0..^((♯‘𝐹) + 1))⟶ℝ) |
38 | inidm 4198 | . . 3 ⊢ ((0..^((♯‘𝐹) + 1)) ∩ (0..^((♯‘𝐹) + 1))) = (0..^((♯‘𝐹) + 1)) | |
39 | 2, 22, 37, 34, 34, 38 | off 7427 | . 2 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → ((〈“0”〉 ++ 𝐹) ∘f − ((𝐹 ++ 〈“0”〉) ∘f/c · 𝐶)):(0..^((♯‘𝐹) + 1))⟶ℝ) |
40 | signs.h | . . 3 ⊢ 𝐻 = ((〈“0”〉 ++ 𝐹) ∘f − ((𝐹 ++ 〈“0”〉) ∘f/c · 𝐶)) | |
41 | 40 | feq1i 6508 | . 2 ⊢ (𝐻:(0..^((♯‘𝐹) + 1))⟶ℝ ↔ ((〈“0”〉 ++ 𝐹) ∘f − ((𝐹 ++ 〈“0”〉) ∘f/c · 𝐶)):(0..^((♯‘𝐹) + 1))⟶ℝ) |
42 | 39, 41 | sylibr 236 | 1 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → 𝐻:(0..^((♯‘𝐹) + 1))⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 Vcvv 3497 ifcif 4470 {cpr 4572 {ctp 4574 〈cop 4576 ↦ cmpt 5149 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ∈ cmpo 7161 ∘f cof 7410 ℝcr 10539 0cc0 10540 1c1 10541 + caddc 10543 · cmul 10545 − cmin 10873 -cneg 10874 ℝ+crp 12392 ...cfz 12895 ..^cfzo 13036 ♯chash 13693 Word cword 13864 ++ cconcat 13925 〈“cs1 13952 sgncsgn 14448 Σcsu 15045 ndxcnx 16483 Basecbs 16486 +gcplusg 16568 Σg cgsu 16717 ∘f/c cofc 31358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13926 df-s1 13953 df-ofc 31359 |
This theorem is referenced by: signshwrd 31863 signshlen 31864 |
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