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Mathbox for Thierry Arnoux |
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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 11529 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 − 𝑦) ∈ ℝ) | |
2 | 1 | adantl 481 | . . 3 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 − 𝑦) ∈ ℝ) |
3 | 0re 11221 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
4 | s1cl 14557 | . . . . . . . 8 ⊢ (0 ∈ ℝ → 〈“0”〉 ∈ Word ℝ) | |
5 | 3, 4 | ax-mp 5 | . . . . . . 7 ⊢ 〈“0”〉 ∈ Word ℝ |
6 | ccatcl 14529 | . . . . . . 7 ⊢ ((〈“0”〉 ∈ Word ℝ ∧ 𝐹 ∈ Word ℝ) → (〈“0”〉 ++ 𝐹) ∈ Word ℝ) | |
7 | 5, 6 | mpan 687 | . . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (〈“0”〉 ++ 𝐹) ∈ Word ℝ) |
8 | wrdf 14474 | . . . . . 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 11214 | . . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → 1 ∈ ℂ) | |
11 | lencl 14488 | . . . . . . . . 9 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℕ0) | |
12 | 11 | nn0cnd 12539 | . . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℂ) |
13 | ccatlen 14530 | . . . . . . . . . 10 ⊢ ((〈“0”〉 ∈ Word ℝ ∧ 𝐹 ∈ Word ℝ) → (♯‘(〈“0”〉 ++ 𝐹)) = ((♯‘〈“0”〉) + (♯‘𝐹))) | |
14 | 5, 13 | mpan 687 | . . . . . . . . 9 ⊢ (𝐹 ∈ Word ℝ → (♯‘(〈“0”〉 ++ 𝐹)) = ((♯‘〈“0”〉) + (♯‘𝐹))) |
15 | s1len 14561 | . . . . . . . . . 10 ⊢ (♯‘〈“0”〉) = 1 | |
16 | 15 | oveq1i 7422 | . . . . . . . . 9 ⊢ ((♯‘〈“0”〉) + (♯‘𝐹)) = (1 + (♯‘𝐹)) |
17 | 14, 16 | eqtrdi 2787 | . . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → (♯‘(〈“0”〉 ++ 𝐹)) = (1 + (♯‘𝐹))) |
18 | 10, 12, 17 | comraddd 11433 | . . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (♯‘(〈“0”〉 ++ 𝐹)) = ((♯‘𝐹) + 1)) |
19 | 18 | oveq2d 7428 | . . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (0..^(♯‘(〈“0”〉 ++ 𝐹))) = (0..^((♯‘𝐹) + 1))) |
20 | 19 | feq2d 6703 | . . . . 5 ⊢ (𝐹 ∈ Word ℝ → ((〈“0”〉 ++ 𝐹):(0..^(♯‘(〈“0”〉 ++ 𝐹)))⟶ℝ ↔ (〈“0”〉 ++ 𝐹):(0..^((♯‘𝐹) + 1))⟶ℝ)) |
21 | 9, 20 | mpbid 231 | . . . 4 ⊢ (𝐹 ∈ Word ℝ → (〈“0”〉 ++ 𝐹):(0..^((♯‘𝐹) + 1))⟶ℝ) |
22 | 21 | adantr 480 | . . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → (〈“0”〉 ++ 𝐹):(0..^((♯‘𝐹) + 1))⟶ℝ) |
23 | remulcl 11199 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ) | |
24 | 23 | adantl 481 | . . . 4 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ) |
25 | ccatcl 14529 | . . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 〈“0”〉 ∈ Word ℝ) → (𝐹 ++ 〈“0”〉) ∈ Word ℝ) | |
26 | 5, 25 | mpan2 688 | . . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (𝐹 ++ 〈“0”〉) ∈ Word ℝ) |
27 | wrdf 14474 | . . . . . . 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 14575 | . . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → (♯‘(𝐹 ++ 〈“0”〉)) = ((♯‘𝐹) + 1)) | |
30 | 29 | oveq2d 7428 | . . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (0..^(♯‘(𝐹 ++ 〈“0”〉))) = (0..^((♯‘𝐹) + 1))) |
31 | 30 | feq2d 6703 | . . . . . 6 ⊢ (𝐹 ∈ Word ℝ → ((𝐹 ++ 〈“0”〉):(0..^(♯‘(𝐹 ++ 〈“0”〉)))⟶ℝ ↔ (𝐹 ++ 〈“0”〉):(0..^((♯‘𝐹) + 1))⟶ℝ)) |
32 | 28, 31 | mpbid 231 | . . . . 5 ⊢ (𝐹 ∈ Word ℝ → (𝐹 ++ 〈“0”〉):(0..^((♯‘𝐹) + 1))⟶ℝ) |
33 | 32 | adantr 480 | . . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → (𝐹 ++ 〈“0”〉):(0..^((♯‘𝐹) + 1))⟶ℝ) |
34 | ovexd 7447 | . . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → (0..^((♯‘𝐹) + 1)) ∈ V) | |
35 | rpre 12987 | . . . . 5 ⊢ (𝐶 ∈ ℝ+ → 𝐶 ∈ ℝ) | |
36 | 35 | adantl 481 | . . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → 𝐶 ∈ ℝ) |
37 | 24, 33, 34, 36 | ofcf 33400 | . . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → ((𝐹 ++ 〈“0”〉) ∘f/c · 𝐶):(0..^((♯‘𝐹) + 1))⟶ℝ) |
38 | inidm 4218 | . . 3 ⊢ ((0..^((♯‘𝐹) + 1)) ∩ (0..^((♯‘𝐹) + 1))) = (0..^((♯‘𝐹) + 1)) | |
39 | 2, 22, 37, 34, 34, 38 | off 7692 | . 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 6708 | . 2 ⊢ (𝐻:(0..^((♯‘𝐹) + 1))⟶ℝ ↔ ((〈“0”〉 ++ 𝐹) ∘f − ((𝐹 ++ 〈“0”〉) ∘f/c · 𝐶)):(0..^((♯‘𝐹) + 1))⟶ℝ) |
42 | 39, 41 | sylibr 233 | 1 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → 𝐻:(0..^((♯‘𝐹) + 1))⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 Vcvv 3473 ifcif 4528 {cpr 4630 {ctp 4632 〈cop 4634 ↦ cmpt 5231 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 ∘f cof 7672 ℝcr 11113 0cc0 11114 1c1 11115 + caddc 11117 · cmul 11119 − cmin 11449 -cneg 11450 ℝ+crp 12979 ...cfz 13489 ..^cfzo 13632 ♯chash 14295 Word cword 14469 ++ cconcat 14525 〈“cs1 14550 sgncsgn 15038 Σcsu 15637 ndxcnx 17131 Basecbs 17149 +gcplusg 17202 Σg cgsu 17391 ∘f/c cofc 33392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-fz 13490 df-fzo 13633 df-hash 14296 df-word 14470 df-concat 14526 df-s1 14551 df-ofc 33393 |
This theorem is referenced by: signshwrd 33899 signshlen 33900 |
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