Mathbox for Asger C. Ipsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppf | Structured version Visualization version GIF version |
Description: Knopp's function is a function. (Contributed by Asger C. Ipsen, 25-Aug-2021.) |
Ref | Expression |
---|---|
knoppf.t | ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) |
knoppf.f | ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) |
knoppf.w | ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) |
knoppf.c | ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) |
knoppf.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Ref | Expression |
---|---|
knoppf | ⊢ (𝜑 → 𝑊:ℝ⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 12441 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 12153 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → 0 ∈ ℤ) | |
3 | eqidd 2737 | . . 3 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑖 ∈ ℕ0) → ((𝐹‘𝑤)‘𝑖) = ((𝐹‘𝑤)‘𝑖)) | |
4 | knoppf.t | . . . 4 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | |
5 | knoppf.f | . . . 4 ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) | |
6 | knoppf.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
7 | 6 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → 𝑁 ∈ ℕ) |
8 | 7 | adantr 484 | . . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑖 ∈ ℕ0) → 𝑁 ∈ ℕ) |
9 | knoppf.c | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) | |
10 | 9 | knoppndvlem3 34380 | . . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1)) |
11 | 10 | simpld 498 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
12 | 11 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → 𝐶 ∈ ℝ) |
13 | 12 | adantr 484 | . . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑖 ∈ ℕ0) → 𝐶 ∈ ℝ) |
14 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → 𝑤 ∈ ℝ) | |
15 | 14 | adantr 484 | . . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑖 ∈ ℕ0) → 𝑤 ∈ ℝ) |
16 | simpr 488 | . . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0) | |
17 | 4, 5, 8, 13, 15, 16 | knoppcnlem3 34361 | . . 3 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑖 ∈ ℕ0) → ((𝐹‘𝑤)‘𝑖) ∈ ℝ) |
18 | knoppf.w | . . . . . 6 ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) | |
19 | fveq2 6695 | . . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝐹‘𝑤) = (𝐹‘𝑧)) | |
20 | 19 | fveq1d 6697 | . . . . . . . 8 ⊢ (𝑤 = 𝑧 → ((𝐹‘𝑤)‘𝑖) = ((𝐹‘𝑧)‘𝑖)) |
21 | 20 | sumeq2sdv 15233 | . . . . . . 7 ⊢ (𝑤 = 𝑧 → Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖) = Σ𝑖 ∈ ℕ0 ((𝐹‘𝑧)‘𝑖)) |
22 | 21 | cbvmptv 5143 | . . . . . 6 ⊢ (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) = (𝑧 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑧)‘𝑖)) |
23 | 18, 22 | eqtri 2759 | . . . . 5 ⊢ 𝑊 = (𝑧 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑧)‘𝑖)) |
24 | 9 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → 𝐶 ∈ (-1(,)1)) |
25 | 4, 5, 23, 14, 24, 7 | knoppndvlem4 34381 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → seq0( + , (𝐹‘𝑤)) ⇝ (𝑊‘𝑤)) |
26 | seqex 13541 | . . . . 5 ⊢ seq0( + , (𝐹‘𝑤)) ∈ V | |
27 | fvex 6708 | . . . . 5 ⊢ (𝑊‘𝑤) ∈ V | |
28 | 26, 27 | breldm 5762 | . . . 4 ⊢ (seq0( + , (𝐹‘𝑤)) ⇝ (𝑊‘𝑤) → seq0( + , (𝐹‘𝑤)) ∈ dom ⇝ ) |
29 | 25, 28 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → seq0( + , (𝐹‘𝑤)) ∈ dom ⇝ ) |
30 | 1, 2, 3, 17, 29 | isumrecl 15292 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖) ∈ ℝ) |
31 | 30, 18 | fmptd 6909 | 1 ⊢ (𝜑 → 𝑊:ℝ⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 ↦ cmpt 5120 dom cdm 5536 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ℝcr 10693 0cc0 10694 1c1 10695 + caddc 10697 · cmul 10699 < clt 10832 − cmin 11027 -cneg 11028 / cdiv 11454 ℕcn 11795 2c2 11850 ℕ0cn0 12055 (,)cioo 12900 ⌊cfl 13330 seqcseq 13539 ↑cexp 13600 abscabs 14762 ⇝ cli 15010 Σcsu 15214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-pm 8489 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-inf 9037 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-n0 12056 df-z 12142 df-uz 12404 df-rp 12552 df-ioo 12904 df-ico 12906 df-fz 13061 df-fzo 13204 df-fl 13332 df-seq 13540 df-exp 13601 df-hash 13862 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-limsup 14997 df-clim 15014 df-rlim 15015 df-sum 15215 df-ulm 25223 |
This theorem is referenced by: knoppcn2 34402 |
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