Mathbox for Asger C. Ipsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndv | Structured version Visualization version GIF version |
Description: The continuous nowhere differentiable function 𝑊 ( Knopp, K. (1918). Math. Z. 2, 1-26 ) is, in fact, nowhere differentiable. (Contributed by Asger C. Ipsen, 19-Aug-2021.) |
Ref | Expression |
---|---|
knoppndv.t | ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) |
knoppndv.f | ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) |
knoppndv.w | ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) |
knoppndv.c | ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) |
knoppndv.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
knoppndv.1 | ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) |
Ref | Expression |
---|---|
knoppndv | ⊢ (𝜑 → dom (ℝ D 𝑊) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 486 | . . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → 𝜑) | |
2 | ax-resscn 10675 | . . . . . . . . . 10 ⊢ ℝ ⊆ ℂ | |
3 | 2 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℂ) |
4 | knoppndv.t | . . . . . . . . . . 11 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | |
5 | knoppndv.f | . . . . . . . . . . 11 ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) | |
6 | knoppndv.w | . . . . . . . . . . 11 ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) | |
7 | knoppndv.n | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
8 | knoppndv.c | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) | |
9 | 8 | knoppndvlem3 34340 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1)) |
10 | 9 | simpld 498 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
11 | 9 | simprd 499 | . . . . . . . . . . 11 ⊢ (𝜑 → (abs‘𝐶) < 1) |
12 | 4, 5, 6, 7, 10, 11 | knoppcn 34330 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ (ℝ–cn→ℂ)) |
13 | cncff 23648 | . . . . . . . . . 10 ⊢ (𝑊 ∈ (ℝ–cn→ℂ) → 𝑊:ℝ⟶ℂ) | |
14 | 12, 13 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑊:ℝ⟶ℂ) |
15 | ssidd 3901 | . . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℝ) | |
16 | 3, 14, 15 | dvbss 24656 | . . . . . . . 8 ⊢ (𝜑 → dom (ℝ D 𝑊) ⊆ ℝ) |
17 | 16 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → dom (ℝ D 𝑊) ⊆ ℝ) |
18 | simpr 488 | . . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → ℎ ∈ dom (ℝ D 𝑊)) | |
19 | 17, 18 | sseldd 3879 | . . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → ℎ ∈ ℝ) |
20 | 1, 19 | jca 515 | . . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → (𝜑 ∧ ℎ ∈ ℝ)) |
21 | ssidd 3901 | . . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ ℝ) → ℝ ⊆ ℝ) | |
22 | 14 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ ℝ) → 𝑊:ℝ⟶ℂ) |
23 | 8 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 𝐶 ∈ (-1(,)1)) |
24 | simprr 773 | . . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 𝑑 ∈ ℝ+) | |
25 | simprl 771 | . . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 𝑒 ∈ ℝ+) | |
26 | simplr 769 | . . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → ℎ ∈ ℝ) | |
27 | 7 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 𝑁 ∈ ℕ) |
28 | knoppndv.1 | . . . . . . . . 9 ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) | |
29 | 28 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 1 < (𝑁 · (abs‘𝐶))) |
30 | 4, 5, 6, 23, 24, 25, 26, 27, 29 | knoppndvlem22 34359 | . . . . . . 7 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ ((𝑎 ≤ ℎ ∧ ℎ ≤ 𝑏) ∧ ((𝑏 − 𝑎) < 𝑑 ∧ 𝑎 ≠ 𝑏) ∧ 𝑒 ≤ ((abs‘((𝑊‘𝑏) − (𝑊‘𝑎))) / (𝑏 − 𝑎)))) |
31 | 30 | ralrimivva 3104 | . . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ ℝ) → ∀𝑒 ∈ ℝ+ ∀𝑑 ∈ ℝ+ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ ((𝑎 ≤ ℎ ∧ ℎ ≤ 𝑏) ∧ ((𝑏 − 𝑎) < 𝑑 ∧ 𝑎 ≠ 𝑏) ∧ 𝑒 ≤ ((abs‘((𝑊‘𝑏) − (𝑊‘𝑎))) / (𝑏 − 𝑎)))) |
32 | 21, 22, 31 | unbdqndv2 34337 | . . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ ℝ) → ¬ ℎ ∈ dom (ℝ D 𝑊)) |
33 | 20, 32 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → ¬ ℎ ∈ dom (ℝ D 𝑊)) |
34 | 33 | pm2.01da 799 | . . 3 ⊢ (𝜑 → ¬ ℎ ∈ dom (ℝ D 𝑊)) |
35 | 34 | alrimiv 1934 | . 2 ⊢ (𝜑 → ∀ℎ ¬ ℎ ∈ dom (ℝ D 𝑊)) |
36 | eq0 4233 | . 2 ⊢ (dom (ℝ D 𝑊) = ∅ ↔ ∀ℎ ¬ ℎ ∈ dom (ℝ D 𝑊)) | |
37 | 35, 36 | sylibr 237 | 1 ⊢ (𝜑 → dom (ℝ D 𝑊) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1088 ∀wal 1540 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 ∃wrex 3055 ⊆ wss 3844 ∅c0 4212 class class class wbr 5031 ↦ cmpt 5111 dom cdm 5526 ⟶wf 6336 ‘cfv 6340 (class class class)co 7173 ℂcc 10616 ℝcr 10617 1c1 10619 + caddc 10621 · cmul 10623 < clt 10756 ≤ cle 10757 − cmin 10951 -cneg 10952 / cdiv 11378 ℕcn 11719 2c2 11774 ℕ0cn0 11979 ℝ+crp 12475 (,)cioo 12824 ⌊cfl 13254 ↑cexp 13524 abscabs 14686 Σcsu 15138 –cn→ccncf 23631 D cdv 24618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-inf2 9180 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 ax-pre-sup 10696 ax-addf 10697 ax-mulf 10698 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-iin 4885 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-of 7428 df-om 7603 df-1st 7717 df-2nd 7718 df-supp 7860 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-2o 8135 df-er 8323 df-map 8442 df-pm 8443 df-ixp 8511 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-fsupp 8910 df-fi 8951 df-sup 8982 df-inf 8983 df-oi 9050 df-card 9444 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-div 11379 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-5 11785 df-6 11786 df-7 11787 df-8 11788 df-9 11789 df-n0 11980 df-z 12066 df-dec 12183 df-uz 12328 df-q 12434 df-rp 12476 df-xneg 12593 df-xadd 12594 df-xmul 12595 df-ioo 12828 df-ico 12830 df-icc 12831 df-fz 12985 df-fzo 13128 df-fl 13256 df-seq 13464 df-exp 13525 df-hash 13786 df-cj 14551 df-re 14552 df-im 14553 df-sqrt 14687 df-abs 14688 df-limsup 14921 df-clim 14938 df-rlim 14939 df-sum 15139 df-dvds 15703 df-struct 16591 df-ndx 16592 df-slot 16593 df-base 16595 df-sets 16596 df-ress 16597 df-plusg 16684 df-mulr 16685 df-starv 16686 df-sca 16687 df-vsca 16688 df-ip 16689 df-tset 16690 df-ple 16691 df-ds 16693 df-unif 16694 df-hom 16695 df-cco 16696 df-rest 16802 df-topn 16803 df-0g 16821 df-gsum 16822 df-topgen 16823 df-pt 16824 df-prds 16827 df-xrs 16881 df-qtop 16886 df-imas 16887 df-xps 16889 df-mre 16963 df-mrc 16964 df-acs 16966 df-mgm 17971 df-sgrp 18020 df-mnd 18031 df-submnd 18076 df-mulg 18346 df-cntz 18568 df-cmn 19029 df-psmet 20212 df-xmet 20213 df-met 20214 df-bl 20215 df-mopn 20216 df-cnfld 20221 df-top 21648 df-topon 21665 df-topsp 21687 df-bases 21700 df-ntr 21774 df-cn 21981 df-cnp 21982 df-tx 22316 df-hmeo 22509 df-xms 23076 df-ms 23077 df-tms 23078 df-cncf 23633 df-limc 24621 df-dv 24622 df-ulm 25127 |
This theorem is referenced by: cnndvlem1 34363 |
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