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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 485 | . . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → 𝜑) | |
2 | ax-resscn 10588 | . . . . . . . . . 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 33848 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1)) |
10 | 9 | simpld 497 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
11 | 9 | simprd 498 | . . . . . . . . . . 11 ⊢ (𝜑 → (abs‘𝐶) < 1) |
12 | 4, 5, 6, 7, 10, 11 | knoppcn 33838 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ (ℝ–cn→ℂ)) |
13 | cncff 23495 | . . . . . . . . . 10 ⊢ (𝑊 ∈ (ℝ–cn→ℂ) → 𝑊:ℝ⟶ℂ) | |
14 | 12, 13 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑊:ℝ⟶ℂ) |
15 | ssidd 3989 | . . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℝ) | |
16 | 3, 14, 15 | dvbss 24493 | . . . . . . . 8 ⊢ (𝜑 → dom (ℝ D 𝑊) ⊆ ℝ) |
17 | 16 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → dom (ℝ D 𝑊) ⊆ ℝ) |
18 | simpr 487 | . . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → ℎ ∈ dom (ℝ D 𝑊)) | |
19 | 17, 18 | sseldd 3967 | . . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → ℎ ∈ ℝ) |
20 | 1, 19 | jca 514 | . . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → (𝜑 ∧ ℎ ∈ ℝ)) |
21 | ssidd 3989 | . . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ ℝ) → ℝ ⊆ ℝ) | |
22 | 14 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ ℝ) → 𝑊:ℝ⟶ℂ) |
23 | 8 | ad2antrr 724 | . . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 𝐶 ∈ (-1(,)1)) |
24 | simprr 771 | . . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 𝑑 ∈ ℝ+) | |
25 | simprl 769 | . . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 𝑒 ∈ ℝ+) | |
26 | simplr 767 | . . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → ℎ ∈ ℝ) | |
27 | 7 | ad2antrr 724 | . . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 𝑁 ∈ ℕ) |
28 | knoppndv.1 | . . . . . . . . 9 ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) | |
29 | 28 | ad2antrr 724 | . . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+)) → 1 < (𝑁 · (abs‘𝐶))) |
30 | 4, 5, 6, 23, 24, 25, 26, 27, 29 | knoppndvlem22 33867 | . . . . . . 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 3191 | . . . . . 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 33845 | . . . . 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 797 | . . 3 ⊢ (𝜑 → ¬ ℎ ∈ dom (ℝ D 𝑊)) |
35 | 34 | alrimiv 1924 | . 2 ⊢ (𝜑 → ∀ℎ ¬ ℎ ∈ dom (ℝ D 𝑊)) |
36 | eq0 4307 | . 2 ⊢ (dom (ℝ D 𝑊) = ∅ ↔ ∀ℎ ¬ ℎ ∈ dom (ℝ D 𝑊)) | |
37 | 35, 36 | sylibr 236 | 1 ⊢ (𝜑 → dom (ℝ D 𝑊) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 ∀wal 1531 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 ⊆ wss 3935 ∅c0 4290 class class class wbr 5058 ↦ cmpt 5138 dom cdm 5549 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℝcr 10530 1c1 10532 + caddc 10534 · cmul 10536 < clt 10669 ≤ cle 10670 − cmin 10864 -cneg 10865 / cdiv 11291 ℕcn 11632 2c2 11686 ℕ0cn0 11891 ℝ+crp 12383 (,)cioo 12732 ⌊cfl 13154 ↑cexp 13423 abscabs 14587 Σcsu 15036 –cn→ccncf 23478 D cdv 24455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-dvds 15602 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-ntr 21622 df-cn 21829 df-cnp 21830 df-tx 22164 df-hmeo 22357 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 df-limc 24458 df-dv 24459 df-ulm 24959 |
This theorem is referenced by: cnndvlem1 33871 |
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