knoppndv - Mathbox for Asger C. Ipsen

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Theorem knoppndv 32862

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.)

**Hypotheses**
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)))

**Assertion**
Ref	Expression
knoppndv	⊢ (𝜑 → dom (ℝ D 𝑊) = ∅)

Distinct variable groups: 𝐶,𝑖,𝑛,𝑤,𝑦 𝑖,𝐹,𝑤 𝑖,𝑁,𝑛,𝑤,𝑦 𝑥,𝑁,𝑖,𝑤 𝑇,𝑛,𝑦 𝜑,𝑖,𝑛,𝑤,𝑦 Allowed substitution hints: 𝜑(𝑥) 𝐶(𝑥) 𝑇(𝑥,𝑤,𝑖) 𝐹(𝑥,𝑦,𝑛) 𝑊(𝑥,𝑦,𝑤,𝑖,𝑛)

Proof of Theorem knoppndv Dummy variables 𝑎 𝑏 𝑑 𝑒 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 468	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → 𝜑)
2		ax-resscn 10199	. . . . . . . . . 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 32842	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
10	9	simpld 482	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℝ)
11	9	simprd 483	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘𝐶) < 1)
12	4, 5, 6, 7, 10, 11	knoppcn 32831	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ (ℝ–cn→ℂ))
13		cncff 22916	. . . . . . . . . 10 ⊢ (𝑊 ∈ (ℝ–cn→ℂ) → 𝑊:ℝ⟶ℂ)
14	12, 13	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑊:ℝ⟶ℂ)
15		ssid 3773	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ
16	15	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℝ)
17	3, 14, 16	dvbss 23885	. . . . . . . 8 ⊢ (𝜑 → dom (ℝ D 𝑊) ⊆ ℝ)
18	17	adantr 466	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → dom (ℝ D 𝑊) ⊆ ℝ)
19		simpr 471	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → ℎ ∈ dom (ℝ D 𝑊))
20	18, 19	sseldd 3753	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → ℎ ∈ ℝ)
21	1, 20	jca 501	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → (𝜑 ∧ ℎ ∈ ℝ))
22	15	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ ℝ) → ℝ ⊆ ℝ)
23	14	adantr 466	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ ℝ) → 𝑊:ℝ⟶ℂ)
24	8	ad2antrr 705	. . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) → 𝐶 ∈ (-1(,)1))
25		simprr 756	. . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) → 𝑑 ∈ ℝ⁺)
26		simprl 754	. . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) → 𝑒 ∈ ℝ⁺)
27		simplr 752	. . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) → ℎ ∈ ℝ)
28	7	ad2antrr 705	. . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) → 𝑁 ∈ ℕ)
29		knoppndv.1	. . . . . . . . 9 ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶)))
30	29	ad2antrr 705	. . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) → 1 < (𝑁 · (abs‘𝐶)))
31	4, 5, 6, 24, 25, 26, 27, 28, 30	knoppndvlem22 32861	. . . . . . 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))))
32	31	ralrimivva 3120	. . . . . 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))))
33	22, 23, 32	unbdqndv2 32839	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ ℝ) → ¬ ℎ ∈ dom (ℝ D 𝑊))
34	21, 33	syl 17	. . . 4 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → ¬ ℎ ∈ dom (ℝ D 𝑊))
35	34	pm2.01da 800	. . 3 ⊢ (𝜑 → ¬ ℎ ∈ dom (ℝ D 𝑊))
36	35	alrimiv 2007	. 2 ⊢ (𝜑 → ∀ℎ ¬ ℎ ∈ dom (ℝ D 𝑊))
37		eq0 4077	. 2 ⊢ (dom (ℝ D 𝑊) = ∅ ↔ ∀ℎ ¬ ℎ ∈ dom (ℝ D 𝑊))
38	36, 37	sylibr 224	1 ⊢ (𝜑 → dom (ℝ D 𝑊) = ∅)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 382 ∧ w3a 1071 ∀wal 1629 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∃wrex 3062 ⊆ wss 3723 ∅c0 4063 class class class wbr 4787 ↦ cmpt 4864 dom cdm 5250 ⟶wf 6026 ‘cfv 6030 (class class class)co 6796 ℂcc 10140 ℝcr 10141 1c1 10143 + caddc 10145 · cmul 10147 < clt 10280 ≤ cle 10281 − cmin 10472 -cneg 10473 / cdiv 10890 ℕcn 11226 2c2 11276 ℕ₀cn0 11499 ℝ⁺crp 12035 (,)cioo 12380 ⌊cfl 12799 ↑cexp 13067 abscabs 14182 Σcsu 14624 –cn→ccncf 22899 D cdv 23847

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-inf2 8706 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-pre-sup 10220 ax-addf 10221 ax-mulf 10222

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-iin 4658 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-of 7048 df-om 7217 df-1st 7319 df-2nd 7320 df-supp 7451 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-2o 7718 df-oadd 7721 df-er 7900 df-map 8015 df-pm 8016 df-ixp 8067 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-fsupp 8436 df-fi 8477 df-sup 8508 df-inf 8509 df-oi 8575 df-card 8969 df-cda 9196 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-nn 11227 df-2 11285 df-3 11286 df-4 11287 df-5 11288 df-6 11289 df-7 11290 df-8 11291 df-9 11292 df-n0 11500 df-z 11585 df-dec 11701 df-uz 11894 df-q 11997 df-rp 12036 df-xneg 12151 df-xadd 12152 df-xmul 12153 df-ioo 12384 df-ico 12386 df-icc 12387 df-fz 12534 df-fzo 12674 df-fl 12801 df-seq 13009 df-exp 13068 df-hash 13322 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-limsup 14410 df-clim 14427 df-rlim 14428 df-sum 14625 df-dvds 15190 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-starv 16164 df-sca 16165 df-vsca 16166 df-ip 16167 df-tset 16168 df-ple 16169 df-ds 16172 df-unif 16173 df-hom 16174 df-cco 16175 df-rest 16291 df-topn 16292 df-0g 16310 df-gsum 16311 df-topgen 16312 df-pt 16313 df-prds 16316 df-xrs 16370 df-qtop 16375 df-imas 16376 df-xps 16378 df-mre 16454 df-mrc 16455 df-acs 16457 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-submnd 17544 df-mulg 17749 df-cntz 17957 df-cmn 18402 df-psmet 19953 df-xmet 19954 df-met 19955 df-bl 19956 df-mopn 19957 df-cnfld 19962 df-top 20919 df-topon 20936 df-topsp 20958 df-bases 20971 df-ntr 21045 df-cn 21252 df-cnp 21253 df-tx 21586 df-hmeo 21779 df-xms 22345 df-ms 22346 df-tms 22347 df-cncf 22901 df-limc 23850 df-dv 23851 df-ulm 24351

This theorem is referenced by: cnndvlem1 32865

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