knoppndv - Mathbox for Asger C. Ipsen

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Theorem knoppndv 36042

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 481	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → 𝜑)
2		ax-resscn 11203	. . . . . . . . . 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 36022	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
10	9	simpld 493	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℝ)
11	9	simprd 494	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘𝐶) < 1)
12	4, 5, 6, 7, 10, 11	knoppcn 36012	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ (ℝ–cn→ℂ))
13		cncff 24833	. . . . . . . . . 10 ⊢ (𝑊 ∈ (ℝ–cn→ℂ) → 𝑊:ℝ⟶ℂ)
14	12, 13	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑊:ℝ⟶ℂ)
15		ssidd 4005	. . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℝ)
16	3, 14, 15	dvbss 25850	. . . . . . . 8 ⊢ (𝜑 → dom (ℝ D 𝑊) ⊆ ℝ)
17	16	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → dom (ℝ D 𝑊) ⊆ ℝ)
18		simpr 483	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → ℎ ∈ dom (ℝ D 𝑊))
19	17, 18	sseldd 3983	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → ℎ ∈ ℝ)
20	1, 19	jca 510	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → (𝜑 ∧ ℎ ∈ ℝ))
21		ssidd 4005	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ ℝ) → ℝ ⊆ ℝ)
22	14	adantr 479	. . . . . 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 36041	. . . . . . 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 3198	. . . . . 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 36019	. . . . 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 1922	. 2 ⊢ (𝜑 → ∀ℎ ¬ ℎ ∈ dom (ℝ D 𝑊))
36		eq0 4347	. 2 ⊢ (dom (ℝ D 𝑊) = ∅ ↔ ∀ℎ ¬ ℎ ∈ dom (ℝ D 𝑊))
37	35, 36	sylibr 233	1 ⊢ (𝜑 → dom (ℝ D 𝑊) = ∅)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1084 ∀wal 1531 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∃wrex 3067 ⊆ wss 3949 ∅c0 4326 class class class wbr 5152 ↦ cmpt 5235 dom cdm 5682 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 ℝcr 11145 1c1 11147 + caddc 11149 · cmul 11151 < clt 11286 ≤ cle 11287 − cmin 11482 -cneg 11483 / cdiv 11909 ℕcn 12250 2c2 12305 ℕ₀cn0 12510 ℝ⁺crp 13014 (,)cioo 13364 ⌊cfl 13795 ↑cexp 14066 abscabs 15221 Σcsu 15672 –cn→ccncf 24816 D cdv 25812

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-limsup 15455 df-clim 15472 df-rlim 15473 df-sum 15673 df-dvds 16239 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-mulg 19031 df-cntz 19275 df-cmn 19744 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-ntr 22944 df-cn 23151 df-cnp 23152 df-tx 23486 df-hmeo 23679 df-xms 24246 df-ms 24247 df-tms 24248 df-cncf 24818 df-limc 25815 df-dv 25816 df-ulm 26333

This theorem is referenced by: cnndvlem1 36045

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