knoppndv - Mathbox for Asger C. Ipsen

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Theorem knoppndv 36794

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 482	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → 𝜑)
2		ax-resscn 11095	. . . . . . . . . 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 36774	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
10	9	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℝ)
11	9	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘𝐶) < 1)
12	4, 5, 6, 7, 10, 11	knoppcn 36764	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ (ℝ–cn→ℂ))
13		cncff 24860	. . . . . . . . . 10 ⊢ (𝑊 ∈ (ℝ–cn→ℂ) → 𝑊:ℝ⟶ℂ)
14	12, 13	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑊:ℝ⟶ℂ)
15		ssidd 3945	. . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℝ)
16	3, 14, 15	dvbss 25868	. . . . . . . 8 ⊢ (𝜑 → dom (ℝ D 𝑊) ⊆ ℝ)
17	16	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → dom (ℝ D 𝑊) ⊆ ℝ)
18		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → ℎ ∈ dom (ℝ D 𝑊))
19	17, 18	sseldd 3922	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → ℎ ∈ ℝ)
20	1, 19	jca 511	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ dom (ℝ D 𝑊)) → (𝜑 ∧ ℎ ∈ ℝ))
21		ssidd 3945	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ ℝ) → ℝ ⊆ ℝ)
22	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ ℝ) → 𝑊:ℝ⟶ℂ)
23	8	ad2antrr 727	. . . . . . . 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 727	. . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) → 𝑁 ∈ ℕ)
28		knoppndv.1	. . . . . . . . 9 ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶)))
29	28	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ ℝ) ∧ (𝑒 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺)) → 1 < (𝑁 · (abs‘𝐶)))
30	4, 5, 6, 23, 24, 25, 26, 27, 29	knoppndvlem22 36793	. . . . . . 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 3180	. . . . . 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 36771	. . . . 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 1929	. 2 ⊢ (𝜑 → ∀ℎ ¬ ℎ ∈ dom (ℝ D 𝑊))
36		eq0 4290	. 2 ⊢ (dom (ℝ D 𝑊) = ∅ ↔ ∀ℎ ¬ ℎ ∈ dom (ℝ D 𝑊))
37	35, 36	sylibr 234	1 ⊢ (𝜑 → dom (ℝ D 𝑊) = ∅)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 ∀wal 1540 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∃wrex 3061 ⊆ wss 3889 ∅c0 4273 class class class wbr 5085 ↦ cmpt 5166 dom cdm 5631 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ℝcr 11037 1c1 11039 + caddc 11041 · cmul 11043 < clt 11179 ≤ cle 11180 − cmin 11377 -cneg 11378 / cdiv 11807 ℕcn 12174 2c2 12236 ℕ₀cn0 12437 ℝ⁺crp 12942 (,)cioo 13298 ⌊cfl 13749 ↑cexp 14023 abscabs 15196 Σcsu 15648 –cn→ccncf 24843 D cdv 25830

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-dvds 16222 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-ntr 22985 df-cn 23192 df-cnp 23193 df-tx 23527 df-hmeo 23720 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-ulm 26342

This theorem is referenced by: cnndvlem1 36797

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