cnndvlem1 - Mathbox for Asger C. Ipsen

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Theorem cnndvlem1 33036

Description: Lemma for cnndv 33038. (Contributed by Asger C. Ipsen, 25-Aug-2021.)

**Hypotheses**
Ref	Expression
cnndvlem1.t	⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))
cnndvlem1.f	⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ₀ ↦ (((1 / 2)↑𝑛) · (𝑇‘(((2 · 3)↑𝑛) · 𝑦)))))
cnndvlem1.w	⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ₀ ((𝐹‘𝑤)‘𝑖))

**Assertion**
Ref	Expression
cnndvlem1	⊢ (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅)

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

**Proof of Theorem cnndvlem1**
Step	Hyp	Ref	Expression
1		cnndvlem1.t	. . . 4 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))
2		cnndvlem1.f	. . . 4 ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ₀ ↦ (((1 / 2)↑𝑛) · (𝑇‘(((2 · 3)↑𝑛) · 𝑦)))))
3		cnndvlem1.w	. . . 4 ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ₀ ((𝐹‘𝑤)‘𝑖))
4		3nn 11392	. . . . 5 ⊢ 3 ∈ ℕ
5	4	a1i 11	. . . 4 ⊢ (⊤ → 3 ∈ ℕ)
6		neg1rr 11435	. . . . . . . . 9 ⊢ -1 ∈ ℝ
7	6	rexri 10387	. . . . . . . 8 ⊢ -1 ∈ ℝ^*
8		1re 10328	. . . . . . . . 9 ⊢ 1 ∈ ℝ
9	8	rexri 10387	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
10		halfre 11534	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ
11	10	rexri 10387	. . . . . . . 8 ⊢ (1 / 2) ∈ ℝ^*
12	7, 9, 11	3pm3.2i 1439	. . . . . . 7 ⊢ (-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ (1 / 2) ∈ ℝ^*)
13		neg1lt0 11437	. . . . . . . . . 10 ⊢ -1 < 0
14		halfgt0 11536	. . . . . . . . . 10 ⊢ 0 < (1 / 2)
15	13, 14	pm3.2i 463	. . . . . . . . 9 ⊢ (-1 < 0 ∧ 0 < (1 / 2))
16		0re 10330	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
17	6, 16, 10	lttri 10453	. . . . . . . . 9 ⊢ ((-1 < 0 ∧ 0 < (1 / 2)) → -1 < (1 / 2))
18	15, 17	ax-mp 5	. . . . . . . 8 ⊢ -1 < (1 / 2)
19		halflt1 11538	. . . . . . . 8 ⊢ (1 / 2) < 1
20	18, 19	pm3.2i 463	. . . . . . 7 ⊢ (-1 < (1 / 2) ∧ (1 / 2) < 1)
21	12, 20	pm3.2i 463	. . . . . 6 ⊢ ((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ (1 / 2) ∈ ℝ^*) ∧ (-1 < (1 / 2) ∧ (1 / 2) < 1))
22		elioo3g 12453	. . . . . 6 ⊢ ((1 / 2) ∈ (-1(,)1) ↔ ((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ (1 / 2) ∈ ℝ^*) ∧ (-1 < (1 / 2) ∧ (1 / 2) < 1)))
23	21, 22	mpbir 223	. . . . 5 ⊢ (1 / 2) ∈ (-1(,)1)
24	23	a1i 11	. . . 4 ⊢ (⊤ → (1 / 2) ∈ (-1(,)1))
25	1, 2, 3, 5, 24	knoppcn2 33035	. . 3 ⊢ (⊤ → 𝑊 ∈ (ℝ–cn→ℝ))
26	25	mptru 1661	. 2 ⊢ 𝑊 ∈ (ℝ–cn→ℝ)
27		2cn 11388	. . . . . . . . 9 ⊢ 2 ∈ ℂ
28	27	mulid2i 10334	. . . . . . . 8 ⊢ (1 · 2) = 2
29		2lt3 11492	. . . . . . . 8 ⊢ 2 < 3
30	28, 29	eqbrtri 4864	. . . . . . 7 ⊢ (1 · 2) < 3
31		2pos 11423	. . . . . . . 8 ⊢ 0 < 2
32	4	nnrei 11322	. . . . . . . . 9 ⊢ 3 ∈ ℝ
33		2re 11387	. . . . . . . . 9 ⊢ 2 ∈ ℝ
34	8, 32, 33	ltmuldivi 11236	. . . . . . . 8 ⊢ (0 < 2 → ((1 · 2) < 3 ↔ 1 < (3 / 2)))
35	31, 34	ax-mp 5	. . . . . . 7 ⊢ ((1 · 2) < 3 ↔ 1 < (3 / 2))
36	30, 35	mpbi 222	. . . . . 6 ⊢ 1 < (3 / 2)
37	16, 10, 14	ltleii 10450	. . . . . . . . 9 ⊢ 0 ≤ (1 / 2)
38	10	absidi 14458	. . . . . . . . 9 ⊢ (0 ≤ (1 / 2) → (abs‘(1 / 2)) = (1 / 2))
39	37, 38	ax-mp 5	. . . . . . . 8 ⊢ (abs‘(1 / 2)) = (1 / 2)
40	39	oveq2i 6889	. . . . . . 7 ⊢ (3 · (abs‘(1 / 2))) = (3 · (1 / 2))
41	4	nncni 11323	. . . . . . . . 9 ⊢ 3 ∈ ℂ
42		2ne0 11424	. . . . . . . . 9 ⊢ 2 ≠ 0
43	41, 27, 42	divreci 11062	. . . . . . . 8 ⊢ (3 / 2) = (3 · (1 / 2))
44	43	eqcomi 2808	. . . . . . 7 ⊢ (3 · (1 / 2)) = (3 / 2)
45	40, 44	eqtri 2821	. . . . . 6 ⊢ (3 · (abs‘(1 / 2))) = (3 / 2)
46	36, 45	breqtrri 4870	. . . . 5 ⊢ 1 < (3 · (abs‘(1 / 2)))
47	46	a1i 11	. . . 4 ⊢ (⊤ → 1 < (3 · (abs‘(1 / 2))))
48	1, 2, 3, 24, 5, 47	knoppndv 33033	. . 3 ⊢ (⊤ → dom (ℝ D 𝑊) = ∅)
49	48	mptru 1661	. 2 ⊢ dom (ℝ D 𝑊) = ∅
50	26, 49	pm3.2i 463	1 ⊢ (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 198 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ⊤wtru 1654 ∈ wcel 2157 ∅c0 4115 class class class wbr 4843 ↦ cmpt 4922 dom cdm 5312 ‘cfv 6101 (class class class)co 6878 ℝcr 10223 0cc0 10224 1c1 10225 + caddc 10227 · cmul 10229 ℝ^*cxr 10362 < clt 10363 ≤ cle 10364 − cmin 10556 -cneg 10557 / cdiv 10976 ℕcn 11312 2c2 11368 3c3 11369 ℕ₀cn0 11580 (,)cioo 12424 ⌊cfl 12846 ↑cexp 13114 abscabs 14315 Σcsu 14757 –cn→ccncf 23007 D cdv 23968

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 ax-addf 10303 ax-mulf 10304

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-om 7300 df-1st 7401 df-2nd 7402 df-supp 7533 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-2o 7800 df-oadd 7803 df-er 7982 df-map 8097 df-pm 8098 df-ixp 8149 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fsupp 8518 df-fi 8559 df-sup 8590 df-inf 8591 df-oi 8657 df-card 9051 df-cda 9278 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-q 12034 df-rp 12075 df-xneg 12193 df-xadd 12194 df-xmul 12195 df-ioo 12428 df-ico 12430 df-icc 12431 df-fz 12581 df-fzo 12721 df-fl 12848 df-seq 13056 df-exp 13115 df-hash 13371 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-limsup 14543 df-clim 14560 df-rlim 14561 df-sum 14758 df-dvds 15320 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-mulr 16281 df-starv 16282 df-sca 16283 df-vsca 16284 df-ip 16285 df-tset 16286 df-ple 16287 df-ds 16289 df-unif 16290 df-hom 16291 df-cco 16292 df-rest 16398 df-topn 16399 df-0g 16417 df-gsum 16418 df-topgen 16419 df-pt 16420 df-prds 16423 df-xrs 16477 df-qtop 16482 df-imas 16483 df-xps 16485 df-mre 16561 df-mrc 16562 df-acs 16564 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-submnd 17651 df-mulg 17857 df-cntz 18062 df-cmn 18510 df-psmet 20060 df-xmet 20061 df-met 20062 df-bl 20063 df-mopn 20064 df-cnfld 20069 df-top 21027 df-topon 21044 df-topsp 21066 df-bases 21079 df-ntr 21153 df-cn 21360 df-cnp 21361 df-tx 21694 df-hmeo 21887 df-xms 22453 df-ms 22454 df-tms 22455 df-cncf 23009 df-limc 23971 df-dv 23972 df-ulm 24472

This theorem is referenced by: cnndvlem2 33037

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