cnndvlem1 - Mathbox for Asger C. Ipsen

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Theorem cnndvlem1 32653

Description: Lemma for cnndv 32655. (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 11224	. . . . 5 ⊢ 3 ∈ ℕ
5	4	a1i 11	. . . 4 ⊢ (⊤ → 3 ∈ ℕ)
6		neg1rr 11163	. . . . . . . . 9 ⊢ -1 ∈ ℝ
7	6	rexri 10135	. . . . . . . 8 ⊢ -1 ∈ ℝ^*
8		1re 10077	. . . . . . . . 9 ⊢ 1 ∈ ℝ
9	8	rexri 10135	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
10		halfre 11284	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ
11	10	rexri 10135	. . . . . . . 8 ⊢ (1 / 2) ∈ ℝ^*
12	7, 9, 11	3pm3.2i 1259	. . . . . . 7 ⊢ (-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ (1 / 2) ∈ ℝ^*)
13		neg1lt0 11165	. . . . . . . . . 10 ⊢ -1 < 0
14		halfgt0 11286	. . . . . . . . . 10 ⊢ 0 < (1 / 2)
15	13, 14	pm3.2i 470	. . . . . . . . 9 ⊢ (-1 < 0 ∧ 0 < (1 / 2))
16		0re 10078	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
17	6, 16, 10	lttri 10201	. . . . . . . . 9 ⊢ ((-1 < 0 ∧ 0 < (1 / 2)) → -1 < (1 / 2))
18	15, 17	ax-mp 5	. . . . . . . 8 ⊢ -1 < (1 / 2)
19		halflt1 11288	. . . . . . . 8 ⊢ (1 / 2) < 1
20	18, 19	pm3.2i 470	. . . . . . 7 ⊢ (-1 < (1 / 2) ∧ (1 / 2) < 1)
21	12, 20	pm3.2i 470	. . . . . 6 ⊢ ((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ (1 / 2) ∈ ℝ^*) ∧ (-1 < (1 / 2) ∧ (1 / 2) < 1))
22		elioo3g 12242	. . . . . 6 ⊢ ((1 / 2) ∈ (-1(,)1) ↔ ((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ (1 / 2) ∈ ℝ^*) ∧ (-1 < (1 / 2) ∧ (1 / 2) < 1)))
23	21, 22	mpbir 221	. . . . 5 ⊢ (1 / 2) ∈ (-1(,)1)
24	23	a1i 11	. . . 4 ⊢ (⊤ → (1 / 2) ∈ (-1(,)1))
25	1, 2, 3, 5, 24	knoppcn2 32652	. . 3 ⊢ (⊤ → 𝑊 ∈ (ℝ–cn→ℝ))
26	25	trud 1533	. 2 ⊢ 𝑊 ∈ (ℝ–cn→ℝ)
27		2cn 11129	. . . . . . . . 9 ⊢ 2 ∈ ℂ
28	27	mulid2i 10081	. . . . . . . 8 ⊢ (1 · 2) = 2
29		2lt3 11233	. . . . . . . 8 ⊢ 2 < 3
30	28, 29	eqbrtri 4706	. . . . . . 7 ⊢ (1 · 2) < 3
31		2pos 11150	. . . . . . . 8 ⊢ 0 < 2
32	4	nnrei 11067	. . . . . . . . 9 ⊢ 3 ∈ ℝ
33		2re 11128	. . . . . . . . 9 ⊢ 2 ∈ ℝ
34	8, 32, 33	ltmuldivi 10982	. . . . . . . 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 220	. . . . . 6 ⊢ 1 < (3 / 2)
37	16, 10, 14	ltleii 10198	. . . . . . . . 9 ⊢ 0 ≤ (1 / 2)
38	10	absidi 14161	. . . . . . . . 9 ⊢ (0 ≤ (1 / 2) → (abs‘(1 / 2)) = (1 / 2))
39	37, 38	ax-mp 5	. . . . . . . 8 ⊢ (abs‘(1 / 2)) = (1 / 2)
40	39	oveq2i 6701	. . . . . . 7 ⊢ (3 · (abs‘(1 / 2))) = (3 · (1 / 2))
41	4	nncni 11068	. . . . . . . . 9 ⊢ 3 ∈ ℂ
42		2ne0 11151	. . . . . . . . 9 ⊢ 2 ≠ 0
43	41, 27, 42	divreci 10808	. . . . . . . 8 ⊢ (3 / 2) = (3 · (1 / 2))
44	43	eqcomi 2660	. . . . . . 7 ⊢ (3 · (1 / 2)) = (3 / 2)
45	40, 44	eqtri 2673	. . . . . 6 ⊢ (3 · (abs‘(1 / 2))) = (3 / 2)
46	36, 45	breqtrri 4712	. . . . 5 ⊢ 1 < (3 · (abs‘(1 / 2)))
47	46	a1i 11	. . . 4 ⊢ (⊤ → 1 < (3 · (abs‘(1 / 2))))
48	1, 2, 3, 24, 5, 47	knoppndv 32650	. . 3 ⊢ (⊤ → dom (ℝ D 𝑊) = ∅)
49	48	trud 1533	. 2 ⊢ dom (ℝ D 𝑊) = ∅
50	26, 49	pm3.2i 470	1 ⊢ (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 196 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ⊤wtru 1524 ∈ wcel 2030 ∅c0 3948 class class class wbr 4685 ↦ cmpt 4762 dom cdm 5143 ‘cfv 5926 (class class class)co 6690 ℝcr 9973 0cc0 9974 1c1 9975 + caddc 9977 · cmul 9979 ℝ^*cxr 10111 < clt 10112 ≤ cle 10113 − cmin 10304 -cneg 10305 / cdiv 10722 ℕcn 11058 2c2 11108 3c3 11109 ℕ₀cn0 11330 (,)cioo 12213 ⌊cfl 12631 ↑cexp 12900 abscabs 14018 Σcsu 14460 –cn→ccncf 22726 D cdv 23672

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ioo 12217 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-seq 12842 df-exp 12901 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-limsup 14246 df-clim 14263 df-rlim 14264 df-sum 14461 df-dvds 15028 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-pt 16152 df-prds 16155 df-xrs 16209 df-qtop 16214 df-imas 16215 df-xps 16217 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-mulg 17588 df-cntz 17796 df-cmn 18241 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-cnfld 19795 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-ntr 20872 df-cn 21079 df-cnp 21080 df-tx 21413 df-hmeo 21606 df-xms 22172 df-ms 22173 df-tms 22174 df-cncf 22728 df-limc 23675 df-dv 23676 df-ulm 24176

This theorem is referenced by: cnndvlem2 32654

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