cnndvlem1 - Mathbox for Asger C. Ipsen

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Theorem cnndvlem1 36581

Description: Lemma for cnndv 36583. (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 12204	. . . . 5 ⊢ 3 ∈ ℕ
5	4	a1i 11	. . . 4 ⊢ (⊤ → 3 ∈ ℕ)
6		neg1rr 12111	. . . . . . . . 9 ⊢ -1 ∈ ℝ
7	6	rexri 11170	. . . . . . . 8 ⊢ -1 ∈ ℝ^*
8		1re 11112	. . . . . . . . 9 ⊢ 1 ∈ ℝ
9	8	rexri 11170	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
10		halfre 12334	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ
11	10	rexri 11170	. . . . . . . 8 ⊢ (1 / 2) ∈ ℝ^*
12	7, 9, 11	3pm3.2i 1340	. . . . . . 7 ⊢ (-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ (1 / 2) ∈ ℝ^*)
13		neg1lt0 12113	. . . . . . . . . 10 ⊢ -1 < 0
14		halfgt0 12336	. . . . . . . . . 10 ⊢ 0 < (1 / 2)
15	13, 14	pm3.2i 470	. . . . . . . . 9 ⊢ (-1 < 0 ∧ 0 < (1 / 2))
16		0re 11114	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
17	6, 16, 10	lttri 11239	. . . . . . . . 9 ⊢ ((-1 < 0 ∧ 0 < (1 / 2)) → -1 < (1 / 2))
18	15, 17	ax-mp 5	. . . . . . . 8 ⊢ -1 < (1 / 2)
19		halflt1 12338	. . . . . . . 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 13274	. . . . . 6 ⊢ ((1 / 2) ∈ (-1(,)1) ↔ ((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ (1 / 2) ∈ ℝ^*) ∧ (-1 < (1 / 2) ∧ (1 / 2) < 1)))
23	21, 22	mpbir 231	. . . . 5 ⊢ (1 / 2) ∈ (-1(,)1)
24	23	a1i 11	. . . 4 ⊢ (⊤ → (1 / 2) ∈ (-1(,)1))
25	1, 2, 3, 5, 24	knoppcn2 36580	. . 3 ⊢ (⊤ → 𝑊 ∈ (ℝ–cn→ℝ))
26	25	mptru 1548	. 2 ⊢ 𝑊 ∈ (ℝ–cn→ℝ)
27		2cn 12200	. . . . . . . . 9 ⊢ 2 ∈ ℂ
28	27	mullidi 11117	. . . . . . . 8 ⊢ (1 · 2) = 2
29		2lt3 12292	. . . . . . . 8 ⊢ 2 < 3
30	28, 29	eqbrtri 5110	. . . . . . 7 ⊢ (1 · 2) < 3
31		2pos 12228	. . . . . . . 8 ⊢ 0 < 2
32	4	nnrei 12134	. . . . . . . . 9 ⊢ 3 ∈ ℝ
33		2re 12199	. . . . . . . . 9 ⊢ 2 ∈ ℝ
34	8, 32, 33	ltmuldivi 12042	. . . . . . . 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 230	. . . . . 6 ⊢ 1 < (3 / 2)
37	16, 10, 14	ltleii 11236	. . . . . . . . 9 ⊢ 0 ≤ (1 / 2)
38	10	absidi 15285	. . . . . . . . 9 ⊢ (0 ≤ (1 / 2) → (abs‘(1 / 2)) = (1 / 2))
39	37, 38	ax-mp 5	. . . . . . . 8 ⊢ (abs‘(1 / 2)) = (1 / 2)
40	39	oveq2i 7357	. . . . . . 7 ⊢ (3 · (abs‘(1 / 2))) = (3 · (1 / 2))
41	4	nncni 12135	. . . . . . . . 9 ⊢ 3 ∈ ℂ
42		2ne0 12229	. . . . . . . . 9 ⊢ 2 ≠ 0
43	41, 27, 42	divreci 11866	. . . . . . . 8 ⊢ (3 / 2) = (3 · (1 / 2))
44	43	eqcomi 2740	. . . . . . 7 ⊢ (3 · (1 / 2)) = (3 / 2)
45	40, 44	eqtri 2754	. . . . . 6 ⊢ (3 · (abs‘(1 / 2))) = (3 / 2)
46	36, 45	breqtrri 5116	. . . . 5 ⊢ 1 < (3 · (abs‘(1 / 2)))
47	46	a1i 11	. . . 4 ⊢ (⊤ → 1 < (3 · (abs‘(1 / 2))))
48	1, 2, 3, 24, 5, 47	knoppndv 36578	. . 3 ⊢ (⊤ → dom (ℝ D 𝑊) = ∅)
49	48	mptru 1548	. 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ⊤wtru 1542 ∈ wcel 2111 ∅c0 4280 class class class wbr 5089 ↦ cmpt 5170 dom cdm 5614 ‘cfv 6481 (class class class)co 7346 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 · cmul 11011 ℝ^*cxr 11145 < clt 11146 ≤ cle 11147 − cmin 11344 -cneg 11345 / cdiv 11774 ℕcn 12125 2c2 12180 3c3 12181 ℕ₀cn0 12381 (,)cioo 13245 ⌊cfl 13694 ↑cexp 13968 abscabs 15141 Σcsu 15593 –cn→ccncf 24796 D cdv 25791

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-dvds 16164 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-ntr 22935 df-cn 23142 df-cnp 23143 df-tx 23477 df-hmeo 23670 df-xms 24235 df-ms 24236 df-tms 24237 df-cncf 24798 df-limc 25794 df-dv 25795 df-ulm 26313

This theorem is referenced by: cnndvlem2 36582

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