cnndvlem1 - Mathbox for Asger C. Ipsen

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Theorem cnndvlem1 36518

Description: Lemma for cnndv 36520. (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 12241	. . . . 5 ⊢ 3 ∈ ℕ
5	4	a1i 11	. . . 4 ⊢ (⊤ → 3 ∈ ℕ)
6		neg1rr 12148	. . . . . . . . 9 ⊢ -1 ∈ ℝ
7	6	rexri 11208	. . . . . . . 8 ⊢ -1 ∈ ℝ^*
8		1re 11150	. . . . . . . . 9 ⊢ 1 ∈ ℝ
9	8	rexri 11208	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
10		halfre 12371	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ
11	10	rexri 11208	. . . . . . . 8 ⊢ (1 / 2) ∈ ℝ^*
12	7, 9, 11	3pm3.2i 1340	. . . . . . 7 ⊢ (-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ (1 / 2) ∈ ℝ^*)
13		neg1lt0 12150	. . . . . . . . . 10 ⊢ -1 < 0
14		halfgt0 12373	. . . . . . . . . 10 ⊢ 0 < (1 / 2)
15	13, 14	pm3.2i 470	. . . . . . . . 9 ⊢ (-1 < 0 ∧ 0 < (1 / 2))
16		0re 11152	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
17	6, 16, 10	lttri 11276	. . . . . . . . 9 ⊢ ((-1 < 0 ∧ 0 < (1 / 2)) → -1 < (1 / 2))
18	15, 17	ax-mp 5	. . . . . . . 8 ⊢ -1 < (1 / 2)
19		halflt1 12375	. . . . . . . 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 13311	. . . . . 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 36517	. . 3 ⊢ (⊤ → 𝑊 ∈ (ℝ–cn→ℝ))
26	25	mptru 1547	. 2 ⊢ 𝑊 ∈ (ℝ–cn→ℝ)
27		2cn 12237	. . . . . . . . 9 ⊢ 2 ∈ ℂ
28	27	mullidi 11155	. . . . . . . 8 ⊢ (1 · 2) = 2
29		2lt3 12329	. . . . . . . 8 ⊢ 2 < 3
30	28, 29	eqbrtri 5123	. . . . . . 7 ⊢ (1 · 2) < 3
31		2pos 12265	. . . . . . . 8 ⊢ 0 < 2
32	4	nnrei 12171	. . . . . . . . 9 ⊢ 3 ∈ ℝ
33		2re 12236	. . . . . . . . 9 ⊢ 2 ∈ ℝ
34	8, 32, 33	ltmuldivi 12079	. . . . . . . 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 11273	. . . . . . . . 9 ⊢ 0 ≤ (1 / 2)
38	10	absidi 15320	. . . . . . . . 9 ⊢ (0 ≤ (1 / 2) → (abs‘(1 / 2)) = (1 / 2))
39	37, 38	ax-mp 5	. . . . . . . 8 ⊢ (abs‘(1 / 2)) = (1 / 2)
40	39	oveq2i 7380	. . . . . . 7 ⊢ (3 · (abs‘(1 / 2))) = (3 · (1 / 2))
41	4	nncni 12172	. . . . . . . . 9 ⊢ 3 ∈ ℂ
42		2ne0 12266	. . . . . . . . 9 ⊢ 2 ≠ 0
43	41, 27, 42	divreci 11903	. . . . . . . 8 ⊢ (3 / 2) = (3 · (1 / 2))
44	43	eqcomi 2738	. . . . . . 7 ⊢ (3 · (1 / 2)) = (3 / 2)
45	40, 44	eqtri 2752	. . . . . 6 ⊢ (3 · (abs‘(1 / 2))) = (3 / 2)
46	36, 45	breqtrri 5129	. . . . 5 ⊢ 1 < (3 · (abs‘(1 / 2)))
47	46	a1i 11	. . . 4 ⊢ (⊤ → 1 < (3 · (abs‘(1 / 2))))
48	1, 2, 3, 24, 5, 47	knoppndv 36515	. . 3 ⊢ (⊤ → dom (ℝ D 𝑊) = ∅)
49	48	mptru 1547	. 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 1540 ⊤wtru 1541 ∈ wcel 2109 ∅c0 4292 class class class wbr 5102 ↦ cmpt 5183 dom cdm 5631 ‘cfv 6499 (class class class)co 7369 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 · cmul 11049 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 − cmin 11381 -cneg 11382 / cdiv 11811 ℕcn 12162 2c2 12217 3c3 12218 ℕ₀cn0 12418 (,)cioo 13282 ⌊cfl 13728 ↑cexp 14002 abscabs 15176 Σcsu 15628 –cn→ccncf 24802 D cdv 25797

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 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 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15629 df-dvds 16199 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-submnd 18693 df-mulg 18982 df-cntz 19231 df-cmn 19696 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-cnfld 21297 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22866 df-ntr 22940 df-cn 23147 df-cnp 23148 df-tx 23482 df-hmeo 23675 df-xms 24241 df-ms 24242 df-tms 24243 df-cncf 24804 df-limc 25800 df-dv 25801 df-ulm 26319

This theorem is referenced by: cnndvlem2 36519

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