cnndvlem1 - Mathbox for Asger C. Ipsen

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Theorem cnndvlem1 37074

Description: Lemma for cnndv 37076. (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 12323	. . . . 5 ⊢ 3 ∈ ℕ
5	4	a1i 11	. . . 4 ⊢ (⊤ → 3 ∈ ℕ)
6		neg1rr 12207	. . . . . . . . 9 ⊢ -1 ∈ ℝ
7	6	rexri 11270	. . . . . . . 8 ⊢ -1 ∈ ℝ^*
8		1re 11211	. . . . . . . . 9 ⊢ 1 ∈ ℝ
9	8	rexri 11270	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
10		halfre 12460	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ
11	10	rexri 11270	. . . . . . . 8 ⊢ (1 / 2) ∈ ℝ^*
12	7, 9, 11	3pm3.2i 1356	. . . . . . 7 ⊢ (-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ (1 / 2) ∈ ℝ^*)
13		neg1lt0 12209	. . . . . . . . . 10 ⊢ -1 < 0
14		halfgt0 12462	. . . . . . . . . 10 ⊢ 0 < (1 / 2)
15	13, 14	pm3.2i 475	. . . . . . . . 9 ⊢ (-1 < 0 ∧ 0 < (1 / 2))
16		0re 11213	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
17	6, 16, 10	lttri 11339	. . . . . . . . 9 ⊢ ((-1 < 0 ∧ 0 < (1 / 2)) → -1 < (1 / 2))
18	15, 17	ax-mp 5	. . . . . . . 8 ⊢ -1 < (1 / 2)
19		halflt1 12464	. . . . . . . 8 ⊢ (1 / 2) < 1
20	18, 19	pm3.2i 475	. . . . . . 7 ⊢ (-1 < (1 / 2) ∧ (1 / 2) < 1)
21	12, 20	pm3.2i 475	. . . . . 6 ⊢ ((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ (1 / 2) ∈ ℝ^*) ∧ (-1 < (1 / 2) ∧ (1 / 2) < 1))
22		elioo3g 13404	. . . . . 6 ⊢ ((1 / 2) ∈ (-1(,)1) ↔ ((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ (1 / 2) ∈ ℝ^*) ∧ (-1 < (1 / 2) ∧ (1 / 2) < 1)))
23	21, 22	mpbir 234	. . . . 5 ⊢ (1 / 2) ∈ (-1(,)1)
24	23	a1i 11	. . . 4 ⊢ (⊤ → (1 / 2) ∈ (-1(,)1))
25	1, 2, 3, 5, 24	knoppcn2 37073	. . 3 ⊢ (⊤ → 𝑊 ∈ (ℝ–cn→ℝ))
26	25	mptru 1575	. 2 ⊢ 𝑊 ∈ (ℝ–cn→ℝ)
27		2cn 12319	. . . . . . . . 9 ⊢ 2 ∈ ℂ
28	27	mullidi 11217	. . . . . . . 8 ⊢ (1 · 2) = 2
29		2lt3 12417	. . . . . . . 8 ⊢ 2 < 3
30	28, 29	eqbrtri 5137	. . . . . . 7 ⊢ (1 · 2) < 3
31		2pos 12348	. . . . . . . 8 ⊢ 0 < 2
32	4	nnrei 12245	. . . . . . . . 9 ⊢ 3 ∈ ℝ
33		2re 12318	. . . . . . . . 9 ⊢ 2 ∈ ℝ
34	8, 32, 33	ltmuldivi 12138	. . . . . . . 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 233	. . . . . 6 ⊢ 1 < (3 / 2)
37	16, 10, 14	ltleii 11336	. . . . . . . . 9 ⊢ 0 ≤ (1 / 2)
38	10	absidi 15432	. . . . . . . . 9 ⊢ (0 ≤ (1 / 2) → (abs‘(1 / 2)) = (1 / 2))
39	37, 38	ax-mp 5	. . . . . . . 8 ⊢ (abs‘(1 / 2)) = (1 / 2)
40	39	oveq2i 7425	. . . . . . 7 ⊢ (3 · (abs‘(1 / 2))) = (3 · (1 / 2))
41	4	nncni 12246	. . . . . . . . 9 ⊢ 3 ∈ ℂ
42		2ne0 12350	. . . . . . . . 9 ⊢ 2 ≠ 0
43	41, 27, 42	divreci 11963	. . . . . . . 8 ⊢ (3 / 2) = (3 · (1 / 2))
44	43	eqcomi 2779	. . . . . . 7 ⊢ (3 · (1 / 2)) = (3 / 2)
45	40, 44	eqtri 2793	. . . . . 6 ⊢ (3 · (abs‘(1 / 2))) = (3 / 2)
46	36, 45	breqtrri 5143	. . . . 5 ⊢ 1 < (3 · (abs‘(1 / 2)))
47	46	a1i 11	. . . 4 ⊢ (⊤ → 1 < (3 · (abs‘(1 / 2))))
48	1, 2, 3, 24, 5, 47	knoppndv 37071	. . 3 ⊢ (⊤ → dom (ℝ D 𝑊) = ∅)
49	48	mptru 1575	. 2 ⊢ dom (ℝ D 𝑊) = ∅
50	26, 49	pm3.2i 475	1 ⊢ (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1568 ⊤wtru 1569 ∈ wcel 2150 ∅c0 4294 class class class wbr 5114 ↦ cmpt 5197 dom cdm 5665 ‘cfv 6540 (class class class)co 7414 ℝcr 11102 0cc0 11103 1c1 11104 + caddc 11106 · cmul 11108 ℝ^*cxr 11245 < clt 11246 ≤ cle 11247 − cmin 11444 -cneg 11445 / cdiv 11874 ℕcn 12236 2c2 12298 3c3 12299 ℕ₀cn0 12507 (,)cioo 13375 ⌊cfl 13826 ↑cexp 14100 abscabs 15288 Σcsu 15740 –cn→ccncf 25018 D cdv 26005

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2152 ax-9 2160 ax-10 2183 ax-11 2199 ax-12 2220 ax-ext 2742 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5340 ax-pr 5408 ax-un 7736 ax-inf2 9613 ax-cnex 11159 ax-resscn 11160 ax-1cn 11161 ax-icn 11162 ax-addcl 11163 ax-addrcl 11164 ax-mulcl 11165 ax-mulrcl 11166 ax-mulcom 11167 ax-addass 11168 ax-mulass 11169 ax-distr 11170 ax-i2m1 11171 ax-1ne0 11172 ax-1rid 11173 ax-rnegex 11174 ax-rrecex 11175 ax-cnre 11176 ax-pre-lttri 11177 ax-pre-lttrn 11178 ax-pre-ltadd 11179 ax-pre-mulgt0 11180 ax-pre-sup 11181 ax-addf 11182

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2099 df-mo 2574 df-eu 2604 df-clab 2749 df-cleq 2762 df-clel 2845 df-nfc 2919 df-ne 2966 df-nel 3072 df-ral 3087 df-rex 3097 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3464 df-sbc 3753 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5560 df-eprel 5565 df-po 5573 df-so 5574 df-fr 5618 df-se 5619 df-we 5620 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7678 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-er 8697 df-map 8829 df-pm 8830 df-ixp 8899 df-en 8947 df-dom 8948 df-sdom 8949 df-fin 8950 df-fsupp 9325 df-fi 9374 df-sup 9405 df-inf 9406 df-oi 9475 df-card 9928 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11446 df-neg 11447 df-div 11875 df-nn 12237 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12508 df-z 12595 df-dec 12715 df-uz 12866 df-q 12976 df-rp 13020 df-xneg 13140 df-xadd 13141 df-xmul 13142 df-ioo 13379 df-ico 13381 df-icc 13382 df-fz 13539 df-fzo 13686 df-fl 13828 df-seq 14041 df-exp 14101 df-hash 14370 df-cj 15153 df-re 15154 df-im 15155 df-sqrt 15289 df-abs 15290 df-limsup 15525 df-clim 15542 df-rlim 15543 df-sum 15741 df-dvds 16314 df-struct 17210 df-sets 17227 df-slot 17245 df-ndx 17257 df-base 17273 df-ress 17294 df-plusg 17326 df-mulr 17327 df-starv 17328 df-sca 17329 df-vsca 17330 df-ip 17331 df-tset 17332 df-ple 17333 df-ds 17335 df-unif 17336 df-hom 17337 df-cco 17338 df-rest 17478 df-topn 17479 df-0g 17497 df-gsum 17498 df-topgen 17499 df-pt 17500 df-prds 17503 df-xrs 17559 df-qtop 17564 df-imas 17565 df-xps 17567 df-mre 17641 df-mrc 17642 df-acs 17644 df-mgm 18701 df-sgrp 18780 df-mnd 18796 df-submnd 18845 df-mulg 19137 df-cntz 19390 df-cmn 19855 df-psmet 21497 df-xmet 21498 df-met 21499 df-bl 21500 df-mopn 21501 df-cnfld 21506 df-top 23034 df-topon 23051 df-topsp 23073 df-bases 23086 df-ntr 23160 df-cn 23367 df-cnp 23368 df-tx 23702 df-hmeo 23895 df-xms 24460 df-ms 24461 df-tms 24462 df-cncf 25020 df-limc 26008 df-dv 26009 df-ulm 26520

This theorem is referenced by: cnndvlem2 37075

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