cnndvlem1 - Mathbox for Asger C. Ipsen

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Theorem cnndvlem1 37580

Description: Lemma for cnndv 37582. (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 12446	. . . . 5 ⊢ 3 ∈ ℕ
5	4	a1i 11	. . . 4 ⊢ (⊤ → 3 ∈ ℕ)
6		neg1rr 12330	. . . . . . . . 9 ⊢ -1 ∈ ℝ
7	6	rexri 11391	. . . . . . . 8 ⊢ -1 ∈ ℝ^*
8		1re 11332	. . . . . . . . 9 ⊢ 1 ∈ ℝ
9	8	rexri 11391	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
10		halfre 12583	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ
11	10	rexri 11391	. . . . . . . 8 ⊢ (1 / 2) ∈ ℝ^*
12	7, 9, 11	3pm3.2i 1358	. . . . . . 7 ⊢ ( -1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ (1 / 2) ∈ ℝ^*)
13		neg1lt0 12332	. . . . . . . . . 10 ⊢ -1 < 0
14		halfgt0 12585	. . . . . . . . . 10 ⊢ 0 < (1 / 2)
15	13, 14	pm3.2i 476	. . . . . . . . 9 ⊢ ( -1 < 0 ∧ 0 < (1 / 2))
16		0re 11334	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
17	6, 16, 10	lttri 11460	. . . . . . . . 9 ⊢ (( -1 < 0 ∧ 0 < (1 / 2)) → -1 < (1 / 2))
18	15, 17	ax-mp 5	. . . . . . . 8 ⊢ -1 < (1 / 2)
19		halflt1 12587	. . . . . . . 8 ⊢ (1 / 2) < 1
20	18, 19	pm3.2i 476	. . . . . . 7 ⊢ ( -1 < (1 / 2) ∧ (1 / 2) < 1)
21	12, 20	pm3.2i 476	. . . . . 6 ⊢ (( -1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ (1 / 2) ∈ ℝ^*) ∧ ( -1 < (1 / 2) ∧ (1 / 2) < 1))
22		elioo3g 13529	. . . . . 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 37579	. . 3 ⊢ (⊤ → 𝑊 ∈ (ℝ–cn→ℝ))
26	25	mptru 1577	. 2 ⊢ 𝑊 ∈ (ℝ–cn→ℝ)
27		2cn 12442	. . . . . . . . 9 ⊢ 2 ∈ ℂ
28	27	mullidi 11338	. . . . . . . 8 ⊢ (1 · 2) = 2
29		2lt3 12540	. . . . . . . 8 ⊢ 2 < 3
30	28, 29	eqbrtri 5129	. . . . . . 7 ⊢ (1 · 2) < 3
31		2pos 12471	. . . . . . . 8 ⊢ 0 < 2
32	4	nnrei 12368	. . . . . . . . 9 ⊢ 3 ∈ ℝ
33		2re 12441	. . . . . . . . 9 ⊢ 2 ∈ ℝ
34	8, 32, 33	ltmuldivi 12261	. . . . . . . 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 11457	. . . . . . . . 9 ⊢ 0 ≤ (1 / 2)
38	10	absidi 15571	. . . . . . . . 9 ⊢ (0 ≤ (1 / 2) → (abs‘(1 / 2)) = (1 / 2))
39	37, 38	ax-mp 5	. . . . . . . 8 ⊢ (abs‘(1 / 2)) = (1 / 2)
40	39	oveq2i 7451	. . . . . . 7 ⊢ (3 · (abs‘(1 / 2))) = (3 · (1 / 2))
41	4	nncni 12369	. . . . . . . . 9 ⊢ 3 ∈ ℂ
42		2ne0 12473	. . . . . . . . 9 ⊢ 2 ≠ 0
43	41, 27, 42	divreci 12086	. . . . . . . 8 ⊢ (3 / 2) = (3 · (1 / 2))
44	43	eqcomi 2772	. . . . . . 7 ⊢ (3 · (1 / 2)) = (3 / 2)
45	40, 44	eqtri 2786	. . . . . 6 ⊢ (3 · (abs‘(1 / 2))) = (3 / 2)
46	36, 45	breqtrri 5135	. . . . 5 ⊢ 1 < (3 · (abs‘(1 / 2)))
47	46	a1i 11	. . . 4 ⊢ (⊤ → 1 < (3 · (abs‘(1 / 2))))
48	1, 2, 3, 24, 5, 47	knoppndv 37577	. . 3 ⊢ (⊤ → dom (ℝ D 𝑊) = ∅)
49	48	mptru 1577	. 2 ⊢ dom (ℝ D 𝑊) = ∅
50	26, 49	pm3.2i 476	1 ⊢ (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ⊤wtru 1571 ∈ wcel 2147 ∅c0 4281 class class class wbr 5106 ↦ cmpt 5189 dom cdm 5655 ‘cfv 6555 (class class class)co 7440 ℝcr 11223 0cc0 11224 1c1 11225 + caddc 11227 · cmul 11229 ℝ^*cxr 11366 < clt 11367 ≤ cle 11368 − cmin 11565 -cneg 11566 / cdiv 11997 ℕcn 12359 2c2 12421 3c3 12422 ℕ₀cn0 12630 (,)cioo 13500 ⌊cfl 13954 ↑cexp 14228 abscabs 15427 Σcsu 15879 –cn→ccncf 25224 D cdv 26210

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-inf2 9666 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301 ax-pre-sup 11302 ax-addf 11303

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-isom 6564 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-of 7714 df-om 7899 df-1st 8022 df-2nd 8023 df-supp 8202 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-2o 8502 df-er 8742 df-map 8874 df-pm 8875 df-ixp 8951 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-fsupp 9379 df-fi 9427 df-sup 9458 df-inf 9459 df-oi 9528 df-card 10044 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-div 11998 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-9 12436 df-n0 12631 df-z 12718 df-dec 12839 df-uz 12990 df-q 13100 df-rp 13145 df-xneg 13265 df-xadd 13266 df-xmul 13267 df-ioo 13504 df-ico 13506 df-icc 13507 df-fz 13664 df-fzo 13813 df-fl 13956 df-seq 14169 df-exp 14229 df-hash 14499 df-cj 15292 df-re 15293 df-im 15294 df-sqrt 15428 df-abs 15429 df-limsup 15664 df-clim 15681 df-rlim 15682 df-sum 15880 df-dvds 16449 df-struct 17351 df-sets 17368 df-slot 17386 df-ndx 17398 df-base 17414 df-ress 17435 df-plusg 17467 df-mulr 17468 df-starv 17469 df-sca 17470 df-vsca 17471 df-ip 17472 df-tset 17473 df-ple 17474 df-ds 17476 df-unif 17477 df-hom 17478 df-cco 17479 df-rest 17619 df-topn 17620 df-0g 17638 df-gsum 17639 df-topgen 17640 df-pt 17641 df-prds 17644 df-xrs 17700 df-qtop 17705 df-imas 17706 df-xps 17708 df-mre 17782 df-mrc 17783 df-acs 17785 df-mgm 18842 df-sgrp 18934 df-mnd 18950 df-submnd 19005 df-mulg 19304 df-cntz 19558 df-cmn 20023 df-psmet 21697 df-xmet 21698 df-met 21699 df-bl 21700 df-mopn 21701 df-cnfld 21706 df-top 23239 df-topon 23256 df-topsp 23278 df-bases 23291 df-ntr 23365 df-cn 23572 df-cnp 23573 df-tx 23908 df-hmeo 24101 df-xms 24666 df-ms 24667 df-tms 24668 df-cncf 25226 df-limc 26213 df-dv 26214 df-ulm 26731

This theorem is used by: cnndvlem2 37581

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