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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnndvlem1 | Structured version Visualization version GIF version |
Description: Lemma for cnndv 33038. (Contributed by Asger C. Ipsen, 25-Aug-2021.) |
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)) |
Ref | Expression |
---|---|
cnndvlem1 | ⊢ (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅) |
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 11392 | . . . . 5 ⊢ 3 ∈ ℕ | |
5 | 4 | a1i 11 | . . . 4 ⊢ (⊤ → 3 ∈ ℕ) |
6 | neg1rr 11435 | . . . . . . . . 9 ⊢ -1 ∈ ℝ | |
7 | 6 | rexri 10387 | . . . . . . . 8 ⊢ -1 ∈ ℝ* |
8 | 1re 10328 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
9 | 8 | rexri 10387 | . . . . . . . 8 ⊢ 1 ∈ ℝ* |
10 | halfre 11534 | . . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ | |
11 | 10 | rexri 10387 | . . . . . . . 8 ⊢ (1 / 2) ∈ ℝ* |
12 | 7, 9, 11 | 3pm3.2i 1439 | . . . . . . 7 ⊢ (-1 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ (1 / 2) ∈ ℝ*) |
13 | neg1lt0 11437 | . . . . . . . . . 10 ⊢ -1 < 0 | |
14 | halfgt0 11536 | . . . . . . . . . 10 ⊢ 0 < (1 / 2) | |
15 | 13, 14 | pm3.2i 463 | . . . . . . . . 9 ⊢ (-1 < 0 ∧ 0 < (1 / 2)) |
16 | 0re 10330 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
17 | 6, 16, 10 | lttri 10453 | . . . . . . . . 9 ⊢ ((-1 < 0 ∧ 0 < (1 / 2)) → -1 < (1 / 2)) |
18 | 15, 17 | ax-mp 5 | . . . . . . . 8 ⊢ -1 < (1 / 2) |
19 | halflt1 11538 | . . . . . . . 8 ⊢ (1 / 2) < 1 | |
20 | 18, 19 | pm3.2i 463 | . . . . . . 7 ⊢ (-1 < (1 / 2) ∧ (1 / 2) < 1) |
21 | 12, 20 | pm3.2i 463 | . . . . . 6 ⊢ ((-1 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ (1 / 2) ∈ ℝ*) ∧ (-1 < (1 / 2) ∧ (1 / 2) < 1)) |
22 | elioo3g 12453 | . . . . . 6 ⊢ ((1 / 2) ∈ (-1(,)1) ↔ ((-1 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ (1 / 2) ∈ ℝ*) ∧ (-1 < (1 / 2) ∧ (1 / 2) < 1))) | |
23 | 21, 22 | mpbir 223 | . . . . 5 ⊢ (1 / 2) ∈ (-1(,)1) |
24 | 23 | a1i 11 | . . . 4 ⊢ (⊤ → (1 / 2) ∈ (-1(,)1)) |
25 | 1, 2, 3, 5, 24 | knoppcn2 33035 | . . 3 ⊢ (⊤ → 𝑊 ∈ (ℝ–cn→ℝ)) |
26 | 25 | mptru 1661 | . 2 ⊢ 𝑊 ∈ (ℝ–cn→ℝ) |
27 | 2cn 11388 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
28 | 27 | mulid2i 10334 | . . . . . . . 8 ⊢ (1 · 2) = 2 |
29 | 2lt3 11492 | . . . . . . . 8 ⊢ 2 < 3 | |
30 | 28, 29 | eqbrtri 4864 | . . . . . . 7 ⊢ (1 · 2) < 3 |
31 | 2pos 11423 | . . . . . . . 8 ⊢ 0 < 2 | |
32 | 4 | nnrei 11322 | . . . . . . . . 9 ⊢ 3 ∈ ℝ |
33 | 2re 11387 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
34 | 8, 32, 33 | ltmuldivi 11236 | . . . . . . . 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 222 | . . . . . 6 ⊢ 1 < (3 / 2) |
37 | 16, 10, 14 | ltleii 10450 | . . . . . . . . 9 ⊢ 0 ≤ (1 / 2) |
38 | 10 | absidi 14458 | . . . . . . . . 9 ⊢ (0 ≤ (1 / 2) → (abs‘(1 / 2)) = (1 / 2)) |
39 | 37, 38 | ax-mp 5 | . . . . . . . 8 ⊢ (abs‘(1 / 2)) = (1 / 2) |
40 | 39 | oveq2i 6889 | . . . . . . 7 ⊢ (3 · (abs‘(1 / 2))) = (3 · (1 / 2)) |
41 | 4 | nncni 11323 | . . . . . . . . 9 ⊢ 3 ∈ ℂ |
42 | 2ne0 11424 | . . . . . . . . 9 ⊢ 2 ≠ 0 | |
43 | 41, 27, 42 | divreci 11062 | . . . . . . . 8 ⊢ (3 / 2) = (3 · (1 / 2)) |
44 | 43 | eqcomi 2808 | . . . . . . 7 ⊢ (3 · (1 / 2)) = (3 / 2) |
45 | 40, 44 | eqtri 2821 | . . . . . 6 ⊢ (3 · (abs‘(1 / 2))) = (3 / 2) |
46 | 36, 45 | breqtrri 4870 | . . . . 5 ⊢ 1 < (3 · (abs‘(1 / 2))) |
47 | 46 | a1i 11 | . . . 4 ⊢ (⊤ → 1 < (3 · (abs‘(1 / 2)))) |
48 | 1, 2, 3, 24, 5, 47 | knoppndv 33033 | . . 3 ⊢ (⊤ → dom (ℝ D 𝑊) = ∅) |
49 | 48 | mptru 1661 | . 2 ⊢ dom (ℝ D 𝑊) = ∅ |
50 | 26, 49 | pm3.2i 463 | 1 ⊢ (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ⊤wtru 1654 ∈ wcel 2157 ∅c0 4115 class class class wbr 4843 ↦ cmpt 4922 dom cdm 5312 ‘cfv 6101 (class class class)co 6878 ℝcr 10223 0cc0 10224 1c1 10225 + caddc 10227 · cmul 10229 ℝ*cxr 10362 < clt 10363 ≤ cle 10364 − cmin 10556 -cneg 10557 / cdiv 10976 ℕcn 11312 2c2 11368 3c3 11369 ℕ0cn0 11580 (,)cioo 12424 ⌊cfl 12846 ↑cexp 13114 abscabs 14315 Σcsu 14757 –cn→ccncf 23007 D cdv 23968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 ax-addf 10303 ax-mulf 10304 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-om 7300 df-1st 7401 df-2nd 7402 df-supp 7533 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-2o 7800 df-oadd 7803 df-er 7982 df-map 8097 df-pm 8098 df-ixp 8149 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fsupp 8518 df-fi 8559 df-sup 8590 df-inf 8591 df-oi 8657 df-card 9051 df-cda 9278 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-q 12034 df-rp 12075 df-xneg 12193 df-xadd 12194 df-xmul 12195 df-ioo 12428 df-ico 12430 df-icc 12431 df-fz 12581 df-fzo 12721 df-fl 12848 df-seq 13056 df-exp 13115 df-hash 13371 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-limsup 14543 df-clim 14560 df-rlim 14561 df-sum 14758 df-dvds 15320 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-mulr 16281 df-starv 16282 df-sca 16283 df-vsca 16284 df-ip 16285 df-tset 16286 df-ple 16287 df-ds 16289 df-unif 16290 df-hom 16291 df-cco 16292 df-rest 16398 df-topn 16399 df-0g 16417 df-gsum 16418 df-topgen 16419 df-pt 16420 df-prds 16423 df-xrs 16477 df-qtop 16482 df-imas 16483 df-xps 16485 df-mre 16561 df-mrc 16562 df-acs 16564 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-submnd 17651 df-mulg 17857 df-cntz 18062 df-cmn 18510 df-psmet 20060 df-xmet 20061 df-met 20062 df-bl 20063 df-mopn 20064 df-cnfld 20069 df-top 21027 df-topon 21044 df-topsp 21066 df-bases 21079 df-ntr 21153 df-cn 21360 df-cnp 21361 df-tx 21694 df-hmeo 21887 df-xms 22453 df-ms 22454 df-tms 22455 df-cncf 23009 df-limc 23971 df-dv 23972 df-ulm 24472 |
This theorem is referenced by: cnndvlem2 33037 |
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