![]() |
Mathbox for Asger C. Ipsen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > cnndvlem1 | Structured version Visualization version GIF version |
Description: Lemma for cnndv 33991. (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 11704 | . . . . 5 ⊢ 3 ∈ ℕ | |
5 | 4 | a1i 11 | . . . 4 ⊢ (⊤ → 3 ∈ ℕ) |
6 | neg1rr 11740 | . . . . . . . . 9 ⊢ -1 ∈ ℝ | |
7 | 6 | rexri 10688 | . . . . . . . 8 ⊢ -1 ∈ ℝ* |
8 | 1re 10630 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
9 | 8 | rexri 10688 | . . . . . . . 8 ⊢ 1 ∈ ℝ* |
10 | halfre 11839 | . . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ | |
11 | 10 | rexri 10688 | . . . . . . . 8 ⊢ (1 / 2) ∈ ℝ* |
12 | 7, 9, 11 | 3pm3.2i 1336 | . . . . . . 7 ⊢ (-1 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ (1 / 2) ∈ ℝ*) |
13 | neg1lt0 11742 | . . . . . . . . . 10 ⊢ -1 < 0 | |
14 | halfgt0 11841 | . . . . . . . . . 10 ⊢ 0 < (1 / 2) | |
15 | 13, 14 | pm3.2i 474 | . . . . . . . . 9 ⊢ (-1 < 0 ∧ 0 < (1 / 2)) |
16 | 0re 10632 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
17 | 6, 16, 10 | lttri 10755 | . . . . . . . . 9 ⊢ ((-1 < 0 ∧ 0 < (1 / 2)) → -1 < (1 / 2)) |
18 | 15, 17 | ax-mp 5 | . . . . . . . 8 ⊢ -1 < (1 / 2) |
19 | halflt1 11843 | . . . . . . . 8 ⊢ (1 / 2) < 1 | |
20 | 18, 19 | pm3.2i 474 | . . . . . . 7 ⊢ (-1 < (1 / 2) ∧ (1 / 2) < 1) |
21 | 12, 20 | pm3.2i 474 | . . . . . 6 ⊢ ((-1 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ (1 / 2) ∈ ℝ*) ∧ (-1 < (1 / 2) ∧ (1 / 2) < 1)) |
22 | elioo3g 12755 | . . . . . 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 33988 | . . 3 ⊢ (⊤ → 𝑊 ∈ (ℝ–cn→ℝ)) |
26 | 25 | mptru 1545 | . 2 ⊢ 𝑊 ∈ (ℝ–cn→ℝ) |
27 | 2cn 11700 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
28 | 27 | mulid2i 10635 | . . . . . . . 8 ⊢ (1 · 2) = 2 |
29 | 2lt3 11797 | . . . . . . . 8 ⊢ 2 < 3 | |
30 | 28, 29 | eqbrtri 5051 | . . . . . . 7 ⊢ (1 · 2) < 3 |
31 | 2pos 11728 | . . . . . . . 8 ⊢ 0 < 2 | |
32 | 4 | nnrei 11634 | . . . . . . . . 9 ⊢ 3 ∈ ℝ |
33 | 2re 11699 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
34 | 8, 32, 33 | ltmuldivi 11549 | . . . . . . . 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 10752 | . . . . . . . . 9 ⊢ 0 ≤ (1 / 2) |
38 | 10 | absidi 14729 | . . . . . . . . 9 ⊢ (0 ≤ (1 / 2) → (abs‘(1 / 2)) = (1 / 2)) |
39 | 37, 38 | ax-mp 5 | . . . . . . . 8 ⊢ (abs‘(1 / 2)) = (1 / 2) |
40 | 39 | oveq2i 7146 | . . . . . . 7 ⊢ (3 · (abs‘(1 / 2))) = (3 · (1 / 2)) |
41 | 4 | nncni 11635 | . . . . . . . . 9 ⊢ 3 ∈ ℂ |
42 | 2ne0 11729 | . . . . . . . . 9 ⊢ 2 ≠ 0 | |
43 | 41, 27, 42 | divreci 11374 | . . . . . . . 8 ⊢ (3 / 2) = (3 · (1 / 2)) |
44 | 43 | eqcomi 2807 | . . . . . . 7 ⊢ (3 · (1 / 2)) = (3 / 2) |
45 | 40, 44 | eqtri 2821 | . . . . . 6 ⊢ (3 · (abs‘(1 / 2))) = (3 / 2) |
46 | 36, 45 | breqtrri 5057 | . . . . 5 ⊢ 1 < (3 · (abs‘(1 / 2))) |
47 | 46 | a1i 11 | . . . 4 ⊢ (⊤ → 1 < (3 · (abs‘(1 / 2)))) |
48 | 1, 2, 3, 24, 5, 47 | knoppndv 33986 | . . 3 ⊢ (⊤ → dom (ℝ D 𝑊) = ∅) |
49 | 48 | mptru 1545 | . 2 ⊢ dom (ℝ D 𝑊) = ∅ |
50 | 26, 49 | pm3.2i 474 | 1 ⊢ (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ⊤wtru 1539 ∈ wcel 2111 ∅c0 4243 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 − cmin 10859 -cneg 10860 / cdiv 11286 ℕcn 11625 2c2 11680 3c3 11681 ℕ0cn0 11885 (,)cioo 12726 ⌊cfl 13155 ↑cexp 13425 abscabs 14585 Σcsu 15034 –cn→ccncf 23481 D cdv 24466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-dvds 15600 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-ntr 21625 df-cn 21832 df-cnp 21833 df-tx 22167 df-hmeo 22360 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470 df-ulm 24972 |
This theorem is referenced by: cnndvlem2 33990 |
Copyright terms: Public domain | W3C validator |