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 31700. (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 11063 | . . . . 5 ⊢ 3 ∈ ℕ | |
5 | 4 | a1i 11 | . . . 4 ⊢ (⊤ → 3 ∈ ℕ) |
6 | neg1rr 11002 | . . . . . . . . 9 ⊢ -1 ∈ ℝ | |
7 | 6 | rexri 9976 | . . . . . . . 8 ⊢ -1 ∈ ℝ* |
8 | 1re 9918 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
9 | 8 | rexri 9976 | . . . . . . . 8 ⊢ 1 ∈ ℝ* |
10 | halfre 11123 | . . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ | |
11 | 10 | rexri 9976 | . . . . . . . 8 ⊢ (1 / 2) ∈ ℝ* |
12 | 7, 9, 11 | 3pm3.2i 1232 | . . . . . . 7 ⊢ (-1 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ (1 / 2) ∈ ℝ*) |
13 | neg1lt0 11004 | . . . . . . . . . 10 ⊢ -1 < 0 | |
14 | halfgt0 11125 | . . . . . . . . . 10 ⊢ 0 < (1 / 2) | |
15 | 13, 14 | pm3.2i 470 | . . . . . . . . 9 ⊢ (-1 < 0 ∧ 0 < (1 / 2)) |
16 | 0re 9919 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
17 | 6, 16, 10 | lttri 10042 | . . . . . . . . 9 ⊢ ((-1 < 0 ∧ 0 < (1 / 2)) → -1 < (1 / 2)) |
18 | 15, 17 | ax-mp 5 | . . . . . . . 8 ⊢ -1 < (1 / 2) |
19 | halflt1 11127 | . . . . . . . 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 12075 | . . . . . 6 ⊢ ((1 / 2) ∈ (-1(,)1) ↔ ((-1 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ (1 / 2) ∈ ℝ*) ∧ (-1 < (1 / 2) ∧ (1 / 2) < 1))) | |
23 | 21, 22 | mpbir 220 | . . . . 5 ⊢ (1 / 2) ∈ (-1(,)1) |
24 | 23 | a1i 11 | . . . 4 ⊢ (⊤ → (1 / 2) ∈ (-1(,)1)) |
25 | 1, 2, 3, 5, 24 | knoppcn2 31697 | . . 3 ⊢ (⊤ → 𝑊 ∈ (ℝ–cn→ℝ)) |
26 | 25 | trud 1484 | . 2 ⊢ 𝑊 ∈ (ℝ–cn→ℝ) |
27 | 2cn 10968 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
28 | 27 | mulid2i 9922 | . . . . . . . 8 ⊢ (1 · 2) = 2 |
29 | 2lt3 11072 | . . . . . . . 8 ⊢ 2 < 3 | |
30 | 28, 29 | eqbrtri 4604 | . . . . . . 7 ⊢ (1 · 2) < 3 |
31 | 2pos 10989 | . . . . . . . 8 ⊢ 0 < 2 | |
32 | 4 | nnrei 10906 | . . . . . . . . 9 ⊢ 3 ∈ ℝ |
33 | 2re 10967 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
34 | 8, 32, 33 | ltmuldivi 10823 | . . . . . . . 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 219 | . . . . . 6 ⊢ 1 < (3 / 2) |
37 | 16, 10, 14 | ltleii 10039 | . . . . . . . . 9 ⊢ 0 ≤ (1 / 2) |
38 | 10 | absidi 13965 | . . . . . . . . 9 ⊢ (0 ≤ (1 / 2) → (abs‘(1 / 2)) = (1 / 2)) |
39 | 37, 38 | ax-mp 5 | . . . . . . . 8 ⊢ (abs‘(1 / 2)) = (1 / 2) |
40 | 39 | oveq2i 6560 | . . . . . . 7 ⊢ (3 · (abs‘(1 / 2))) = (3 · (1 / 2)) |
41 | 4 | nncni 10907 | . . . . . . . . 9 ⊢ 3 ∈ ℂ |
42 | 2ne0 10990 | . . . . . . . . 9 ⊢ 2 ≠ 0 | |
43 | 41, 27, 42 | divreci 10649 | . . . . . . . 8 ⊢ (3 / 2) = (3 · (1 / 2)) |
44 | 43 | eqcomi 2619 | . . . . . . 7 ⊢ (3 · (1 / 2)) = (3 / 2) |
45 | 40, 44 | eqtri 2632 | . . . . . 6 ⊢ (3 · (abs‘(1 / 2))) = (3 / 2) |
46 | 36, 45 | breqtrri 4610 | . . . . 5 ⊢ 1 < (3 · (abs‘(1 / 2))) |
47 | 46 | a1i 11 | . . . 4 ⊢ (⊤ → 1 < (3 · (abs‘(1 / 2)))) |
48 | 1, 2, 3, 24, 5, 47 | knoppndv 31695 | . . 3 ⊢ (⊤ → dom (ℝ D 𝑊) = ∅) |
49 | 48 | trud 1484 | . 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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ∅c0 3874 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 − cmin 10145 -cneg 10146 / cdiv 10563 ℕcn 10897 2c2 10947 3c3 10948 ℕ0cn0 11169 (,)cioo 12046 ⌊cfl 12453 ↑cexp 12722 abscabs 13822 Σcsu 14264 –cn→ccncf 22487 D cdv 23433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-dvds 14822 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-ntr 20634 df-cn 20841 df-cnp 20842 df-tx 21175 df-hmeo 21368 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 df-ulm 23935 |
This theorem is referenced by: cnndvlem2 31699 |
Copyright terms: Public domain | W3C validator |