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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnndvlem1 | Structured version Visualization version GIF version |
Description: Lemma for cnndv 33878. (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 11717 | . . . . 5 ⊢ 3 ∈ ℕ | |
5 | 4 | a1i 11 | . . . 4 ⊢ (⊤ → 3 ∈ ℕ) |
6 | neg1rr 11753 | . . . . . . . . 9 ⊢ -1 ∈ ℝ | |
7 | 6 | rexri 10699 | . . . . . . . 8 ⊢ -1 ∈ ℝ* |
8 | 1re 10641 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
9 | 8 | rexri 10699 | . . . . . . . 8 ⊢ 1 ∈ ℝ* |
10 | halfre 11852 | . . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ | |
11 | 10 | rexri 10699 | . . . . . . . 8 ⊢ (1 / 2) ∈ ℝ* |
12 | 7, 9, 11 | 3pm3.2i 1335 | . . . . . . 7 ⊢ (-1 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ (1 / 2) ∈ ℝ*) |
13 | neg1lt0 11755 | . . . . . . . . . 10 ⊢ -1 < 0 | |
14 | halfgt0 11854 | . . . . . . . . . 10 ⊢ 0 < (1 / 2) | |
15 | 13, 14 | pm3.2i 473 | . . . . . . . . 9 ⊢ (-1 < 0 ∧ 0 < (1 / 2)) |
16 | 0re 10643 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
17 | 6, 16, 10 | lttri 10766 | . . . . . . . . 9 ⊢ ((-1 < 0 ∧ 0 < (1 / 2)) → -1 < (1 / 2)) |
18 | 15, 17 | ax-mp 5 | . . . . . . . 8 ⊢ -1 < (1 / 2) |
19 | halflt1 11856 | . . . . . . . 8 ⊢ (1 / 2) < 1 | |
20 | 18, 19 | pm3.2i 473 | . . . . . . 7 ⊢ (-1 < (1 / 2) ∧ (1 / 2) < 1) |
21 | 12, 20 | pm3.2i 473 | . . . . . 6 ⊢ ((-1 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ (1 / 2) ∈ ℝ*) ∧ (-1 < (1 / 2) ∧ (1 / 2) < 1)) |
22 | elioo3g 12768 | . . . . . 6 ⊢ ((1 / 2) ∈ (-1(,)1) ↔ ((-1 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ (1 / 2) ∈ ℝ*) ∧ (-1 < (1 / 2) ∧ (1 / 2) < 1))) | |
23 | 21, 22 | mpbir 233 | . . . . 5 ⊢ (1 / 2) ∈ (-1(,)1) |
24 | 23 | a1i 11 | . . . 4 ⊢ (⊤ → (1 / 2) ∈ (-1(,)1)) |
25 | 1, 2, 3, 5, 24 | knoppcn2 33875 | . . 3 ⊢ (⊤ → 𝑊 ∈ (ℝ–cn→ℝ)) |
26 | 25 | mptru 1544 | . 2 ⊢ 𝑊 ∈ (ℝ–cn→ℝ) |
27 | 2cn 11713 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
28 | 27 | mulid2i 10646 | . . . . . . . 8 ⊢ (1 · 2) = 2 |
29 | 2lt3 11810 | . . . . . . . 8 ⊢ 2 < 3 | |
30 | 28, 29 | eqbrtri 5087 | . . . . . . 7 ⊢ (1 · 2) < 3 |
31 | 2pos 11741 | . . . . . . . 8 ⊢ 0 < 2 | |
32 | 4 | nnrei 11647 | . . . . . . . . 9 ⊢ 3 ∈ ℝ |
33 | 2re 11712 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
34 | 8, 32, 33 | ltmuldivi 11560 | . . . . . . . 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 232 | . . . . . 6 ⊢ 1 < (3 / 2) |
37 | 16, 10, 14 | ltleii 10763 | . . . . . . . . 9 ⊢ 0 ≤ (1 / 2) |
38 | 10 | absidi 14737 | . . . . . . . . 9 ⊢ (0 ≤ (1 / 2) → (abs‘(1 / 2)) = (1 / 2)) |
39 | 37, 38 | ax-mp 5 | . . . . . . . 8 ⊢ (abs‘(1 / 2)) = (1 / 2) |
40 | 39 | oveq2i 7167 | . . . . . . 7 ⊢ (3 · (abs‘(1 / 2))) = (3 · (1 / 2)) |
41 | 4 | nncni 11648 | . . . . . . . . 9 ⊢ 3 ∈ ℂ |
42 | 2ne0 11742 | . . . . . . . . 9 ⊢ 2 ≠ 0 | |
43 | 41, 27, 42 | divreci 11385 | . . . . . . . 8 ⊢ (3 / 2) = (3 · (1 / 2)) |
44 | 43 | eqcomi 2830 | . . . . . . 7 ⊢ (3 · (1 / 2)) = (3 / 2) |
45 | 40, 44 | eqtri 2844 | . . . . . 6 ⊢ (3 · (abs‘(1 / 2))) = (3 / 2) |
46 | 36, 45 | breqtrri 5093 | . . . . 5 ⊢ 1 < (3 · (abs‘(1 / 2))) |
47 | 46 | a1i 11 | . . . 4 ⊢ (⊤ → 1 < (3 · (abs‘(1 / 2)))) |
48 | 1, 2, 3, 24, 5, 47 | knoppndv 33873 | . . 3 ⊢ (⊤ → dom (ℝ D 𝑊) = ∅) |
49 | 48 | mptru 1544 | . 2 ⊢ dom (ℝ D 𝑊) = ∅ |
50 | 26, 49 | pm3.2i 473 | 1 ⊢ (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ∅c0 4291 class class class wbr 5066 ↦ cmpt 5146 dom cdm 5555 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 ℝ*cxr 10674 < clt 10675 ≤ cle 10676 − cmin 10870 -cneg 10871 / cdiv 11297 ℕcn 11638 2c2 11693 3c3 11694 ℕ0cn0 11898 (,)cioo 12739 ⌊cfl 13161 ↑cexp 13430 abscabs 14593 Σcsu 15042 –cn→ccncf 23484 D cdv 24461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-dvds 15608 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-ntr 21628 df-cn 21835 df-cnp 21836 df-tx 22170 df-hmeo 22363 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-ulm 24965 |
This theorem is referenced by: cnndvlem2 33877 |
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