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Mirrors > Home > MPE Home > Th. List > Mathboxes > aks4d1p1p3 | Structured version Visualization version GIF version |
Description: Bound of a ceiling of the binary logarithm to the fifth power. (Contributed by metakunt, 19-Aug-2024.) |
Ref | Expression |
---|---|
aks4d1p1p3.1 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
aks4d1p1p3.2 | ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) |
aks4d1p1p3.3 | ⊢ (𝜑 → 3 ≤ 𝑁) |
Ref | Expression |
---|---|
aks4d1p1p3 | ⊢ (𝜑 → (𝑁↑𝑐(⌊‘(2 logb 𝐵))) < (𝑁↑𝑐(2 logb (((2 logb 𝑁)↑5) + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 12047 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ) |
3 | 2pos 12076 | . . . . . . 7 ⊢ 0 < 2 | |
4 | 3 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 < 2) |
5 | aks4d1p1p3.1 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
6 | 5 | nnred 11988 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
7 | 5 | nngt0d 12022 | . . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝑁) |
8 | 1red 10976 | . . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℝ) | |
9 | 1lt2 12144 | . . . . . . . . . . . . . 14 ⊢ 1 < 2 | |
10 | 9 | a1i 11 | . . . . . . . . . . . . 13 ⊢ (𝜑 → 1 < 2) |
11 | 8, 10 | ltned 11111 | . . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≠ 2) |
12 | 11 | necomd 2999 | . . . . . . . . . . 11 ⊢ (𝜑 → 2 ≠ 1) |
13 | 2, 4, 6, 7, 12 | relogbcld 39981 | . . . . . . . . . 10 ⊢ (𝜑 → (2 logb 𝑁) ∈ ℝ) |
14 | 5nn0 12253 | . . . . . . . . . . 11 ⊢ 5 ∈ ℕ0 | |
15 | 14 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → 5 ∈ ℕ0) |
16 | 13, 15 | reexpcld 13881 | . . . . . . . . 9 ⊢ (𝜑 → ((2 logb 𝑁)↑5) ∈ ℝ) |
17 | ceilcl 13562 | . . . . . . . . 9 ⊢ (((2 logb 𝑁)↑5) ∈ ℝ → (⌈‘((2 logb 𝑁)↑5)) ∈ ℤ) | |
18 | 16, 17 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (⌈‘((2 logb 𝑁)↑5)) ∈ ℤ) |
19 | 18 | zred 12426 | . . . . . . 7 ⊢ (𝜑 → (⌈‘((2 logb 𝑁)↑5)) ∈ ℝ) |
20 | aks4d1p1p3.2 | . . . . . . . . 9 ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) | |
21 | 20 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 = (⌈‘((2 logb 𝑁)↑5))) |
22 | 21 | eleq1d 2823 | . . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ ℝ ↔ (⌈‘((2 logb 𝑁)↑5)) ∈ ℝ)) |
23 | 19, 22 | mpbird 256 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
24 | 0red 10978 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ) | |
25 | 7re 12066 | . . . . . . . 8 ⊢ 7 ∈ ℝ | |
26 | 25 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 7 ∈ ℝ) |
27 | 7pos 12084 | . . . . . . . 8 ⊢ 0 < 7 | |
28 | 27 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 < 7) |
29 | aks4d1p1p3.3 | . . . . . . . . . 10 ⊢ (𝜑 → 3 ≤ 𝑁) | |
30 | 6, 29 | 3lexlogpow5ineq3 40065 | . . . . . . . . 9 ⊢ (𝜑 → 7 < ((2 logb 𝑁)↑5)) |
31 | ceilge 13565 | . . . . . . . . . 10 ⊢ (((2 logb 𝑁)↑5) ∈ ℝ → ((2 logb 𝑁)↑5) ≤ (⌈‘((2 logb 𝑁)↑5))) | |
32 | 16, 31 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → ((2 logb 𝑁)↑5) ≤ (⌈‘((2 logb 𝑁)↑5))) |
33 | 26, 16, 19, 30, 32 | ltletrd 11135 | . . . . . . . 8 ⊢ (𝜑 → 7 < (⌈‘((2 logb 𝑁)↑5))) |
34 | 21 | eqcomd 2744 | . . . . . . . 8 ⊢ (𝜑 → (⌈‘((2 logb 𝑁)↑5)) = 𝐵) |
35 | 33, 34 | breqtrd 5100 | . . . . . . 7 ⊢ (𝜑 → 7 < 𝐵) |
36 | 24, 26, 23, 28, 35 | lttrd 11136 | . . . . . 6 ⊢ (𝜑 → 0 < 𝐵) |
37 | 2, 4, 23, 36, 12 | relogbcld 39981 | . . . . 5 ⊢ (𝜑 → (2 logb 𝐵) ∈ ℝ) |
38 | 37 | flcld 13518 | . . . 4 ⊢ (𝜑 → (⌊‘(2 logb 𝐵)) ∈ ℤ) |
39 | 38 | zred 12426 | . . 3 ⊢ (𝜑 → (⌊‘(2 logb 𝐵)) ∈ ℝ) |
40 | 16, 8 | readdcld 11004 | . . . 4 ⊢ (𝜑 → (((2 logb 𝑁)↑5) + 1) ∈ ℝ) |
41 | 16 | ltp1d 11905 | . . . . . 6 ⊢ (𝜑 → ((2 logb 𝑁)↑5) < (((2 logb 𝑁)↑5) + 1)) |
42 | 26, 16, 40, 30, 41 | lttrd 11136 | . . . . 5 ⊢ (𝜑 → 7 < (((2 logb 𝑁)↑5) + 1)) |
43 | 24, 26, 40, 28, 42 | lttrd 11136 | . . . 4 ⊢ (𝜑 → 0 < (((2 logb 𝑁)↑5) + 1)) |
44 | 2, 4, 40, 43, 12 | relogbcld 39981 | . . 3 ⊢ (𝜑 → (2 logb (((2 logb 𝑁)↑5) + 1)) ∈ ℝ) |
45 | flle 13519 | . . . 4 ⊢ ((2 logb 𝐵) ∈ ℝ → (⌊‘(2 logb 𝐵)) ≤ (2 logb 𝐵)) | |
46 | 37, 45 | syl 17 | . . 3 ⊢ (𝜑 → (⌊‘(2 logb 𝐵)) ≤ (2 logb 𝐵)) |
47 | ceilm1lt 13568 | . . . . . . 7 ⊢ (((2 logb 𝑁)↑5) ∈ ℝ → ((⌈‘((2 logb 𝑁)↑5)) − 1) < ((2 logb 𝑁)↑5)) | |
48 | 16, 47 | syl 17 | . . . . . 6 ⊢ (𝜑 → ((⌈‘((2 logb 𝑁)↑5)) − 1) < ((2 logb 𝑁)↑5)) |
49 | 19, 8, 16 | ltsubaddd 11571 | . . . . . 6 ⊢ (𝜑 → (((⌈‘((2 logb 𝑁)↑5)) − 1) < ((2 logb 𝑁)↑5) ↔ (⌈‘((2 logb 𝑁)↑5)) < (((2 logb 𝑁)↑5) + 1))) |
50 | 48, 49 | mpbid 231 | . . . . 5 ⊢ (𝜑 → (⌈‘((2 logb 𝑁)↑5)) < (((2 logb 𝑁)↑5) + 1)) |
51 | 21, 50 | eqbrtrd 5096 | . . . 4 ⊢ (𝜑 → 𝐵 < (((2 logb 𝑁)↑5) + 1)) |
52 | 2z 12352 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
53 | 52 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℤ) |
54 | 53 | uzidd 12598 | . . . . 5 ⊢ (𝜑 → 2 ∈ (ℤ≥‘2)) |
55 | 23, 36 | elrpd 12769 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
56 | 40, 43 | elrpd 12769 | . . . . 5 ⊢ (𝜑 → (((2 logb 𝑁)↑5) + 1) ∈ ℝ+) |
57 | logblt 25934 | . . . . 5 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ ℝ+ ∧ (((2 logb 𝑁)↑5) + 1) ∈ ℝ+) → (𝐵 < (((2 logb 𝑁)↑5) + 1) ↔ (2 logb 𝐵) < (2 logb (((2 logb 𝑁)↑5) + 1)))) | |
58 | 54, 55, 56, 57 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → (𝐵 < (((2 logb 𝑁)↑5) + 1) ↔ (2 logb 𝐵) < (2 logb (((2 logb 𝑁)↑5) + 1)))) |
59 | 51, 58 | mpbid 231 | . . 3 ⊢ (𝜑 → (2 logb 𝐵) < (2 logb (((2 logb 𝑁)↑5) + 1))) |
60 | 39, 37, 44, 46, 59 | lelttrd 11133 | . 2 ⊢ (𝜑 → (⌊‘(2 logb 𝐵)) < (2 logb (((2 logb 𝑁)↑5) + 1))) |
61 | 3re 12053 | . . . . 5 ⊢ 3 ∈ ℝ | |
62 | 61 | a1i 11 | . . . 4 ⊢ (𝜑 → 3 ∈ ℝ) |
63 | 1lt3 12146 | . . . . 5 ⊢ 1 < 3 | |
64 | 63 | a1i 11 | . . . 4 ⊢ (𝜑 → 1 < 3) |
65 | 8, 62, 6, 64, 29 | ltletrd 11135 | . . 3 ⊢ (𝜑 → 1 < 𝑁) |
66 | 6, 65, 39, 44 | cxpltd 25874 | . 2 ⊢ (𝜑 → ((⌊‘(2 logb 𝐵)) < (2 logb (((2 logb 𝑁)↑5) + 1)) ↔ (𝑁↑𝑐(⌊‘(2 logb 𝐵))) < (𝑁↑𝑐(2 logb (((2 logb 𝑁)↑5) + 1))))) |
67 | 60, 66 | mpbid 231 | 1 ⊢ (𝜑 → (𝑁↑𝑐(⌊‘(2 logb 𝐵))) < (𝑁↑𝑐(2 logb (((2 logb 𝑁)↑5) + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 < clt 11009 ≤ cle 11010 − cmin 11205 ℕcn 11973 2c2 12028 3c3 12029 5c5 12031 7c7 12033 ℕ0cn0 12233 ℤcz 12319 ℤ≥cuz 12582 ℝ+crp 12730 ⌊cfl 13510 ⌈cceil 13511 ↑cexp 13782 ↑𝑐ccxp 25711 logb clogb 25914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-ceil 13513 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-pi 15782 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-log 25712 df-cxp 25713 df-logb 25915 |
This theorem is referenced by: aks4d1p1p2 40078 |
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