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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 12227 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ) |
3 | 2pos 12256 | . . . . . . 7 ⊢ 0 < 2 | |
4 | 3 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 < 2) |
5 | aks4d1p1p3.1 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
6 | 5 | nnred 12168 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
7 | 5 | nngt0d 12202 | . . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝑁) |
8 | 1red 11156 | . . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℝ) | |
9 | 1lt2 12324 | . . . . . . . . . . . . . 14 ⊢ 1 < 2 | |
10 | 9 | a1i 11 | . . . . . . . . . . . . 13 ⊢ (𝜑 → 1 < 2) |
11 | 8, 10 | ltned 11291 | . . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≠ 2) |
12 | 11 | necomd 2999 | . . . . . . . . . . 11 ⊢ (𝜑 → 2 ≠ 1) |
13 | 2, 4, 6, 7, 12 | relogbcld 40430 | . . . . . . . . . 10 ⊢ (𝜑 → (2 logb 𝑁) ∈ ℝ) |
14 | 5nn0 12433 | . . . . . . . . . . 11 ⊢ 5 ∈ ℕ0 | |
15 | 14 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → 5 ∈ ℕ0) |
16 | 13, 15 | reexpcld 14068 | . . . . . . . . 9 ⊢ (𝜑 → ((2 logb 𝑁)↑5) ∈ ℝ) |
17 | ceilcl 13747 | . . . . . . . . 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 12607 | . . . . . . 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 2822 | . . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ ℝ ↔ (⌈‘((2 logb 𝑁)↑5)) ∈ ℝ)) |
23 | 19, 22 | mpbird 256 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
24 | 0red 11158 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ) | |
25 | 7re 12246 | . . . . . . . 8 ⊢ 7 ∈ ℝ | |
26 | 25 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 7 ∈ ℝ) |
27 | 7pos 12264 | . . . . . . . 8 ⊢ 0 < 7 | |
28 | 27 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 < 7) |
29 | aks4d1p1p3.3 | . . . . . . . . . 10 ⊢ (𝜑 → 3 ≤ 𝑁) | |
30 | 6, 29 | 3lexlogpow5ineq3 40514 | . . . . . . . . 9 ⊢ (𝜑 → 7 < ((2 logb 𝑁)↑5)) |
31 | ceilge 13750 | . . . . . . . . . 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 11315 | . . . . . . . 8 ⊢ (𝜑 → 7 < (⌈‘((2 logb 𝑁)↑5))) |
34 | 21 | eqcomd 2742 | . . . . . . . 8 ⊢ (𝜑 → (⌈‘((2 logb 𝑁)↑5)) = 𝐵) |
35 | 33, 34 | breqtrd 5131 | . . . . . . 7 ⊢ (𝜑 → 7 < 𝐵) |
36 | 24, 26, 23, 28, 35 | lttrd 11316 | . . . . . 6 ⊢ (𝜑 → 0 < 𝐵) |
37 | 2, 4, 23, 36, 12 | relogbcld 40430 | . . . . 5 ⊢ (𝜑 → (2 logb 𝐵) ∈ ℝ) |
38 | 37 | flcld 13703 | . . . 4 ⊢ (𝜑 → (⌊‘(2 logb 𝐵)) ∈ ℤ) |
39 | 38 | zred 12607 | . . 3 ⊢ (𝜑 → (⌊‘(2 logb 𝐵)) ∈ ℝ) |
40 | 16, 8 | readdcld 11184 | . . . 4 ⊢ (𝜑 → (((2 logb 𝑁)↑5) + 1) ∈ ℝ) |
41 | 16 | ltp1d 12085 | . . . . . 6 ⊢ (𝜑 → ((2 logb 𝑁)↑5) < (((2 logb 𝑁)↑5) + 1)) |
42 | 26, 16, 40, 30, 41 | lttrd 11316 | . . . . 5 ⊢ (𝜑 → 7 < (((2 logb 𝑁)↑5) + 1)) |
43 | 24, 26, 40, 28, 42 | lttrd 11316 | . . . 4 ⊢ (𝜑 → 0 < (((2 logb 𝑁)↑5) + 1)) |
44 | 2, 4, 40, 43, 12 | relogbcld 40430 | . . 3 ⊢ (𝜑 → (2 logb (((2 logb 𝑁)↑5) + 1)) ∈ ℝ) |
45 | flle 13704 | . . . 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 13753 | . . . . . . 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 11751 | . . . . . 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 5127 | . . . 4 ⊢ (𝜑 → 𝐵 < (((2 logb 𝑁)↑5) + 1)) |
52 | 2z 12535 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
53 | 52 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℤ) |
54 | 53 | uzidd 12779 | . . . . 5 ⊢ (𝜑 → 2 ∈ (ℤ≥‘2)) |
55 | 23, 36 | elrpd 12954 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
56 | 40, 43 | elrpd 12954 | . . . . 5 ⊢ (𝜑 → (((2 logb 𝑁)↑5) + 1) ∈ ℝ+) |
57 | logblt 26134 | . . . . 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 1371 | . . . 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 11313 | . 2 ⊢ (𝜑 → (⌊‘(2 logb 𝐵)) < (2 logb (((2 logb 𝑁)↑5) + 1))) |
61 | 3re 12233 | . . . . 5 ⊢ 3 ∈ ℝ | |
62 | 61 | a1i 11 | . . . 4 ⊢ (𝜑 → 3 ∈ ℝ) |
63 | 1lt3 12326 | . . . . 5 ⊢ 1 < 3 | |
64 | 63 | a1i 11 | . . . 4 ⊢ (𝜑 → 1 < 3) |
65 | 8, 62, 6, 64, 29 | ltletrd 11315 | . . 3 ⊢ (𝜑 → 1 < 𝑁) |
66 | 6, 65, 39, 44 | cxpltd 26074 | . 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 1541 ∈ wcel 2106 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 < clt 11189 ≤ cle 11190 − cmin 11385 ℕcn 12153 2c2 12208 3c3 12209 5c5 12211 7c7 12213 ℕ0cn0 12413 ℤcz 12499 ℤ≥cuz 12763 ℝ+crp 12915 ⌊cfl 13695 ⌈cceil 13696 ↑cexp 13967 ↑𝑐ccxp 25911 logb clogb 26114 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-ceil 13698 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-sin 15952 df-cos 15953 df-pi 15955 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231 df-log 25912 df-cxp 25913 df-logb 26115 |
This theorem is referenced by: aks4d1p1p2 40527 |
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