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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 12234 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ) |
3 | 2pos 12263 | . . . . . . 7 ⊢ 0 < 2 | |
4 | 3 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 < 2) |
5 | aks4d1p1p3.1 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
6 | 5 | nnred 12175 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
7 | 5 | nngt0d 12209 | . . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝑁) |
8 | 1red 11163 | . . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℝ) | |
9 | 1lt2 12331 | . . . . . . . . . . . . . 14 ⊢ 1 < 2 | |
10 | 9 | a1i 11 | . . . . . . . . . . . . 13 ⊢ (𝜑 → 1 < 2) |
11 | 8, 10 | ltned 11298 | . . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≠ 2) |
12 | 11 | necomd 3000 | . . . . . . . . . . 11 ⊢ (𝜑 → 2 ≠ 1) |
13 | 2, 4, 6, 7, 12 | relogbcld 40459 | . . . . . . . . . 10 ⊢ (𝜑 → (2 logb 𝑁) ∈ ℝ) |
14 | 5nn0 12440 | . . . . . . . . . . 11 ⊢ 5 ∈ ℕ0 | |
15 | 14 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → 5 ∈ ℕ0) |
16 | 13, 15 | reexpcld 14075 | . . . . . . . . 9 ⊢ (𝜑 → ((2 logb 𝑁)↑5) ∈ ℝ) |
17 | ceilcl 13754 | . . . . . . . . 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 12614 | . . . . . . 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 257 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
24 | 0red 11165 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ) | |
25 | 7re 12253 | . . . . . . . 8 ⊢ 7 ∈ ℝ | |
26 | 25 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 7 ∈ ℝ) |
27 | 7pos 12271 | . . . . . . . 8 ⊢ 0 < 7 | |
28 | 27 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 < 7) |
29 | aks4d1p1p3.3 | . . . . . . . . . 10 ⊢ (𝜑 → 3 ≤ 𝑁) | |
30 | 6, 29 | 3lexlogpow5ineq3 40543 | . . . . . . . . 9 ⊢ (𝜑 → 7 < ((2 logb 𝑁)↑5)) |
31 | ceilge 13757 | . . . . . . . . . 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 11322 | . . . . . . . 8 ⊢ (𝜑 → 7 < (⌈‘((2 logb 𝑁)↑5))) |
34 | 21 | eqcomd 2743 | . . . . . . . 8 ⊢ (𝜑 → (⌈‘((2 logb 𝑁)↑5)) = 𝐵) |
35 | 33, 34 | breqtrd 5136 | . . . . . . 7 ⊢ (𝜑 → 7 < 𝐵) |
36 | 24, 26, 23, 28, 35 | lttrd 11323 | . . . . . 6 ⊢ (𝜑 → 0 < 𝐵) |
37 | 2, 4, 23, 36, 12 | relogbcld 40459 | . . . . 5 ⊢ (𝜑 → (2 logb 𝐵) ∈ ℝ) |
38 | 37 | flcld 13710 | . . . 4 ⊢ (𝜑 → (⌊‘(2 logb 𝐵)) ∈ ℤ) |
39 | 38 | zred 12614 | . . 3 ⊢ (𝜑 → (⌊‘(2 logb 𝐵)) ∈ ℝ) |
40 | 16, 8 | readdcld 11191 | . . . 4 ⊢ (𝜑 → (((2 logb 𝑁)↑5) + 1) ∈ ℝ) |
41 | 16 | ltp1d 12092 | . . . . . 6 ⊢ (𝜑 → ((2 logb 𝑁)↑5) < (((2 logb 𝑁)↑5) + 1)) |
42 | 26, 16, 40, 30, 41 | lttrd 11323 | . . . . 5 ⊢ (𝜑 → 7 < (((2 logb 𝑁)↑5) + 1)) |
43 | 24, 26, 40, 28, 42 | lttrd 11323 | . . . 4 ⊢ (𝜑 → 0 < (((2 logb 𝑁)↑5) + 1)) |
44 | 2, 4, 40, 43, 12 | relogbcld 40459 | . . 3 ⊢ (𝜑 → (2 logb (((2 logb 𝑁)↑5) + 1)) ∈ ℝ) |
45 | flle 13711 | . . . 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 13760 | . . . . . . 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 11758 | . . . . . 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 5132 | . . . 4 ⊢ (𝜑 → 𝐵 < (((2 logb 𝑁)↑5) + 1)) |
52 | 2z 12542 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
53 | 52 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℤ) |
54 | 53 | uzidd 12786 | . . . . 5 ⊢ (𝜑 → 2 ∈ (ℤ≥‘2)) |
55 | 23, 36 | elrpd 12961 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
56 | 40, 43 | elrpd 12961 | . . . . 5 ⊢ (𝜑 → (((2 logb 𝑁)↑5) + 1) ∈ ℝ+) |
57 | logblt 26150 | . . . . 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 1372 | . . . 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 11320 | . 2 ⊢ (𝜑 → (⌊‘(2 logb 𝐵)) < (2 logb (((2 logb 𝑁)↑5) + 1))) |
61 | 3re 12240 | . . . . 5 ⊢ 3 ∈ ℝ | |
62 | 61 | a1i 11 | . . . 4 ⊢ (𝜑 → 3 ∈ ℝ) |
63 | 1lt3 12333 | . . . . 5 ⊢ 1 < 3 | |
64 | 63 | a1i 11 | . . . 4 ⊢ (𝜑 → 1 < 3) |
65 | 8, 62, 6, 64, 29 | ltletrd 11322 | . . 3 ⊢ (𝜑 → 1 < 𝑁) |
66 | 6, 65, 39, 44 | cxpltd 26090 | . 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 1542 ∈ wcel 2107 class class class wbr 5110 ‘cfv 6501 (class class class)co 7362 ℝcr 11057 0cc0 11058 1c1 11059 + caddc 11061 < clt 11196 ≤ cle 11197 − cmin 11392 ℕcn 12160 2c2 12215 3c3 12216 5c5 12218 7c7 12220 ℕ0cn0 12420 ℤcz 12506 ℤ≥cuz 12770 ℝ+crp 12922 ⌊cfl 13702 ⌈cceil 13703 ↑cexp 13974 ↑𝑐ccxp 25927 logb clogb 26130 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 ax-addf 11137 ax-mulf 11138 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-map 8774 df-pm 8775 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-q 12881 df-rp 12923 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-ioo 13275 df-ioc 13276 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-ceil 13705 df-mod 13782 df-seq 13914 df-exp 13975 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14959 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-limsup 15360 df-clim 15377 df-rlim 15378 df-sum 15578 df-ef 15957 df-sin 15959 df-cos 15960 df-pi 15962 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-starv 17155 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-unif 17163 df-hom 17164 df-cco 17165 df-rest 17311 df-topn 17312 df-0g 17330 df-gsum 17331 df-topgen 17332 df-pt 17333 df-prds 17336 df-xrs 17391 df-qtop 17396 df-imas 17397 df-xps 17399 df-mre 17473 df-mrc 17474 df-acs 17476 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-submnd 18609 df-mulg 18880 df-cntz 19104 df-cmn 19571 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-fbas 20809 df-fg 20810 df-cnfld 20813 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cld 22386 df-ntr 22387 df-cls 22388 df-nei 22465 df-lp 22503 df-perf 22504 df-cn 22594 df-cnp 22595 df-haus 22682 df-tx 22929 df-hmeo 23122 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-xms 23689 df-ms 23690 df-tms 23691 df-cncf 24257 df-limc 25246 df-dv 25247 df-log 25928 df-cxp 25929 df-logb 26131 |
This theorem is referenced by: aks4d1p1p2 40556 |
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