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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 11790 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ) |
3 | 2pos 11819 | . . . . . . 7 ⊢ 0 < 2 | |
4 | 3 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 < 2) |
5 | aks4d1p1p3.1 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
6 | 5 | nnred 11731 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
7 | 5 | nngt0d 11765 | . . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝑁) |
8 | 1red 10720 | . . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℝ) | |
9 | 1lt2 11887 | . . . . . . . . . . . . . 14 ⊢ 1 < 2 | |
10 | 9 | a1i 11 | . . . . . . . . . . . . 13 ⊢ (𝜑 → 1 < 2) |
11 | 8, 10 | ltned 10854 | . . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≠ 2) |
12 | 11 | necomd 2989 | . . . . . . . . . . 11 ⊢ (𝜑 → 2 ≠ 1) |
13 | 2, 4, 6, 7, 12 | relogbcld 39600 | . . . . . . . . . 10 ⊢ (𝜑 → (2 logb 𝑁) ∈ ℝ) |
14 | 5nn0 11996 | . . . . . . . . . . 11 ⊢ 5 ∈ ℕ0 | |
15 | 14 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → 5 ∈ ℕ0) |
16 | 13, 15 | reexpcld 13619 | . . . . . . . . 9 ⊢ (𝜑 → ((2 logb 𝑁)↑5) ∈ ℝ) |
17 | ceilcl 13303 | . . . . . . . . 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 12168 | . . . . . . 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 2817 | . . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ ℝ ↔ (⌈‘((2 logb 𝑁)↑5)) ∈ ℝ)) |
23 | 19, 22 | mpbird 260 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
24 | 0red 10722 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ) | |
25 | 7re 11809 | . . . . . . . 8 ⊢ 7 ∈ ℝ | |
26 | 25 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 7 ∈ ℝ) |
27 | 7pos 11827 | . . . . . . . 8 ⊢ 0 < 7 | |
28 | 27 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 < 7) |
29 | aks4d1p1p3.3 | . . . . . . . . . 10 ⊢ (𝜑 → 3 ≤ 𝑁) | |
30 | 6, 29 | 3lexlogpow5ineq3 39685 | . . . . . . . . 9 ⊢ (𝜑 → 7 < ((2 logb 𝑁)↑5)) |
31 | ceilge 13305 | . . . . . . . . . 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 10878 | . . . . . . . 8 ⊢ (𝜑 → 7 < (⌈‘((2 logb 𝑁)↑5))) |
34 | 21 | eqcomd 2744 | . . . . . . . 8 ⊢ (𝜑 → (⌈‘((2 logb 𝑁)↑5)) = 𝐵) |
35 | 33, 34 | breqtrd 5056 | . . . . . . 7 ⊢ (𝜑 → 7 < 𝐵) |
36 | 24, 26, 23, 28, 35 | lttrd 10879 | . . . . . 6 ⊢ (𝜑 → 0 < 𝐵) |
37 | 2, 4, 23, 36, 12 | relogbcld 39600 | . . . . 5 ⊢ (𝜑 → (2 logb 𝐵) ∈ ℝ) |
38 | 37 | flcld 13259 | . . . 4 ⊢ (𝜑 → (⌊‘(2 logb 𝐵)) ∈ ℤ) |
39 | 38 | zred 12168 | . . 3 ⊢ (𝜑 → (⌊‘(2 logb 𝐵)) ∈ ℝ) |
40 | 16, 8 | readdcld 10748 | . . . 4 ⊢ (𝜑 → (((2 logb 𝑁)↑5) + 1) ∈ ℝ) |
41 | 16 | ltp1d 11648 | . . . . . 6 ⊢ (𝜑 → ((2 logb 𝑁)↑5) < (((2 logb 𝑁)↑5) + 1)) |
42 | 26, 16, 40, 30, 41 | lttrd 10879 | . . . . 5 ⊢ (𝜑 → 7 < (((2 logb 𝑁)↑5) + 1)) |
43 | 24, 26, 40, 28, 42 | lttrd 10879 | . . . 4 ⊢ (𝜑 → 0 < (((2 logb 𝑁)↑5) + 1)) |
44 | 2, 4, 40, 43, 12 | relogbcld 39600 | . . 3 ⊢ (𝜑 → (2 logb (((2 logb 𝑁)↑5) + 1)) ∈ ℝ) |
45 | flle 13260 | . . . 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 13307 | . . . . . . 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 11314 | . . . . . 6 ⊢ (𝜑 → (((⌈‘((2 logb 𝑁)↑5)) − 1) < ((2 logb 𝑁)↑5) ↔ (⌈‘((2 logb 𝑁)↑5)) < (((2 logb 𝑁)↑5) + 1))) |
50 | 48, 49 | mpbid 235 | . . . . 5 ⊢ (𝜑 → (⌈‘((2 logb 𝑁)↑5)) < (((2 logb 𝑁)↑5) + 1)) |
51 | 21, 50 | eqbrtrd 5052 | . . . 4 ⊢ (𝜑 → 𝐵 < (((2 logb 𝑁)↑5) + 1)) |
52 | 2z 12095 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
53 | 52 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℤ) |
54 | 53 | uzidd 12340 | . . . . 5 ⊢ (𝜑 → 2 ∈ (ℤ≥‘2)) |
55 | 23, 36 | elrpd 12511 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
56 | 40, 43 | elrpd 12511 | . . . . 5 ⊢ (𝜑 → (((2 logb 𝑁)↑5) + 1) ∈ ℝ+) |
57 | logblt 25522 | . . . . 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 235 | . . 3 ⊢ (𝜑 → (2 logb 𝐵) < (2 logb (((2 logb 𝑁)↑5) + 1))) |
60 | 39, 37, 44, 46, 59 | lelttrd 10876 | . 2 ⊢ (𝜑 → (⌊‘(2 logb 𝐵)) < (2 logb (((2 logb 𝑁)↑5) + 1))) |
61 | 3re 11796 | . . . . 5 ⊢ 3 ∈ ℝ | |
62 | 61 | a1i 11 | . . . 4 ⊢ (𝜑 → 3 ∈ ℝ) |
63 | 1lt3 11889 | . . . . 5 ⊢ 1 < 3 | |
64 | 63 | a1i 11 | . . . 4 ⊢ (𝜑 → 1 < 3) |
65 | 8, 62, 6, 64, 29 | ltletrd 10878 | . . 3 ⊢ (𝜑 → 1 < 𝑁) |
66 | 6, 65, 39, 44 | cxpltd 25462 | . 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 235 | 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 209 = wceq 1542 ∈ wcel 2114 class class class wbr 5030 ‘cfv 6339 (class class class)co 7170 ℝcr 10614 0cc0 10615 1c1 10616 + caddc 10618 < clt 10753 ≤ cle 10754 − cmin 10948 ℕcn 11716 2c2 11771 3c3 11772 5c5 11774 7c7 11776 ℕ0cn0 11976 ℤcz 12062 ℤ≥cuz 12324 ℝ+crp 12472 ⌊cfl 13251 ⌈cceil 13252 ↑cexp 13521 ↑𝑐ccxp 25299 logb clogb 25502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-inf2 9177 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 ax-addf 10694 ax-mulf 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-2o 8132 df-er 8320 df-map 8439 df-pm 8440 df-ixp 8508 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fsupp 8907 df-fi 8948 df-sup 8979 df-inf 8980 df-oi 9047 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-q 12431 df-rp 12473 df-xneg 12590 df-xadd 12591 df-xmul 12592 df-ioo 12825 df-ioc 12826 df-ico 12827 df-icc 12828 df-fz 12982 df-fzo 13125 df-fl 13253 df-ceil 13254 df-mod 13329 df-seq 13461 df-exp 13522 df-fac 13726 df-bc 13755 df-hash 13783 df-shft 14516 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-limsup 14918 df-clim 14935 df-rlim 14936 df-sum 15136 df-ef 15513 df-sin 15515 df-cos 15516 df-pi 15518 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-starv 16683 df-sca 16684 df-vsca 16685 df-ip 16686 df-tset 16687 df-ple 16688 df-ds 16690 df-unif 16691 df-hom 16692 df-cco 16693 df-rest 16799 df-topn 16800 df-0g 16818 df-gsum 16819 df-topgen 16820 df-pt 16821 df-prds 16824 df-xrs 16878 df-qtop 16883 df-imas 16884 df-xps 16886 df-mre 16960 df-mrc 16961 df-acs 16963 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-submnd 18073 df-mulg 18343 df-cntz 18565 df-cmn 19026 df-psmet 20209 df-xmet 20210 df-met 20211 df-bl 20212 df-mopn 20213 df-fbas 20214 df-fg 20215 df-cnfld 20218 df-top 21645 df-topon 21662 df-topsp 21684 df-bases 21697 df-cld 21770 df-ntr 21771 df-cls 21772 df-nei 21849 df-lp 21887 df-perf 21888 df-cn 21978 df-cnp 21979 df-haus 22066 df-tx 22313 df-hmeo 22506 df-fil 22597 df-fm 22689 df-flim 22690 df-flf 22691 df-xms 23073 df-ms 23074 df-tms 23075 df-cncf 23630 df-limc 24618 df-dv 24619 df-log 25300 df-cxp 25301 df-logb 25503 |
This theorem is referenced by: aks4d1p1p2 39697 |
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