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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 12338 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ) |
3 | 2pos 12367 | . . . . . . 7 ⊢ 0 < 2 | |
4 | 3 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 < 2) |
5 | aks4d1p1p3.1 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
6 | 5 | nnred 12279 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
7 | 5 | nngt0d 12313 | . . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝑁) |
8 | 1red 11260 | . . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℝ) | |
9 | 1lt2 12435 | . . . . . . . . . . . . . 14 ⊢ 1 < 2 | |
10 | 9 | a1i 11 | . . . . . . . . . . . . 13 ⊢ (𝜑 → 1 < 2) |
11 | 8, 10 | ltned 11395 | . . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≠ 2) |
12 | 11 | necomd 2994 | . . . . . . . . . . 11 ⊢ (𝜑 → 2 ≠ 1) |
13 | 2, 4, 6, 7, 12 | relogbcld 41955 | . . . . . . . . . 10 ⊢ (𝜑 → (2 logb 𝑁) ∈ ℝ) |
14 | 5nn0 12544 | . . . . . . . . . . 11 ⊢ 5 ∈ ℕ0 | |
15 | 14 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → 5 ∈ ℕ0) |
16 | 13, 15 | reexpcld 14200 | . . . . . . . . 9 ⊢ (𝜑 → ((2 logb 𝑁)↑5) ∈ ℝ) |
17 | ceilcl 13879 | . . . . . . . . 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 12720 | . . . . . . 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 2824 | . . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ ℝ ↔ (⌈‘((2 logb 𝑁)↑5)) ∈ ℝ)) |
23 | 19, 22 | mpbird 257 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
24 | 0red 11262 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ) | |
25 | 7re 12357 | . . . . . . . 8 ⊢ 7 ∈ ℝ | |
26 | 25 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 7 ∈ ℝ) |
27 | 7pos 12375 | . . . . . . . 8 ⊢ 0 < 7 | |
28 | 27 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 < 7) |
29 | aks4d1p1p3.3 | . . . . . . . . . 10 ⊢ (𝜑 → 3 ≤ 𝑁) | |
30 | 6, 29 | 3lexlogpow5ineq3 42039 | . . . . . . . . 9 ⊢ (𝜑 → 7 < ((2 logb 𝑁)↑5)) |
31 | ceilge 13882 | . . . . . . . . . 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 11419 | . . . . . . . 8 ⊢ (𝜑 → 7 < (⌈‘((2 logb 𝑁)↑5))) |
34 | 21 | eqcomd 2741 | . . . . . . . 8 ⊢ (𝜑 → (⌈‘((2 logb 𝑁)↑5)) = 𝐵) |
35 | 33, 34 | breqtrd 5174 | . . . . . . 7 ⊢ (𝜑 → 7 < 𝐵) |
36 | 24, 26, 23, 28, 35 | lttrd 11420 | . . . . . 6 ⊢ (𝜑 → 0 < 𝐵) |
37 | 2, 4, 23, 36, 12 | relogbcld 41955 | . . . . 5 ⊢ (𝜑 → (2 logb 𝐵) ∈ ℝ) |
38 | 37 | flcld 13835 | . . . 4 ⊢ (𝜑 → (⌊‘(2 logb 𝐵)) ∈ ℤ) |
39 | 38 | zred 12720 | . . 3 ⊢ (𝜑 → (⌊‘(2 logb 𝐵)) ∈ ℝ) |
40 | 16, 8 | readdcld 11288 | . . . 4 ⊢ (𝜑 → (((2 logb 𝑁)↑5) + 1) ∈ ℝ) |
41 | 16 | ltp1d 12196 | . . . . . 6 ⊢ (𝜑 → ((2 logb 𝑁)↑5) < (((2 logb 𝑁)↑5) + 1)) |
42 | 26, 16, 40, 30, 41 | lttrd 11420 | . . . . 5 ⊢ (𝜑 → 7 < (((2 logb 𝑁)↑5) + 1)) |
43 | 24, 26, 40, 28, 42 | lttrd 11420 | . . . 4 ⊢ (𝜑 → 0 < (((2 logb 𝑁)↑5) + 1)) |
44 | 2, 4, 40, 43, 12 | relogbcld 41955 | . . 3 ⊢ (𝜑 → (2 logb (((2 logb 𝑁)↑5) + 1)) ∈ ℝ) |
45 | flle 13836 | . . . 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 13885 | . . . . . . 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 11857 | . . . . . 6 ⊢ (𝜑 → (((⌈‘((2 logb 𝑁)↑5)) − 1) < ((2 logb 𝑁)↑5) ↔ (⌈‘((2 logb 𝑁)↑5)) < (((2 logb 𝑁)↑5) + 1))) |
50 | 48, 49 | mpbid 232 | . . . . 5 ⊢ (𝜑 → (⌈‘((2 logb 𝑁)↑5)) < (((2 logb 𝑁)↑5) + 1)) |
51 | 21, 50 | eqbrtrd 5170 | . . . 4 ⊢ (𝜑 → 𝐵 < (((2 logb 𝑁)↑5) + 1)) |
52 | 2z 12647 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
53 | 52 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℤ) |
54 | 53 | uzidd 12892 | . . . . 5 ⊢ (𝜑 → 2 ∈ (ℤ≥‘2)) |
55 | 23, 36 | elrpd 13072 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
56 | 40, 43 | elrpd 13072 | . . . . 5 ⊢ (𝜑 → (((2 logb 𝑁)↑5) + 1) ∈ ℝ+) |
57 | logblt 26842 | . . . . 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 232 | . . 3 ⊢ (𝜑 → (2 logb 𝐵) < (2 logb (((2 logb 𝑁)↑5) + 1))) |
60 | 39, 37, 44, 46, 59 | lelttrd 11417 | . 2 ⊢ (𝜑 → (⌊‘(2 logb 𝐵)) < (2 logb (((2 logb 𝑁)↑5) + 1))) |
61 | 3re 12344 | . . . . 5 ⊢ 3 ∈ ℝ | |
62 | 61 | a1i 11 | . . . 4 ⊢ (𝜑 → 3 ∈ ℝ) |
63 | 1lt3 12437 | . . . . 5 ⊢ 1 < 3 | |
64 | 63 | a1i 11 | . . . 4 ⊢ (𝜑 → 1 < 3) |
65 | 8, 62, 6, 64, 29 | ltletrd 11419 | . . 3 ⊢ (𝜑 → 1 < 𝑁) |
66 | 6, 65, 39, 44 | cxpltd 26776 | . 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 232 | 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 206 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 < clt 11293 ≤ cle 11294 − cmin 11490 ℕcn 12264 2c2 12319 3c3 12320 5c5 12322 7c7 12324 ℕ0cn0 12524 ℤcz 12611 ℤ≥cuz 12876 ℝ+crp 13032 ⌊cfl 13827 ⌈cceil 13828 ↑cexp 14099 ↑𝑐ccxp 26612 logb clogb 26822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-ceil 13830 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 df-cxp 26614 df-logb 26823 |
This theorem is referenced by: aks4d1p1p2 42052 |
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