aks4d1p1p3 - Mathbox for metakunt

	Mathbox for metakunt		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > aks4d1p1p3		Structured version Visualization version GIF version

Theorem aks4d1p1p3 42062

Description: Bound of a ceiling of the binary logarithm to the fifth power. (Contributed by metakunt, 19-Aug-2024.)

**Hypotheses**
Ref	Expression
aks4d1p1p3.1	⊢ (𝜑 → 𝑁 ∈ ℕ)
aks4d1p1p3.2	⊢ 𝐵 = (⌈‘((2 log_b 𝑁)↑5))
aks4d1p1p3.3	⊢ (𝜑 → 3 ≤ 𝑁)

**Assertion**
Ref	Expression
aks4d1p1p3	⊢ (𝜑 → (𝑁↑_𝑐(⌊‘(2 log_b 𝐵))) < (𝑁↑_𝑐(2 log_b (((2 log_b 𝑁)↑5) + 1))))

**Proof of Theorem aks4d1p1p3**
Step	Hyp	Ref	Expression
1		2re 12202	. . . . . . 7 ⊢ 2 ∈ ℝ
2	1	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ)
3		2pos 12231	. . . . . . 7 ⊢ 0 < 2
4	3	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 < 2)
5		aks4d1p1p3.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6	5	nnred 12143	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℝ)
7	5	nngt0d 12177	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝑁)
8		1red 11116	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℝ)
9		1lt2 12294	. . . . . . . . . . . . . 14 ⊢ 1 < 2
10	9	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 < 2)
11	8, 10	ltned 11252	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≠ 2)
12	11	necomd 2980	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≠ 1)
13	2, 4, 6, 7, 12	relogbcld 41966	. . . . . . . . . 10 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
14		5nn0 12404	. . . . . . . . . . 11 ⊢ 5 ∈ ℕ₀
15	14	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 5 ∈ ℕ₀)
16	13, 15	reexpcld 14070	. . . . . . . . 9 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ∈ ℝ)
17		ceilcl 13746	. . . . . . . . 9 ⊢ (((2 log_b 𝑁)↑5) ∈ ℝ → (⌈‘((2 log_b 𝑁)↑5)) ∈ ℤ)
18	16, 17	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⌈‘((2 log_b 𝑁)↑5)) ∈ ℤ)
19	18	zred 12580	. . . . . . 7 ⊢ (𝜑 → (⌈‘((2 log_b 𝑁)↑5)) ∈ ℝ)
20		aks4d1p1p3.2	. . . . . . . . 9 ⊢ 𝐵 = (⌈‘((2 log_b 𝑁)↑5))
21	20	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (⌈‘((2 log_b 𝑁)↑5)))
22	21	eleq1d 2813	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ ℝ ↔ (⌈‘((2 log_b 𝑁)↑5)) ∈ ℝ))
23	19, 22	mpbird 257	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
24		0red 11118	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
25		7re 12221	. . . . . . . 8 ⊢ 7 ∈ ℝ
26	25	a1i 11	. . . . . . 7 ⊢ (𝜑 → 7 ∈ ℝ)
27		7pos 12239	. . . . . . . 8 ⊢ 0 < 7
28	27	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 < 7)
29		aks4d1p1p3.3	. . . . . . . . . 10 ⊢ (𝜑 → 3 ≤ 𝑁)
30	6, 29	3lexlogpow5ineq3 42050	. . . . . . . . 9 ⊢ (𝜑 → 7 < ((2 log_b 𝑁)↑5))
31		ceilge 13749	. . . . . . . . . 10 ⊢ (((2 log_b 𝑁)↑5) ∈ ℝ → ((2 log_b 𝑁)↑5) ≤ (⌈‘((2 log_b 𝑁)↑5)))
32	16, 31	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ≤ (⌈‘((2 log_b 𝑁)↑5)))
33	26, 16, 19, 30, 32	ltletrd 11276	. . . . . . . 8 ⊢ (𝜑 → 7 < (⌈‘((2 log_b 𝑁)↑5)))
34	21	eqcomd 2735	. . . . . . . 8 ⊢ (𝜑 → (⌈‘((2 log_b 𝑁)↑5)) = 𝐵)
35	33, 34	breqtrd 5118	. . . . . . 7 ⊢ (𝜑 → 7 < 𝐵)
36	24, 26, 23, 28, 35	lttrd 11277	. . . . . 6 ⊢ (𝜑 → 0 < 𝐵)
37	2, 4, 23, 36, 12	relogbcld 41966	. . . . 5 ⊢ (𝜑 → (2 log_b 𝐵) ∈ ℝ)
38	37	flcld 13702	. . . 4 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℤ)
39	38	zred 12580	. . 3 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℝ)
40	16, 8	readdcld 11144	. . . 4 ⊢ (𝜑 → (((2 log_b 𝑁)↑5) + 1) ∈ ℝ)
41	16	ltp1d 12055	. . . . . 6 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) < (((2 log_b 𝑁)↑5) + 1))
42	26, 16, 40, 30, 41	lttrd 11277	. . . . 5 ⊢ (𝜑 → 7 < (((2 log_b 𝑁)↑5) + 1))
43	24, 26, 40, 28, 42	lttrd 11277	. . . 4 ⊢ (𝜑 → 0 < (((2 log_b 𝑁)↑5) + 1))
44	2, 4, 40, 43, 12	relogbcld 41966	. . 3 ⊢ (𝜑 → (2 log_b (((2 log_b 𝑁)↑5) + 1)) ∈ ℝ)
45		flle 13703	. . . 4 ⊢ ((2 log_b 𝐵) ∈ ℝ → (⌊‘(2 log_b 𝐵)) ≤ (2 log_b 𝐵))
46	37, 45	syl 17	. . 3 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ≤ (2 log_b 𝐵))
47		ceilm1lt 13752	. . . . . . 7 ⊢ (((2 log_b 𝑁)↑5) ∈ ℝ → ((⌈‘((2 log_b 𝑁)↑5)) − 1) < ((2 log_b 𝑁)↑5))
48	16, 47	syl 17	. . . . . 6 ⊢ (𝜑 → ((⌈‘((2 log_b 𝑁)↑5)) − 1) < ((2 log_b 𝑁)↑5))
49	19, 8, 16	ltsubaddd 11716	. . . . . 6 ⊢ (𝜑 → (((⌈‘((2 log_b 𝑁)↑5)) − 1) < ((2 log_b 𝑁)↑5) ↔ (⌈‘((2 log_b 𝑁)↑5)) < (((2 log_b 𝑁)↑5) + 1)))
50	48, 49	mpbid 232	. . . . 5 ⊢ (𝜑 → (⌈‘((2 log_b 𝑁)↑5)) < (((2 log_b 𝑁)↑5) + 1))
51	21, 50	eqbrtrd 5114	. . . 4 ⊢ (𝜑 → 𝐵 < (((2 log_b 𝑁)↑5) + 1))
52		2z 12507	. . . . . . 7 ⊢ 2 ∈ ℤ
53	52	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℤ)
54	53	uzidd 12751	. . . . 5 ⊢ (𝜑 → 2 ∈ (ℤ_≥‘2))
55	23, 36	elrpd 12934	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
56	40, 43	elrpd 12934	. . . . 5 ⊢ (𝜑 → (((2 log_b 𝑁)↑5) + 1) ∈ ℝ⁺)
57		logblt 26692	. . . . 5 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝐵 ∈ ℝ⁺ ∧ (((2 log_b 𝑁)↑5) + 1) ∈ ℝ⁺) → (𝐵 < (((2 log_b 𝑁)↑5) + 1) ↔ (2 log_b 𝐵) < (2 log_b (((2 log_b 𝑁)↑5) + 1))))
58	54, 55, 56, 57	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝐵 < (((2 log_b 𝑁)↑5) + 1) ↔ (2 log_b 𝐵) < (2 log_b (((2 log_b 𝑁)↑5) + 1))))
59	51, 58	mpbid 232	. . 3 ⊢ (𝜑 → (2 log_b 𝐵) < (2 log_b (((2 log_b 𝑁)↑5) + 1)))
60	39, 37, 44, 46, 59	lelttrd 11274	. 2 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) < (2 log_b (((2 log_b 𝑁)↑5) + 1)))
61		3re 12208	. . . . 5 ⊢ 3 ∈ ℝ
62	61	a1i 11	. . . 4 ⊢ (𝜑 → 3 ∈ ℝ)
63		1lt3 12296	. . . . 5 ⊢ 1 < 3
64	63	a1i 11	. . . 4 ⊢ (𝜑 → 1 < 3)
65	8, 62, 6, 64, 29	ltletrd 11276	. . 3 ⊢ (𝜑 → 1 < 𝑁)
66	6, 65, 39, 44	cxpltd 26626	. 2 ⊢ (𝜑 → ((⌊‘(2 log_b 𝐵)) < (2 log_b (((2 log_b 𝑁)↑5) + 1)) ↔ (𝑁↑_𝑐(⌊‘(2 log_b 𝐵))) < (𝑁↑_𝑐(2 log_b (((2 log_b 𝑁)↑5) + 1)))))
67	60, 66	mpbid 232	1 ⊢ (𝜑 → (𝑁↑_𝑐(⌊‘(2 log_b 𝐵))) < (𝑁↑_𝑐(2 log_b (((2 log_b 𝑁)↑5) + 1))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2109 class class class wbr 5092 ‘cfv 6482 (class class class)co 7349 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 < clt 11149 ≤ cle 11150 − cmin 11347 ℕcn 12128 2c2 12183 3c3 12184 5c5 12186 7c7 12188 ℕ₀cn0 12384 ℤcz 12471 ℤ_≥cuz 12735 ℝ⁺crp 12893 ⌊cfl 13694 ⌈cceil 13695 ↑cexp 13968 ↑_𝑐ccxp 26462 log_b clogb 26672

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ioc 13253 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-ceil 13697 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-sin 15976 df-cos 15977 df-pi 15979 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-haus 23200 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-limc 25765 df-dv 25766 df-log 26463 df-cxp 26464 df-logb 26673

This theorem is referenced by: aks4d1p1p2 42063

W3C validator