aks4d1p1p3 - Mathbox for metakunt

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Theorem aks4d1p1p3 42161

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 12199	. . . . . . 7 ⊢ 2 ∈ ℝ
2	1	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ)
3		2pos 12228	. . . . . . 7 ⊢ 0 < 2
4	3	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 < 2)
5		aks4d1p1p3.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6	5	nnred 12140	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℝ)
7	5	nngt0d 12174	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝑁)
8		1red 11113	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℝ)
9		1lt2 12291	. . . . . . . . . . . . . 14 ⊢ 1 < 2
10	9	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 < 2)
11	8, 10	ltned 11249	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≠ 2)
12	11	necomd 2983	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≠ 1)
13	2, 4, 6, 7, 12	relogbcld 42065	. . . . . . . . . 10 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
14		5nn0 12401	. . . . . . . . . . 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 12577	. . . . . . 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 2816	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ ℝ ↔ (⌈‘((2 log_b 𝑁)↑5)) ∈ ℝ))
23	19, 22	mpbird 257	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
24		0red 11115	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
25		7re 12218	. . . . . . . 8 ⊢ 7 ∈ ℝ
26	25	a1i 11	. . . . . . 7 ⊢ (𝜑 → 7 ∈ ℝ)
27		7pos 12236	. . . . . . . 8 ⊢ 0 < 7
28	27	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 < 7)
29		aks4d1p1p3.3	. . . . . . . . . 10 ⊢ (𝜑 → 3 ≤ 𝑁)
30	6, 29	3lexlogpow5ineq3 42149	. . . . . . . . 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 11273	. . . . . . . 8 ⊢ (𝜑 → 7 < (⌈‘((2 log_b 𝑁)↑5)))
34	21	eqcomd 2737	. . . . . . . 8 ⊢ (𝜑 → (⌈‘((2 log_b 𝑁)↑5)) = 𝐵)
35	33, 34	breqtrd 5115	. . . . . . 7 ⊢ (𝜑 → 7 < 𝐵)
36	24, 26, 23, 28, 35	lttrd 11274	. . . . . 6 ⊢ (𝜑 → 0 < 𝐵)
37	2, 4, 23, 36, 12	relogbcld 42065	. . . . 5 ⊢ (𝜑 → (2 log_b 𝐵) ∈ ℝ)
38	37	flcld 13702	. . . 4 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℤ)
39	38	zred 12577	. . 3 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℝ)
40	16, 8	readdcld 11141	. . . 4 ⊢ (𝜑 → (((2 log_b 𝑁)↑5) + 1) ∈ ℝ)
41	16	ltp1d 12052	. . . . . 6 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) < (((2 log_b 𝑁)↑5) + 1))
42	26, 16, 40, 30, 41	lttrd 11274	. . . . 5 ⊢ (𝜑 → 7 < (((2 log_b 𝑁)↑5) + 1))
43	24, 26, 40, 28, 42	lttrd 11274	. . . 4 ⊢ (𝜑 → 0 < (((2 log_b 𝑁)↑5) + 1))
44	2, 4, 40, 43, 12	relogbcld 42065	. . 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 11713	. . . . . 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 5111	. . . 4 ⊢ (𝜑 → 𝐵 < (((2 log_b 𝑁)↑5) + 1))
52		2z 12504	. . . . . . 7 ⊢ 2 ∈ ℤ
53	52	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℤ)
54	53	uzidd 12748	. . . . 5 ⊢ (𝜑 → 2 ∈ (ℤ_≥‘2))
55	23, 36	elrpd 12931	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
56	40, 43	elrpd 12931	. . . . 5 ⊢ (𝜑 → (((2 log_b 𝑁)↑5) + 1) ∈ ℝ⁺)
57		logblt 26721	. . . . 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 11271	. 2 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) < (2 log_b (((2 log_b 𝑁)↑5) + 1)))
61		3re 12205	. . . . 5 ⊢ 3 ∈ ℝ
62	61	a1i 11	. . . 4 ⊢ (𝜑 → 3 ∈ ℝ)
63		1lt3 12293	. . . . 5 ⊢ 1 < 3
64	63	a1i 11	. . . 4 ⊢ (𝜑 → 1 < 3)
65	8, 62, 6, 64, 29	ltletrd 11273	. . 3 ⊢ (𝜑 → 1 < 𝑁)
66	6, 65, 39, 44	cxpltd 26655	. 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 1541 ∈ wcel 2111 class class class wbr 5089 ‘cfv 6481 (class class class)co 7346 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 < clt 11146 ≤ cle 11147 − cmin 11344 ℕcn 12125 2c2 12180 3c3 12181 5c5 12183 7c7 12185 ℕ₀cn0 12381 ℤcz 12468 ℤ_≥cuz 12732 ℝ⁺crp 12890 ⌊cfl 13694 ⌈cceil 13695 ↑cexp 13968 ↑_𝑐ccxp 26491 log_b clogb 26701

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ioc 13250 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 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 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-fbas 21288 df-fg 21289 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-ntr 22935 df-cls 22936 df-nei 23013 df-lp 23051 df-perf 23052 df-cn 23142 df-cnp 23143 df-haus 23230 df-tx 23477 df-hmeo 23670 df-fil 23761 df-fm 23853 df-flim 23854 df-flf 23855 df-xms 24235 df-ms 24236 df-tms 24237 df-cncf 24798 df-limc 25794 df-dv 25795 df-log 26492 df-cxp 26493 df-logb 26702

This theorem is referenced by: aks4d1p1p2 42162

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