aks4d1p1p3 - Mathbox for metakunt

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Theorem aks4d1p1p3 42051

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 12238	. . . . . . 7 ⊢ 2 ∈ ℝ
2	1	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ)
3		2pos 12267	. . . . . . 7 ⊢ 0 < 2
4	3	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 < 2)
5		aks4d1p1p3.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6	5	nnred 12179	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℝ)
7	5	nngt0d 12213	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝑁)
8		1red 11153	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℝ)
9		1lt2 12330	. . . . . . . . . . . . . 14 ⊢ 1 < 2
10	9	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 < 2)
11	8, 10	ltned 11288	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≠ 2)
12	11	necomd 2980	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≠ 1)
13	2, 4, 6, 7, 12	relogbcld 41955	. . . . . . . . . 10 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
14		5nn0 12440	. . . . . . . . . . 11 ⊢ 5 ∈ ℕ₀
15	14	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 5 ∈ ℕ₀)
16	13, 15	reexpcld 14106	. . . . . . . . 9 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ∈ ℝ)
17		ceilcl 13782	. . . . . . . . 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 12616	. . . . . . 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 11155	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
25		7re 12257	. . . . . . . 8 ⊢ 7 ∈ ℝ
26	25	a1i 11	. . . . . . 7 ⊢ (𝜑 → 7 ∈ ℝ)
27		7pos 12275	. . . . . . . 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 log_b 𝑁)↑5))
31		ceilge 13785	. . . . . . . . . 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 11312	. . . . . . . 8 ⊢ (𝜑 → 7 < (⌈‘((2 log_b 𝑁)↑5)))
34	21	eqcomd 2735	. . . . . . . 8 ⊢ (𝜑 → (⌈‘((2 log_b 𝑁)↑5)) = 𝐵)
35	33, 34	breqtrd 5128	. . . . . . 7 ⊢ (𝜑 → 7 < 𝐵)
36	24, 26, 23, 28, 35	lttrd 11313	. . . . . 6 ⊢ (𝜑 → 0 < 𝐵)
37	2, 4, 23, 36, 12	relogbcld 41955	. . . . 5 ⊢ (𝜑 → (2 log_b 𝐵) ∈ ℝ)
38	37	flcld 13738	. . . 4 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℤ)
39	38	zred 12616	. . 3 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℝ)
40	16, 8	readdcld 11181	. . . 4 ⊢ (𝜑 → (((2 log_b 𝑁)↑5) + 1) ∈ ℝ)
41	16	ltp1d 12091	. . . . . 6 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) < (((2 log_b 𝑁)↑5) + 1))
42	26, 16, 40, 30, 41	lttrd 11313	. . . . 5 ⊢ (𝜑 → 7 < (((2 log_b 𝑁)↑5) + 1))
43	24, 26, 40, 28, 42	lttrd 11313	. . . 4 ⊢ (𝜑 → 0 < (((2 log_b 𝑁)↑5) + 1))
44	2, 4, 40, 43, 12	relogbcld 41955	. . 3 ⊢ (𝜑 → (2 log_b (((2 log_b 𝑁)↑5) + 1)) ∈ ℝ)
45		flle 13739	. . . 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 13788	. . . . . . 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 11752	. . . . . 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 5124	. . . 4 ⊢ (𝜑 → 𝐵 < (((2 log_b 𝑁)↑5) + 1))
52		2z 12543	. . . . . . 7 ⊢ 2 ∈ ℤ
53	52	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℤ)
54	53	uzidd 12787	. . . . 5 ⊢ (𝜑 → 2 ∈ (ℤ_≥‘2))
55	23, 36	elrpd 12970	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
56	40, 43	elrpd 12970	. . . . 5 ⊢ (𝜑 → (((2 log_b 𝑁)↑5) + 1) ∈ ℝ⁺)
57		logblt 26728	. . . . 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 11310	. 2 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) < (2 log_b (((2 log_b 𝑁)↑5) + 1)))
61		3re 12244	. . . . 5 ⊢ 3 ∈ ℝ
62	61	a1i 11	. . . 4 ⊢ (𝜑 → 3 ∈ ℝ)
63		1lt3 12332	. . . . 5 ⊢ 1 < 3
64	63	a1i 11	. . . 4 ⊢ (𝜑 → 1 < 3)
65	8, 62, 6, 64, 29	ltletrd 11312	. . 3 ⊢ (𝜑 → 1 < 𝑁)
66	6, 65, 39, 44	cxpltd 26662	. 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 5102 ‘cfv 6499 (class class class)co 7369 ℝcr 11045 0cc0 11046 1c1 11047 + caddc 11049 < clt 11186 ≤ cle 11187 − cmin 11383 ℕcn 12164 2c2 12219 3c3 12220 5c5 12222 7c7 12224 ℕ₀cn0 12420 ℤcz 12507 ℤ_≥cuz 12771 ℝ⁺crp 12929 ⌊cfl 13730 ⌈cceil 13731 ↑cexp 14004 ↑_𝑐ccxp 26498 log_b clogb 26708

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9572 ax-cnex 11102 ax-resscn 11103 ax-1cn 11104 ax-icn 11105 ax-addcl 11106 ax-addrcl 11107 ax-mulcl 11108 ax-mulrcl 11109 ax-mulcom 11110 ax-addass 11111 ax-mulass 11112 ax-distr 11113 ax-i2m1 11114 ax-1ne0 11115 ax-1rid 11116 ax-rnegex 11117 ax-rrecex 11118 ax-cnre 11119 ax-pre-lttri 11120 ax-pre-lttrn 11121 ax-pre-ltadd 11122 ax-pre-mulgt0 11123 ax-pre-sup 11124 ax-addf 11125

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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9870 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11385 df-neg 11386 df-div 11814 df-nn 12165 df-2 12227 df-3 12228 df-4 12229 df-5 12230 df-6 12231 df-7 12232 df-8 12233 df-9 12234 df-n0 12421 df-z 12508 df-dec 12628 df-uz 12772 df-q 12886 df-rp 12930 df-xneg 13050 df-xadd 13051 df-xmul 13052 df-ioo 13288 df-ioc 13289 df-ico 13290 df-icc 13291 df-fz 13447 df-fzo 13594 df-fl 13732 df-ceil 13733 df-mod 13810 df-seq 13945 df-exp 14005 df-fac 14217 df-bc 14246 df-hash 14274 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15414 df-clim 15431 df-rlim 15432 df-sum 15630 df-ef 16010 df-sin 16012 df-cos 16013 df-pi 16015 df-struct 17094 df-sets 17111 df-slot 17129 df-ndx 17141 df-base 17157 df-ress 17178 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17362 df-topn 17363 df-0g 17381 df-gsum 17382 df-topgen 17383 df-pt 17384 df-prds 17387 df-xrs 17442 df-qtop 17447 df-imas 17448 df-xps 17450 df-mre 17524 df-mrc 17525 df-acs 17527 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-submnd 18694 df-mulg 18983 df-cntz 19232 df-cmn 19697 df-psmet 21289 df-xmet 21290 df-met 21291 df-bl 21292 df-mopn 21293 df-fbas 21294 df-fg 21295 df-cnfld 21298 df-top 22815 df-topon 22832 df-topsp 22854 df-bases 22867 df-cld 22940 df-ntr 22941 df-cls 22942 df-nei 23019 df-lp 23057 df-perf 23058 df-cn 23148 df-cnp 23149 df-haus 23236 df-tx 23483 df-hmeo 23676 df-fil 23767 df-fm 23859 df-flim 23860 df-flf 23861 df-xms 24242 df-ms 24243 df-tms 24244 df-cncf 24805 df-limc 25801 df-dv 25802 df-log 26499 df-cxp 26500 df-logb 26709

This theorem is referenced by: aks4d1p1p2 42052

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