aks4d1p1p3 - Mathbox for metakunt

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Theorem aks4d1p1p3 40922

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 12282	. . . . . . 7 ⊢ 2 ∈ ℝ
2	1	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ)
3		2pos 12311	. . . . . . 7 ⊢ 0 < 2
4	3	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 < 2)
5		aks4d1p1p3.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6	5	nnred 12223	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℝ)
7	5	nngt0d 12257	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝑁)
8		1red 11211	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℝ)
9		1lt2 12379	. . . . . . . . . . . . . 14 ⊢ 1 < 2
10	9	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 < 2)
11	8, 10	ltned 11346	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≠ 2)
12	11	necomd 2996	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≠ 1)
13	2, 4, 6, 7, 12	relogbcld 40826	. . . . . . . . . 10 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
14		5nn0 12488	. . . . . . . . . . 11 ⊢ 5 ∈ ℕ₀
15	14	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 5 ∈ ℕ₀)
16	13, 15	reexpcld 14124	. . . . . . . . 9 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ∈ ℝ)
17		ceilcl 13803	. . . . . . . . 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 12662	. . . . . . 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 2818	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ ℝ ↔ (⌈‘((2 log_b 𝑁)↑5)) ∈ ℝ))
23	19, 22	mpbird 256	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
24		0red 11213	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
25		7re 12301	. . . . . . . 8 ⊢ 7 ∈ ℝ
26	25	a1i 11	. . . . . . 7 ⊢ (𝜑 → 7 ∈ ℝ)
27		7pos 12319	. . . . . . . 8 ⊢ 0 < 7
28	27	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 < 7)
29		aks4d1p1p3.3	. . . . . . . . . 10 ⊢ (𝜑 → 3 ≤ 𝑁)
30	6, 29	3lexlogpow5ineq3 40910	. . . . . . . . 9 ⊢ (𝜑 → 7 < ((2 log_b 𝑁)↑5))
31		ceilge 13806	. . . . . . . . . 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 11370	. . . . . . . 8 ⊢ (𝜑 → 7 < (⌈‘((2 log_b 𝑁)↑5)))
34	21	eqcomd 2738	. . . . . . . 8 ⊢ (𝜑 → (⌈‘((2 log_b 𝑁)↑5)) = 𝐵)
35	33, 34	breqtrd 5173	. . . . . . 7 ⊢ (𝜑 → 7 < 𝐵)
36	24, 26, 23, 28, 35	lttrd 11371	. . . . . 6 ⊢ (𝜑 → 0 < 𝐵)
37	2, 4, 23, 36, 12	relogbcld 40826	. . . . 5 ⊢ (𝜑 → (2 log_b 𝐵) ∈ ℝ)
38	37	flcld 13759	. . . 4 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℤ)
39	38	zred 12662	. . 3 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℝ)
40	16, 8	readdcld 11239	. . . 4 ⊢ (𝜑 → (((2 log_b 𝑁)↑5) + 1) ∈ ℝ)
41	16	ltp1d 12140	. . . . . 6 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) < (((2 log_b 𝑁)↑5) + 1))
42	26, 16, 40, 30, 41	lttrd 11371	. . . . 5 ⊢ (𝜑 → 7 < (((2 log_b 𝑁)↑5) + 1))
43	24, 26, 40, 28, 42	lttrd 11371	. . . 4 ⊢ (𝜑 → 0 < (((2 log_b 𝑁)↑5) + 1))
44	2, 4, 40, 43, 12	relogbcld 40826	. . 3 ⊢ (𝜑 → (2 log_b (((2 log_b 𝑁)↑5) + 1)) ∈ ℝ)
45		flle 13760	. . . 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 13809	. . . . . . 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 11806	. . . . . 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 231	. . . . 5 ⊢ (𝜑 → (⌈‘((2 log_b 𝑁)↑5)) < (((2 log_b 𝑁)↑5) + 1))
51	21, 50	eqbrtrd 5169	. . . 4 ⊢ (𝜑 → 𝐵 < (((2 log_b 𝑁)↑5) + 1))
52		2z 12590	. . . . . . 7 ⊢ 2 ∈ ℤ
53	52	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℤ)
54	53	uzidd 12834	. . . . 5 ⊢ (𝜑 → 2 ∈ (ℤ_≥‘2))
55	23, 36	elrpd 13009	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
56	40, 43	elrpd 13009	. . . . 5 ⊢ (𝜑 → (((2 log_b 𝑁)↑5) + 1) ∈ ℝ⁺)
57		logblt 26278	. . . . 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 1371	. . . 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 231	. . 3 ⊢ (𝜑 → (2 log_b 𝐵) < (2 log_b (((2 log_b 𝑁)↑5) + 1)))
60	39, 37, 44, 46, 59	lelttrd 11368	. 2 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) < (2 log_b (((2 log_b 𝑁)↑5) + 1)))
61		3re 12288	. . . . 5 ⊢ 3 ∈ ℝ
62	61	a1i 11	. . . 4 ⊢ (𝜑 → 3 ∈ ℝ)
63		1lt3 12381	. . . . 5 ⊢ 1 < 3
64	63	a1i 11	. . . 4 ⊢ (𝜑 → 1 < 3)
65	8, 62, 6, 64, 29	ltletrd 11370	. . 3 ⊢ (𝜑 → 1 < 𝑁)
66	6, 65, 39, 44	cxpltd 26218	. 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 231	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 205 = wceq 1541 ∈ wcel 2106 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 < clt 11244 ≤ cle 11245 − cmin 11440 ℕcn 12208 2c2 12263 3c3 12264 5c5 12266 7c7 12268 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ℝ⁺crp 12970 ⌊cfl 13751 ⌈cceil 13752 ↑cexp 14023 ↑_𝑐ccxp 26055 log_b clogb 26258

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-ceil 13754 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375 df-log 26056 df-cxp 26057 df-logb 26259

This theorem is referenced by: aks4d1p1p2 40923

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