aks4d1p1p3 - Mathbox for metakunt

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Theorem aks4d1p1p3 41595

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 12314	. . . . . . 7 ⊢ 2 ∈ ℝ
2	1	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ)
3		2pos 12343	. . . . . . 7 ⊢ 0 < 2
4	3	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 < 2)
5		aks4d1p1p3.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6	5	nnred 12255	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℝ)
7	5	nngt0d 12289	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝑁)
8		1red 11243	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℝ)
9		1lt2 12411	. . . . . . . . . . . . . 14 ⊢ 1 < 2
10	9	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 < 2)
11	8, 10	ltned 11378	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≠ 2)
12	11	necomd 2986	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≠ 1)
13	2, 4, 6, 7, 12	relogbcld 41498	. . . . . . . . . 10 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
14		5nn0 12520	. . . . . . . . . . 11 ⊢ 5 ∈ ℕ₀
15	14	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 5 ∈ ℕ₀)
16	13, 15	reexpcld 14157	. . . . . . . . 9 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ∈ ℝ)
17		ceilcl 13837	. . . . . . . . 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 12694	. . . . . . 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 2810	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ ℝ ↔ (⌈‘((2 log_b 𝑁)↑5)) ∈ ℝ))
23	19, 22	mpbird 256	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
24		0red 11245	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
25		7re 12333	. . . . . . . 8 ⊢ 7 ∈ ℝ
26	25	a1i 11	. . . . . . 7 ⊢ (𝜑 → 7 ∈ ℝ)
27		7pos 12351	. . . . . . . 8 ⊢ 0 < 7
28	27	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 < 7)
29		aks4d1p1p3.3	. . . . . . . . . 10 ⊢ (𝜑 → 3 ≤ 𝑁)
30	6, 29	3lexlogpow5ineq3 41583	. . . . . . . . 9 ⊢ (𝜑 → 7 < ((2 log_b 𝑁)↑5))
31		ceilge 13840	. . . . . . . . . 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 11402	. . . . . . . 8 ⊢ (𝜑 → 7 < (⌈‘((2 log_b 𝑁)↑5)))
34	21	eqcomd 2731	. . . . . . . 8 ⊢ (𝜑 → (⌈‘((2 log_b 𝑁)↑5)) = 𝐵)
35	33, 34	breqtrd 5169	. . . . . . 7 ⊢ (𝜑 → 7 < 𝐵)
36	24, 26, 23, 28, 35	lttrd 11403	. . . . . 6 ⊢ (𝜑 → 0 < 𝐵)
37	2, 4, 23, 36, 12	relogbcld 41498	. . . . 5 ⊢ (𝜑 → (2 log_b 𝐵) ∈ ℝ)
38	37	flcld 13793	. . . 4 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℤ)
39	38	zred 12694	. . 3 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℝ)
40	16, 8	readdcld 11271	. . . 4 ⊢ (𝜑 → (((2 log_b 𝑁)↑5) + 1) ∈ ℝ)
41	16	ltp1d 12172	. . . . . 6 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) < (((2 log_b 𝑁)↑5) + 1))
42	26, 16, 40, 30, 41	lttrd 11403	. . . . 5 ⊢ (𝜑 → 7 < (((2 log_b 𝑁)↑5) + 1))
43	24, 26, 40, 28, 42	lttrd 11403	. . . 4 ⊢ (𝜑 → 0 < (((2 log_b 𝑁)↑5) + 1))
44	2, 4, 40, 43, 12	relogbcld 41498	. . 3 ⊢ (𝜑 → (2 log_b (((2 log_b 𝑁)↑5) + 1)) ∈ ℝ)
45		flle 13794	. . . 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 13843	. . . . . . 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 11838	. . . . . 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 5165	. . . 4 ⊢ (𝜑 → 𝐵 < (((2 log_b 𝑁)↑5) + 1))
52		2z 12622	. . . . . . 7 ⊢ 2 ∈ ℤ
53	52	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℤ)
54	53	uzidd 12866	. . . . 5 ⊢ (𝜑 → 2 ∈ (ℤ_≥‘2))
55	23, 36	elrpd 13043	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
56	40, 43	elrpd 13043	. . . . 5 ⊢ (𝜑 → (((2 log_b 𝑁)↑5) + 1) ∈ ℝ⁺)
57		logblt 26732	. . . . 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 1368	. . . 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 11400	. 2 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) < (2 log_b (((2 log_b 𝑁)↑5) + 1)))
61		3re 12320	. . . . 5 ⊢ 3 ∈ ℝ
62	61	a1i 11	. . . 4 ⊢ (𝜑 → 3 ∈ ℝ)
63		1lt3 12413	. . . . 5 ⊢ 1 < 3
64	63	a1i 11	. . . 4 ⊢ (𝜑 → 1 < 3)
65	8, 62, 6, 64, 29	ltletrd 11402	. . 3 ⊢ (𝜑 → 1 < 𝑁)
66	6, 65, 39, 44	cxpltd 26669	. 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 1533 ∈ wcel 2098 class class class wbr 5143 ‘cfv 6542 (class class class)co 7415 ℝcr 11135 0cc0 11136 1c1 11137 + caddc 11139 < clt 11276 ≤ cle 11277 − cmin 11472 ℕcn 12240 2c2 12295 3c3 12296 5c5 12298 7c7 12300 ℕ₀cn0 12500 ℤcz 12586 ℤ_≥cuz 12850 ℝ⁺crp 13004 ⌊cfl 13785 ⌈cceil 13786 ↑cexp 14056 ↑_𝑐ccxp 26505 log_b clogb 26712

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 ax-addf 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-er 8721 df-map 8843 df-pm 8844 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-fi 9432 df-sup 9463 df-inf 9464 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-q 12961 df-rp 13005 df-xneg 13122 df-xadd 13123 df-xmul 13124 df-ioo 13358 df-ioc 13359 df-ico 13360 df-icc 13361 df-fz 13515 df-fzo 13658 df-fl 13787 df-ceil 13788 df-mod 13865 df-seq 13997 df-exp 14057 df-fac 14263 df-bc 14292 df-hash 14320 df-shft 15044 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-limsup 15445 df-clim 15462 df-rlim 15463 df-sum 15663 df-ef 16041 df-sin 16043 df-cos 16044 df-pi 16046 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-hom 17254 df-cco 17255 df-rest 17401 df-topn 17402 df-0g 17420 df-gsum 17421 df-topgen 17422 df-pt 17423 df-prds 17426 df-xrs 17481 df-qtop 17486 df-imas 17487 df-xps 17489 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18738 df-mulg 19026 df-cntz 19270 df-cmn 19739 df-psmet 21273 df-xmet 21274 df-met 21275 df-bl 21276 df-mopn 21277 df-fbas 21278 df-fg 21279 df-cnfld 21282 df-top 22812 df-topon 22829 df-topsp 22851 df-bases 22865 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-lp 23056 df-perf 23057 df-cn 23147 df-cnp 23148 df-haus 23235 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-xms 24242 df-ms 24243 df-tms 24244 df-cncf 24814 df-limc 25811 df-dv 25812 df-log 26506 df-cxp 26507 df-logb 26713

This theorem is referenced by: aks4d1p1p2 41596

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