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Theorem chtleppi 26702

Description: Upper bound on the θ function. (Contributed by Mario Carneiro, 22-Sep-2014.)

**Assertion**
Ref	Expression
chtleppi	⊢ (𝐴 ∈ ℝ⁺ → (θ‘𝐴) ≤ ((π‘𝐴) · (log‘𝐴)))

Proof of Theorem chtleppi Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpre 12978	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		ppifi 26599	. . . 4 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
3	1, 2	syl 17	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
4		simpr 485	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
5	4	elin2d 4198	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
6		prmnn 16607	. . . . . 6 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
7	5, 6	syl 17	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
8	7	nnrpd 13010	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
9	8	relogcld 26122	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
10		relogcl 26075	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
11	10	adantr 481	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝐴) ∈ ℝ)
12	4	elin1d 4197	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (0[,]𝐴))
13		0re 11212	. . . . . . . . 9 ⊢ 0 ∈ ℝ
14		elicc2 13385	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
15	13, 1, 14	sylancr 587	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
16	15	biimpa 477	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑝 ∈ (0[,]𝐴)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴))
17	12, 16	syldan 591	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴))
18	17	simp3d 1144	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≤ 𝐴)
19	8	reeflogd 26123	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (exp‘(log‘𝑝)) = 𝑝)
20		reeflog 26080	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(log‘𝐴)) = 𝐴)
21	20	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (exp‘(log‘𝐴)) = 𝐴)
22	18, 19, 21	3brtr4d 5179	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (exp‘(log‘𝑝)) ≤ (exp‘(log‘𝐴)))
23		efle 16057	. . . . 5 ⊢ (((log‘𝑝) ∈ ℝ ∧ (log‘𝐴) ∈ ℝ) → ((log‘𝑝) ≤ (log‘𝐴) ↔ (exp‘(log‘𝑝)) ≤ (exp‘(log‘𝐴))))
24	9, 11, 23	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) ≤ (log‘𝐴) ↔ (exp‘(log‘𝑝)) ≤ (exp‘(log‘𝐴))))
25	22, 24	mpbird 256	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ≤ (log‘𝐴))
26	3, 9, 11, 25	fsumle 15741	. 2 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝) ≤ Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝐴))
27		chtval 26603	. . 3 ⊢ (𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝))
28	1, 27	syl 17	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝))
29		ppival 26620	. . . . 5 ⊢ (𝐴 ∈ ℝ → (π‘𝐴) = (♯‘((0[,]𝐴) ∩ ℙ)))
30	1, 29	syl 17	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (π‘𝐴) = (♯‘((0[,]𝐴) ∩ ℙ)))
31	30	oveq1d 7420	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((π‘𝐴) · (log‘𝐴)) = ((♯‘((0[,]𝐴) ∩ ℙ)) · (log‘𝐴)))
32	10	recnd 11238	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
33		fsumconst 15732	. . . 4 ⊢ ((((0[,]𝐴) ∩ ℙ) ∈ Fin ∧ (log‘𝐴) ∈ ℂ) → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝐴) = ((♯‘((0[,]𝐴) ∩ ℙ)) · (log‘𝐴)))
34	3, 32, 33	syl2anc 584	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝐴) = ((♯‘((0[,]𝐴) ∩ ℙ)) · (log‘𝐴)))
35	31, 34	eqtr4d 2775	. 2 ⊢ (𝐴 ∈ ℝ⁺ → ((π‘𝐴) · (log‘𝐴)) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝐴))
36	26, 28, 35	3brtr4d 5179	1 ⊢ (𝐴 ∈ ℝ⁺ → (θ‘𝐴) ≤ ((π‘𝐴) · (log‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∩ cin 3946 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 Fincfn 8935 ℂcc 11104 ℝcr 11105 0cc0 11106 · cmul 11111 ≤ cle 11245 ℕcn 12208 ℝ⁺crp 12970 [,]cicc 13323 ♯chash 14286 Σcsu 15628 expce 16001 ℙcprime 16604 logclog 26054 θccht 26584 πcppi 26587

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-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-dvds 16194 df-prm 16605 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-cht 26590 df-ppi 26593

This theorem is referenced by: chtppilim 26967

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