chtqleppi - Intuitionistic Logic Explorer

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Theorem chtqleppi 16327

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

**Assertion**
Ref	Expression
chtqleppi	⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → (θ‘𝐴) ≤ ((π‘𝐴) · (log‘𝐴)))

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

Step	Hyp	Ref	Expression
1		ppiqfi 16275	. . . 4 ⊢ (𝐴 ∈ ℚ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
2	1	adantr 276	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
3		simpr 110	. . . . . . 7 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
4	3	elin2d 3419	. . . . . 6 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
5		prmnn 12913	. . . . . 6 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
6	4, 5	syl 14	. . . . 5 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
7	6	nnrpd 10112	. . . 4 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
8	7	relogcld 16145	. . 3 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
9		qre 10041	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
10	9	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → 𝐴 ∈ ℝ)
11		simpr 110	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → 0 < 𝐴)
12	10, 11	elrpd 10111	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → 𝐴 ∈ ℝ⁺)
13	12	relogcld 16145	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → (log‘𝐴) ∈ ℝ)
14	13	adantr 276	. . 3 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝐴) ∈ ℝ)
15	3	elin1d 3418	. . . . . . 7 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (0[,]𝐴))
16		0re 8333	. . . . . . . . 9 ⊢ 0 ∈ ℝ
17		elicc2 10357	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
18	16, 10, 17	sylancr 418	. . . . . . . 8 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
19	18	biimpa 296	. . . . . . 7 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝑝 ∈ (0[,]𝐴)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴))
20	15, 19	syldan 282	. . . . . 6 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴))
21	20	simp3d 1042	. . . . 5 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≤ 𝐴)
22	7	reeflogd 16146	. . . . 5 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (exp‘(log‘𝑝)) = 𝑝)
23	12	reeflogd 16146	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → (exp‘(log‘𝐴)) = 𝐴)
24	23	adantr 276	. . . . 5 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (exp‘(log‘𝐴)) = 𝐴)
25	21, 22, 24	3brtr4d 4163	. . . 4 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (exp‘(log‘𝑝)) ≤ (exp‘(log‘𝐴)))
26		efle 16035	. . . . 5 ⊢ (((log‘𝑝) ∈ ℝ ∧ (log‘𝐴) ∈ ℝ) → ((log‘𝑝) ≤ (log‘𝐴) ↔ (exp‘(log‘𝑝)) ≤ (exp‘(log‘𝐴))))
27	8, 14, 26	syl2anc 415	. . . 4 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) ≤ (log‘𝐴) ↔ (exp‘(log‘𝑝)) ≤ (exp‘(log‘𝐴))))
28	25, 27	mpbird 167	. . 3 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ≤ (log‘𝐴))
29	2, 8, 14, 28	fsumle 12255	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝) ≤ Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝐴))
30		chtqval 16278	. . 3 ⊢ (𝐴 ∈ ℚ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝))
31	30	adantr 276	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝))
32		ppiqval 16281	. . . . 5 ⊢ (𝐴 ∈ ℚ → (π‘𝐴) = (♯‘((0[,]𝐴) ∩ ℙ)))
33	32	adantr 276	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → (π‘𝐴) = (♯‘((0[,]𝐴) ∩ ℙ)))
34	33	oveq1d 6103	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → ((π‘𝐴) · (log‘𝐴)) = ((♯‘((0[,]𝐴) ∩ ℙ)) · (log‘𝐴)))
35		relogcl 16124	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
36	35	recnd 8361	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
37	12, 36	syl 14	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → (log‘𝐴) ∈ ℂ)
38		fsumconst 12246	. . . 4 ⊢ ((((0[,]𝐴) ∩ ℙ) ∈ Fin ∧ (log‘𝐴) ∈ ℂ) → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝐴) = ((♯‘((0[,]𝐴) ∩ ℙ)) · (log‘𝐴)))
39	2, 37, 38	syl2anc 415	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝐴) = ((♯‘((0[,]𝐴) ∩ ℙ)) · (log‘𝐴)))
40	34, 39	eqtr4d 2274	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → ((π‘𝐴) · (log‘𝐴)) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝐴))
41	29, 31, 40	3brtr4d 4163	1 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → (θ‘𝐴) ≤ ((π‘𝐴) · (log‘𝐴)))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 ∩ cin 3219 class class class wbr 4131 ‘cfv 5380 (class class class)co 6088 Fincfn 7028 ℂcc 8184 ℝcr 8185 0cc0 8186 · cmul 8191 < clt 8367 ≤ cle 8368 ℕcn 9314 ℚcq 10035 ℝ⁺crp 10071 [,]cicc 10310 ♯chash 11236 Σcsu 12144 expce 12434 ℙcprime 12910 logclog 16118 θccht 16266 πcppi 16267

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8277 ax-resscn 8278 ax-1cn 8279 ax-1re 8280 ax-icn 8281 ax-addcl 8282 ax-addrcl 8283 ax-mulcl 8284 ax-mulrcl 8285 ax-addcom 8286 ax-mulcom 8287 ax-addass 8288 ax-mulass 8289 ax-distr 8290 ax-i2m1 8291 ax-0lt1 8292 ax-1rid 8293 ax-0id 8294 ax-rnegex 8295 ax-precex 8296 ax-cnre 8297 ax-pre-ltirr 8298 ax-pre-ltwlin 8299 ax-pre-lttrn 8300 ax-pre-apti 8301 ax-pre-ltadd 8302 ax-pre-mulgt0 8303 ax-pre-mulext 8304 ax-arch 8305 ax-caucvg 8306 ax-pre-suploc 8307 ax-addf 8308 ax-mulf 8309

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-disj 4107 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-of 6305 df-1st 6378 df-2nd 6379 df-recs 6581 df-irdg 6646 df-frec 6667 df-1o 6692 df-2o 6693 df-oadd 6696 df-er 6812 df-map 6929 df-pm 6930 df-en 7029 df-dom 7030 df-fin 7031 df-sup 7331 df-inf 7332 df-pnf 8369 df-mnf 8370 df-xr 8371 df-ltxr 8372 df-le 8373 df-sub 8507 df-neg 8508 df-reap 8912 df-ap 8919 df-div 9012 df-inn 9315 df-2 9373 df-3 9374 df-4 9375 df-n0 9576 df-z 9655 df-uz 9938 df-q 10036 df-rp 10072 df-xneg 10191 df-xadd 10192 df-ioo 10311 df-ico 10313 df-icc 10314 df-fz 10429 df-fzo 10567 df-fl 10722 df-mod 10781 df-seqfrec 10906 df-exp 10997 df-fac 11186 df-bc 11208 df-ihash 11237 df-shft 11602 df-cj 11629 df-re 11630 df-im 11631 df-rsqrt 11786 df-abs 11787 df-clim 12070 df-sumdc 12145 df-ef 12440 df-e 12441 df-dvds 12580 df-prm 12911 df-rest 13654 df-topgen 13673 df-psmet 14989 df-xmet 14990 df-met 14991 df-bl 14992 df-mopn 14993 df-top 15257 df-topon 15270 df-bases 15302 df-ntr 15355 df-cn 15447 df-cnp 15448 df-tx 15512 df-cncf 15830 df-limced 15915 df-dvap 15916 df-relog 16120 df-cht 16269 df-ppi 16270

This theorem is used by: (None)

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