chtqleppi - Intuitionistic Logic Explorer

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Theorem chtqleppi 16260

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 16208	. . . 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 12907	. . . . . 6 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
6	4, 5	syl 14	. . . . 5 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
7	6	nnrpd 10106	. . . 4 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
8	7	relogcld 16078	. . 3 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
9		qre 10035	. . . . . . 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 10105	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → 𝐴 ∈ ℝ⁺)
13	12	relogcld 16078	. . . 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 8327	. . . . . . . . 9 ⊢ 0 ∈ ℝ
17		elicc2 10351	. . . . . . . . 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 16079	. . . . 5 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (exp‘(log‘𝑝)) = 𝑝)
23	12	reeflogd 16079	. . . . . 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 4162	. . . 4 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (exp‘(log‘𝑝)) ≤ (exp‘(log‘𝐴)))
26		efle 15968	. . . . 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 12249	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝) ≤ Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝐴))
30		chtqval 16211	. . 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 16214	. . . . 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 6100	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → ((π‘𝐴) · (log‘𝐴)) = ((♯‘((0[,]𝐴) ∩ ℙ)) · (log‘𝐴)))
35		relogcl 16057	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
36	35	recnd 8355	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
37	12, 36	syl 14	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → (log‘𝐴) ∈ ℂ)
38		fsumconst 12240	. . . 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 4162	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 4130 ‘cfv 5377 (class class class)co 6085 Fincfn 7022 ℂcc 8178 ℝcr 8179 0cc0 8180 · cmul 8185 < clt 8361 ≤ cle 8362 ℕcn 9307 ℚcq 10029 ℝ⁺crp 10065 [,]cicc 10304 ♯chash 11230 Σcsu 12138 expce 12428 ℙcprime 12904 logclog 16051 θccht 16199 πcppi 16200

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 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8271 ax-resscn 8272 ax-1cn 8273 ax-1re 8274 ax-icn 8275 ax-addcl 8276 ax-addrcl 8277 ax-mulcl 8278 ax-mulrcl 8279 ax-addcom 8280 ax-mulcom 8281 ax-addass 8282 ax-mulass 8283 ax-distr 8284 ax-i2m1 8285 ax-0lt1 8286 ax-1rid 8287 ax-0id 8288 ax-rnegex 8289 ax-precex 8290 ax-cnre 8291 ax-pre-ltirr 8292 ax-pre-ltwlin 8293 ax-pre-lttrn 8294 ax-pre-apti 8295 ax-pre-ltadd 8296 ax-pre-mulgt0 8297 ax-pre-mulext 8298 ax-arch 8299 ax-caucvg 8300 ax-pre-suploc 8301 ax-addf 8302 ax-mulf 8303

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 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-of 6302 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-2o 6688 df-oadd 6691 df-er 6807 df-map 6924 df-pm 6925 df-en 7023 df-dom 7024 df-fin 7025 df-sup 7325 df-inf 7326 df-pnf 8363 df-mnf 8364 df-xr 8365 df-ltxr 8366 df-le 8367 df-sub 8501 df-neg 8502 df-reap 8906 df-ap 8913 df-div 9006 df-inn 9308 df-2 9366 df-3 9367 df-4 9368 df-n0 9569 df-z 9650 df-uz 9932 df-q 10030 df-rp 10066 df-xneg 10185 df-xadd 10186 df-ioo 10305 df-ico 10307 df-icc 10308 df-fz 10423 df-fzo 10561 df-fl 10716 df-mod 10775 df-seqfrec 10900 df-exp 10991 df-fac 11180 df-bc 11202 df-ihash 11231 df-shft 11596 df-cj 11623 df-re 11624 df-im 11625 df-rsqrt 11780 df-abs 11781 df-clim 12064 df-sumdc 12139 df-ef 12434 df-e 12435 df-dvds 12574 df-prm 12905 df-rest 13648 df-topgen 13667 df-psmet 14964 df-xmet 14965 df-met 14966 df-bl 14967 df-mopn 14968 df-top 15190 df-topon 15203 df-bases 15235 df-ntr 15288 df-cn 15380 df-cnp 15381 df-tx 15445 df-cncf 15763 df-limced 15848 df-dvap 15849 df-relog 16053 df-cht 16202 df-ppi 16203

This theorem is used by: (None)

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