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Mirrors > Home > MPE Home > Th. List > chtlepsi | Structured version Visualization version GIF version |
Description: The first Chebyshev function is less than the second. (Contributed by Mario Carneiro, 7-Apr-2016.) |
Ref | Expression |
---|---|
chtlepsi | ⊢ (𝐴 ∈ ℝ → (θ‘𝐴) ≤ (ψ‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzfid 13344 | . . 3 ⊢ (𝐴 ∈ ℝ → (1...(⌊‘𝐴)) ∈ Fin) | |
2 | elfznn 12939 | . . . . 5 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ) | |
3 | 2 | adantl 484 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ) |
4 | vmacl 25698 | . . . 4 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (Λ‘𝑛) ∈ ℝ) |
6 | vmage0 25701 | . . . 4 ⊢ (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛)) | |
7 | 3, 6 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 0 ≤ (Λ‘𝑛)) |
8 | ppisval 25684 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) = ((2...(⌊‘𝐴)) ∩ ℙ)) | |
9 | inss1 4208 | . . . . 5 ⊢ ((2...(⌊‘𝐴)) ∩ ℙ) ⊆ (2...(⌊‘𝐴)) | |
10 | 2eluzge1 12297 | . . . . . 6 ⊢ 2 ∈ (ℤ≥‘1) | |
11 | fzss1 12949 | . . . . . 6 ⊢ (2 ∈ (ℤ≥‘1) → (2...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴))) | |
12 | 10, 11 | mp1i 13 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (2...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴))) |
13 | 9, 12 | sstrid 3981 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((2...(⌊‘𝐴)) ∩ ℙ) ⊆ (1...(⌊‘𝐴))) |
14 | 8, 13 | eqsstrd 4008 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ⊆ (1...(⌊‘𝐴))) |
15 | 1, 5, 7, 14 | fsumless 15154 | . 2 ⊢ (𝐴 ∈ ℝ → Σ𝑛 ∈ ((0[,]𝐴) ∩ ℙ)(Λ‘𝑛) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛)) |
16 | chtval 25690 | . . 3 ⊢ (𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑛 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑛)) | |
17 | simpr 487 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑛 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑛 ∈ ((0[,]𝐴) ∩ ℙ)) | |
18 | 17 | elin2d 4179 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑛 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑛 ∈ ℙ) |
19 | vmaprm 25697 | . . . . 5 ⊢ (𝑛 ∈ ℙ → (Λ‘𝑛) = (log‘𝑛)) | |
20 | 18, 19 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑛 ∈ ((0[,]𝐴) ∩ ℙ)) → (Λ‘𝑛) = (log‘𝑛)) |
21 | 20 | sumeq2dv 15063 | . . 3 ⊢ (𝐴 ∈ ℝ → Σ𝑛 ∈ ((0[,]𝐴) ∩ ℙ)(Λ‘𝑛) = Σ𝑛 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑛)) |
22 | 16, 21 | eqtr4d 2862 | . 2 ⊢ (𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑛 ∈ ((0[,]𝐴) ∩ ℙ)(Λ‘𝑛)) |
23 | chpval 25702 | . 2 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛)) | |
24 | 15, 22, 23 | 3brtr4d 5101 | 1 ⊢ (𝐴 ∈ ℝ → (θ‘𝐴) ≤ (ψ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∩ cin 3938 ⊆ wss 3939 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 ℝcr 10539 0cc0 10540 1c1 10541 ≤ cle 10679 ℕcn 11641 2c2 11695 ℤ≥cuz 12246 [,]cicc 12744 ...cfz 12895 ⌊cfl 13163 Σcsu 15045 ℙcprime 16018 logclog 25141 θccht 25671 Λcvma 25672 ψcchp 25673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-fi 8878 df-sup 8909 df-inf 8910 df-oi 8977 df-dju 9333 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14429 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-limsup 14831 df-clim 14848 df-rlim 14849 df-sum 15046 df-ef 15424 df-sin 15426 df-cos 15427 df-pi 15429 df-dvds 15611 df-gcd 15847 df-prm 16019 df-pc 16177 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-hom 16592 df-cco 16593 df-rest 16699 df-topn 16700 df-0g 16718 df-gsum 16719 df-topgen 16720 df-pt 16721 df-prds 16724 df-xrs 16778 df-qtop 16783 df-imas 16784 df-xps 16786 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-mulg 18228 df-cntz 18450 df-cmn 18911 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-fbas 20545 df-fg 20546 df-cnfld 20549 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-cld 21630 df-ntr 21631 df-cls 21632 df-nei 21709 df-lp 21747 df-perf 21748 df-cn 21838 df-cnp 21839 df-haus 21926 df-tx 22173 df-hmeo 22366 df-fil 22457 df-fm 22549 df-flim 22550 df-flf 22551 df-xms 22933 df-ms 22934 df-tms 22935 df-cncf 23489 df-limc 24467 df-dv 24468 df-log 25143 df-cht 25677 df-vma 25678 df-chp 25679 |
This theorem is referenced by: chprpcl 25786 chteq0 25788 chpchtlim 26058 |
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