Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > chpdifbndlem2 | Structured version Visualization version GIF version |
Description: Lemma for chpdifbnd 26134. (Contributed by Mario Carneiro, 25-May-2016.) |
Ref | Expression |
---|---|
chpdifbnd.a | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
chpdifbnd.1 | ⊢ (𝜑 → 1 ≤ 𝐴) |
chpdifbnd.b | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
chpdifbnd.2 | ⊢ (𝜑 → ∀𝑧 ∈ (1[,)+∞)(abs‘(((((ψ‘𝑧) · (log‘𝑧)) + Σ𝑚 ∈ (1...(⌊‘𝑧))((Λ‘𝑚) · (ψ‘(𝑧 / 𝑚)))) / 𝑧) − (2 · (log‘𝑧)))) ≤ 𝐵) |
chpdifbnd.c | ⊢ 𝐶 = ((𝐵 · (𝐴 + 1)) + ((2 · 𝐴) · (log‘𝐴))) |
Ref | Expression |
---|---|
chpdifbndlem2 | ⊢ (𝜑 → ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ (1(,)+∞)∀𝑦 ∈ (𝑥[,](𝐴 · 𝑥))((ψ‘𝑦) − (ψ‘𝑥)) ≤ ((2 · (𝑦 − 𝑥)) + (𝑐 · (𝑥 / (log‘𝑥))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chpdifbnd.c | . . 3 ⊢ 𝐶 = ((𝐵 · (𝐴 + 1)) + ((2 · 𝐴) · (log‘𝐴))) | |
2 | chpdifbnd.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
3 | chpdifbnd.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
4 | 1rp 12396 | . . . . . . . 8 ⊢ 1 ∈ ℝ+ | |
5 | rpaddcl 12414 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝐴 + 1) ∈ ℝ+) | |
6 | 3, 4, 5 | sylancl 588 | . . . . . . 7 ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ+) |
7 | 2, 6 | rpmulcld 12450 | . . . . . 6 ⊢ (𝜑 → (𝐵 · (𝐴 + 1)) ∈ ℝ+) |
8 | 7 | rpred 12434 | . . . . 5 ⊢ (𝜑 → (𝐵 · (𝐴 + 1)) ∈ ℝ) |
9 | 2rp 12397 | . . . . . . . 8 ⊢ 2 ∈ ℝ+ | |
10 | rpmulcl 12415 | . . . . . . . 8 ⊢ ((2 ∈ ℝ+ ∧ 𝐴 ∈ ℝ+) → (2 · 𝐴) ∈ ℝ+) | |
11 | 9, 3, 10 | sylancr 589 | . . . . . . 7 ⊢ (𝜑 → (2 · 𝐴) ∈ ℝ+) |
12 | 11 | rpred 12434 | . . . . . 6 ⊢ (𝜑 → (2 · 𝐴) ∈ ℝ) |
13 | 3 | relogcld 25209 | . . . . . 6 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ) |
14 | 12, 13 | remulcld 10674 | . . . . 5 ⊢ (𝜑 → ((2 · 𝐴) · (log‘𝐴)) ∈ ℝ) |
15 | 8, 14 | readdcld 10673 | . . . 4 ⊢ (𝜑 → ((𝐵 · (𝐴 + 1)) + ((2 · 𝐴) · (log‘𝐴))) ∈ ℝ) |
16 | 7 | rpgt0d 12437 | . . . . 5 ⊢ (𝜑 → 0 < (𝐵 · (𝐴 + 1))) |
17 | 11 | rprege0d 12441 | . . . . . 6 ⊢ (𝜑 → ((2 · 𝐴) ∈ ℝ ∧ 0 ≤ (2 · 𝐴))) |
18 | log1 25172 | . . . . . . 7 ⊢ (log‘1) = 0 | |
19 | chpdifbnd.1 | . . . . . . . 8 ⊢ (𝜑 → 1 ≤ 𝐴) | |
20 | logleb 25189 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ+ ∧ 𝐴 ∈ ℝ+) → (1 ≤ 𝐴 ↔ (log‘1) ≤ (log‘𝐴))) | |
21 | 4, 3, 20 | sylancr 589 | . . . . . . . 8 ⊢ (𝜑 → (1 ≤ 𝐴 ↔ (log‘1) ≤ (log‘𝐴))) |
22 | 19, 21 | mpbid 234 | . . . . . . 7 ⊢ (𝜑 → (log‘1) ≤ (log‘𝐴)) |
23 | 18, 22 | eqbrtrrid 5105 | . . . . . 6 ⊢ (𝜑 → 0 ≤ (log‘𝐴)) |
24 | mulge0 11161 | . . . . . 6 ⊢ ((((2 · 𝐴) ∈ ℝ ∧ 0 ≤ (2 · 𝐴)) ∧ ((log‘𝐴) ∈ ℝ ∧ 0 ≤ (log‘𝐴))) → 0 ≤ ((2 · 𝐴) · (log‘𝐴))) | |
25 | 17, 13, 23, 24 | syl12anc 834 | . . . . 5 ⊢ (𝜑 → 0 ≤ ((2 · 𝐴) · (log‘𝐴))) |
26 | 8, 14, 16, 25 | addgtge0d 11217 | . . . 4 ⊢ (𝜑 → 0 < ((𝐵 · (𝐴 + 1)) + ((2 · 𝐴) · (log‘𝐴)))) |
27 | 15, 26 | elrpd 12431 | . . 3 ⊢ (𝜑 → ((𝐵 · (𝐴 + 1)) + ((2 · 𝐴) · (log‘𝐴))) ∈ ℝ+) |
28 | 1, 27 | eqeltrid 2920 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
29 | 3 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 𝑦 ∈ (𝑥[,](𝐴 · 𝑥)))) → 𝐴 ∈ ℝ+) |
30 | 19 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 𝑦 ∈ (𝑥[,](𝐴 · 𝑥)))) → 1 ≤ 𝐴) |
31 | 2 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 𝑦 ∈ (𝑥[,](𝐴 · 𝑥)))) → 𝐵 ∈ ℝ+) |
32 | chpdifbnd.2 | . . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ (1[,)+∞)(abs‘(((((ψ‘𝑧) · (log‘𝑧)) + Σ𝑚 ∈ (1...(⌊‘𝑧))((Λ‘𝑚) · (ψ‘(𝑧 / 𝑚)))) / 𝑧) − (2 · (log‘𝑧)))) ≤ 𝐵) | |
33 | 32 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 𝑦 ∈ (𝑥[,](𝐴 · 𝑥)))) → ∀𝑧 ∈ (1[,)+∞)(abs‘(((((ψ‘𝑧) · (log‘𝑧)) + Σ𝑚 ∈ (1...(⌊‘𝑧))((Λ‘𝑚) · (ψ‘(𝑧 / 𝑚)))) / 𝑧) − (2 · (log‘𝑧)))) ≤ 𝐵) |
34 | simprl 769 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 𝑦 ∈ (𝑥[,](𝐴 · 𝑥)))) → 𝑥 ∈ (1(,)+∞)) | |
35 | simprr 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 𝑦 ∈ (𝑥[,](𝐴 · 𝑥)))) → 𝑦 ∈ (𝑥[,](𝐴 · 𝑥))) | |
36 | 29, 30, 31, 33, 1, 34, 35 | chpdifbndlem1 26132 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 𝑦 ∈ (𝑥[,](𝐴 · 𝑥)))) → ((ψ‘𝑦) − (ψ‘𝑥)) ≤ ((2 · (𝑦 − 𝑥)) + (𝐶 · (𝑥 / (log‘𝑥))))) |
37 | 36 | ralrimivva 3194 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (1(,)+∞)∀𝑦 ∈ (𝑥[,](𝐴 · 𝑥))((ψ‘𝑦) − (ψ‘𝑥)) ≤ ((2 · (𝑦 − 𝑥)) + (𝐶 · (𝑥 / (log‘𝑥))))) |
38 | oveq1 7166 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐 · (𝑥 / (log‘𝑥))) = (𝐶 · (𝑥 / (log‘𝑥)))) | |
39 | 38 | oveq2d 7175 | . . . . 5 ⊢ (𝑐 = 𝐶 → ((2 · (𝑦 − 𝑥)) + (𝑐 · (𝑥 / (log‘𝑥)))) = ((2 · (𝑦 − 𝑥)) + (𝐶 · (𝑥 / (log‘𝑥))))) |
40 | 39 | breq2d 5081 | . . . 4 ⊢ (𝑐 = 𝐶 → (((ψ‘𝑦) − (ψ‘𝑥)) ≤ ((2 · (𝑦 − 𝑥)) + (𝑐 · (𝑥 / (log‘𝑥)))) ↔ ((ψ‘𝑦) − (ψ‘𝑥)) ≤ ((2 · (𝑦 − 𝑥)) + (𝐶 · (𝑥 / (log‘𝑥)))))) |
41 | 40 | 2ralbidv 3202 | . . 3 ⊢ (𝑐 = 𝐶 → (∀𝑥 ∈ (1(,)+∞)∀𝑦 ∈ (𝑥[,](𝐴 · 𝑥))((ψ‘𝑦) − (ψ‘𝑥)) ≤ ((2 · (𝑦 − 𝑥)) + (𝑐 · (𝑥 / (log‘𝑥)))) ↔ ∀𝑥 ∈ (1(,)+∞)∀𝑦 ∈ (𝑥[,](𝐴 · 𝑥))((ψ‘𝑦) − (ψ‘𝑥)) ≤ ((2 · (𝑦 − 𝑥)) + (𝐶 · (𝑥 / (log‘𝑥)))))) |
42 | 41 | rspcev 3626 | . 2 ⊢ ((𝐶 ∈ ℝ+ ∧ ∀𝑥 ∈ (1(,)+∞)∀𝑦 ∈ (𝑥[,](𝐴 · 𝑥))((ψ‘𝑦) − (ψ‘𝑥)) ≤ ((2 · (𝑦 − 𝑥)) + (𝐶 · (𝑥 / (log‘𝑥))))) → ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ (1(,)+∞)∀𝑦 ∈ (𝑥[,](𝐴 · 𝑥))((ψ‘𝑦) − (ψ‘𝑥)) ≤ ((2 · (𝑦 − 𝑥)) + (𝑐 · (𝑥 / (log‘𝑥))))) |
43 | 28, 37, 42 | syl2anc 586 | 1 ⊢ (𝜑 → ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ (1(,)+∞)∀𝑦 ∈ (𝑥[,](𝐴 · 𝑥))((ψ‘𝑦) − (ψ‘𝑥)) ≤ ((2 · (𝑦 − 𝑥)) + (𝑐 · (𝑥 / (log‘𝑥))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ∃wrex 3142 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 ℝcr 10539 0cc0 10540 1c1 10541 + caddc 10543 · cmul 10545 +∞cpnf 10675 ≤ cle 10679 − cmin 10873 / cdiv 11300 2c2 11695 ℝ+crp 12392 (,)cioo 12741 [,)cico 12743 [,]cicc 12744 ...cfz 12895 ⌊cfl 13163 abscabs 14596 Σcsu 15045 logclog 25141 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-vma 25678 df-chp 25679 |
This theorem is referenced by: chpdifbnd 26134 |
Copyright terms: Public domain | W3C validator |