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Mirrors > Home > MPE Home > Th. List > chpdifbndlem2 | Structured version Visualization version GIF version |
Description: Lemma for chpdifbnd 25656. (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 12115 | . . . . . . . 8 ⊢ 1 ∈ ℝ+ | |
5 | rpaddcl 12135 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝐴 + 1) ∈ ℝ+) | |
6 | 3, 4, 5 | sylancl 582 | . . . . . . 7 ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ+) |
7 | 2, 6 | rpmulcld 12171 | . . . . . 6 ⊢ (𝜑 → (𝐵 · (𝐴 + 1)) ∈ ℝ+) |
8 | 7 | rpred 12155 | . . . . 5 ⊢ (𝜑 → (𝐵 · (𝐴 + 1)) ∈ ℝ) |
9 | 2rp 12116 | . . . . . . . 8 ⊢ 2 ∈ ℝ+ | |
10 | rpmulcl 12136 | . . . . . . . 8 ⊢ ((2 ∈ ℝ+ ∧ 𝐴 ∈ ℝ+) → (2 · 𝐴) ∈ ℝ+) | |
11 | 9, 3, 10 | sylancr 583 | . . . . . . 7 ⊢ (𝜑 → (2 · 𝐴) ∈ ℝ+) |
12 | 11 | rpred 12155 | . . . . . 6 ⊢ (𝜑 → (2 · 𝐴) ∈ ℝ) |
13 | 3 | relogcld 24767 | . . . . . 6 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ) |
14 | 12, 13 | remulcld 10386 | . . . . 5 ⊢ (𝜑 → ((2 · 𝐴) · (log‘𝐴)) ∈ ℝ) |
15 | 8, 14 | readdcld 10385 | . . . 4 ⊢ (𝜑 → ((𝐵 · (𝐴 + 1)) + ((2 · 𝐴) · (log‘𝐴))) ∈ ℝ) |
16 | 00id 10529 | . . . . 5 ⊢ (0 + 0) = 0 | |
17 | 0red 10359 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ) | |
18 | 7 | rpgt0d 12158 | . . . . . 6 ⊢ (𝜑 → 0 < (𝐵 · (𝐴 + 1))) |
19 | 11 | rprege0d 12162 | . . . . . . 7 ⊢ (𝜑 → ((2 · 𝐴) ∈ ℝ ∧ 0 ≤ (2 · 𝐴))) |
20 | log1 24730 | . . . . . . . 8 ⊢ (log‘1) = 0 | |
21 | chpdifbnd.1 | . . . . . . . . 9 ⊢ (𝜑 → 1 ≤ 𝐴) | |
22 | logleb 24747 | . . . . . . . . . 10 ⊢ ((1 ∈ ℝ+ ∧ 𝐴 ∈ ℝ+) → (1 ≤ 𝐴 ↔ (log‘1) ≤ (log‘𝐴))) | |
23 | 4, 3, 22 | sylancr 583 | . . . . . . . . 9 ⊢ (𝜑 → (1 ≤ 𝐴 ↔ (log‘1) ≤ (log‘𝐴))) |
24 | 21, 23 | mpbid 224 | . . . . . . . 8 ⊢ (𝜑 → (log‘1) ≤ (log‘𝐴)) |
25 | 20, 24 | syl5eqbrr 4908 | . . . . . . 7 ⊢ (𝜑 → 0 ≤ (log‘𝐴)) |
26 | mulge0 10869 | . . . . . . 7 ⊢ ((((2 · 𝐴) ∈ ℝ ∧ 0 ≤ (2 · 𝐴)) ∧ ((log‘𝐴) ∈ ℝ ∧ 0 ≤ (log‘𝐴))) → 0 ≤ ((2 · 𝐴) · (log‘𝐴))) | |
27 | 19, 13, 25, 26 | syl12anc 872 | . . . . . 6 ⊢ (𝜑 → 0 ≤ ((2 · 𝐴) · (log‘𝐴))) |
28 | 17, 17, 8, 14, 18, 27 | ltleaddd 10972 | . . . . 5 ⊢ (𝜑 → (0 + 0) < ((𝐵 · (𝐴 + 1)) + ((2 · 𝐴) · (log‘𝐴)))) |
29 | 16, 28 | syl5eqbrr 4908 | . . . 4 ⊢ (𝜑 → 0 < ((𝐵 · (𝐴 + 1)) + ((2 · 𝐴) · (log‘𝐴)))) |
30 | 15, 29 | elrpd 12152 | . . 3 ⊢ (𝜑 → ((𝐵 · (𝐴 + 1)) + ((2 · 𝐴) · (log‘𝐴))) ∈ ℝ+) |
31 | 1, 30 | syl5eqel 2909 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
32 | 3 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 𝑦 ∈ (𝑥[,](𝐴 · 𝑥)))) → 𝐴 ∈ ℝ+) |
33 | 21 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 𝑦 ∈ (𝑥[,](𝐴 · 𝑥)))) → 1 ≤ 𝐴) |
34 | 2 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 𝑦 ∈ (𝑥[,](𝐴 · 𝑥)))) → 𝐵 ∈ ℝ+) |
35 | chpdifbnd.2 | . . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ (1[,)+∞)(abs‘(((((ψ‘𝑧) · (log‘𝑧)) + Σ𝑚 ∈ (1...(⌊‘𝑧))((Λ‘𝑚) · (ψ‘(𝑧 / 𝑚)))) / 𝑧) − (2 · (log‘𝑧)))) ≤ 𝐵) | |
36 | 35 | adantr 474 | . . . 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) |
37 | simprl 789 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 𝑦 ∈ (𝑥[,](𝐴 · 𝑥)))) → 𝑥 ∈ (1(,)+∞)) | |
38 | simprr 791 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 𝑦 ∈ (𝑥[,](𝐴 · 𝑥)))) → 𝑦 ∈ (𝑥[,](𝐴 · 𝑥))) | |
39 | 32, 33, 34, 36, 1, 37, 38 | chpdifbndlem1 25654 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 𝑦 ∈ (𝑥[,](𝐴 · 𝑥)))) → ((ψ‘𝑦) − (ψ‘𝑥)) ≤ ((2 · (𝑦 − 𝑥)) + (𝐶 · (𝑥 / (log‘𝑥))))) |
40 | 39 | ralrimivva 3179 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (1(,)+∞)∀𝑦 ∈ (𝑥[,](𝐴 · 𝑥))((ψ‘𝑦) − (ψ‘𝑥)) ≤ ((2 · (𝑦 − 𝑥)) + (𝐶 · (𝑥 / (log‘𝑥))))) |
41 | oveq1 6911 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐 · (𝑥 / (log‘𝑥))) = (𝐶 · (𝑥 / (log‘𝑥)))) | |
42 | 41 | oveq2d 6920 | . . . . 5 ⊢ (𝑐 = 𝐶 → ((2 · (𝑦 − 𝑥)) + (𝑐 · (𝑥 / (log‘𝑥)))) = ((2 · (𝑦 − 𝑥)) + (𝐶 · (𝑥 / (log‘𝑥))))) |
43 | 42 | breq2d 4884 | . . . 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)))))) |
44 | 43 | 2ralbidv 3197 | . . 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)))))) |
45 | 44 | rspcev 3525 | . 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))))) |
46 | 31, 40, 45 | syl2anc 581 | 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 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∀wral 3116 ∃wrex 3117 class class class wbr 4872 ‘cfv 6122 (class class class)co 6904 ℝcr 10250 0cc0 10251 1c1 10252 + caddc 10254 · cmul 10256 +∞cpnf 10387 < clt 10390 ≤ cle 10391 − cmin 10584 / cdiv 11008 2c2 11405 ℝ+crp 12111 (,)cioo 12462 [,)cico 12464 [,]cicc 12465 ...cfz 12618 ⌊cfl 12885 abscabs 14350 Σcsu 14792 logclog 24699 Λcvma 25230 ψcchp 25231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-inf2 8814 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-pre-sup 10329 ax-addf 10330 ax-mulf 10331 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-iin 4742 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-se 5301 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-isom 6131 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-of 7156 df-om 7326 df-1st 7427 df-2nd 7428 df-supp 7559 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-2o 7826 df-oadd 7829 df-er 8008 df-map 8123 df-pm 8124 df-ixp 8175 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-fsupp 8544 df-fi 8585 df-sup 8616 df-inf 8617 df-oi 8683 df-card 9077 df-cda 9304 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-3 11414 df-4 11415 df-5 11416 df-6 11417 df-7 11418 df-8 11419 df-9 11420 df-n0 11618 df-z 11704 df-dec 11821 df-uz 11968 df-q 12071 df-rp 12112 df-xneg 12231 df-xadd 12232 df-xmul 12233 df-ioo 12466 df-ioc 12467 df-ico 12468 df-icc 12469 df-fz 12619 df-fzo 12760 df-fl 12887 df-mod 12963 df-seq 13095 df-exp 13154 df-fac 13353 df-bc 13382 df-hash 13410 df-shft 14183 df-cj 14215 df-re 14216 df-im 14217 df-sqrt 14351 df-abs 14352 df-limsup 14578 df-clim 14595 df-rlim 14596 df-sum 14793 df-ef 15169 df-sin 15171 df-cos 15172 df-pi 15174 df-dvds 15357 df-gcd 15589 df-prm 15757 df-pc 15912 df-struct 16223 df-ndx 16224 df-slot 16225 df-base 16227 df-sets 16228 df-ress 16229 df-plusg 16317 df-mulr 16318 df-starv 16319 df-sca 16320 df-vsca 16321 df-ip 16322 df-tset 16323 df-ple 16324 df-ds 16326 df-unif 16327 df-hom 16328 df-cco 16329 df-rest 16435 df-topn 16436 df-0g 16454 df-gsum 16455 df-topgen 16456 df-pt 16457 df-prds 16460 df-xrs 16514 df-qtop 16519 df-imas 16520 df-xps 16522 df-mre 16598 df-mrc 16599 df-acs 16601 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-submnd 17688 df-mulg 17894 df-cntz 18099 df-cmn 18547 df-psmet 20097 df-xmet 20098 df-met 20099 df-bl 20100 df-mopn 20101 df-fbas 20102 df-fg 20103 df-cnfld 20106 df-top 21068 df-topon 21085 df-topsp 21107 df-bases 21120 df-cld 21193 df-ntr 21194 df-cls 21195 df-nei 21272 df-lp 21310 df-perf 21311 df-cn 21401 df-cnp 21402 df-haus 21489 df-tx 21735 df-hmeo 21928 df-fil 22019 df-fm 22111 df-flim 22112 df-flf 22113 df-xms 22494 df-ms 22495 df-tms 22496 df-cncf 23050 df-limc 24028 df-dv 24029 df-log 24701 df-vma 25236 df-chp 25237 |
This theorem is referenced by: chpdifbnd 25656 |
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