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Mirrors > Home > MPE Home > Th. List > chpdifbndlem2 | Structured version Visualization version GIF version |
Description: Lemma for chpdifbnd 26714. (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 12745 | . . . . . . . 8 ⊢ 1 ∈ ℝ+ | |
5 | rpaddcl 12763 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝐴 + 1) ∈ ℝ+) | |
6 | 3, 4, 5 | sylancl 586 | . . . . . . 7 ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ+) |
7 | 2, 6 | rpmulcld 12799 | . . . . . 6 ⊢ (𝜑 → (𝐵 · (𝐴 + 1)) ∈ ℝ+) |
8 | 7 | rpred 12783 | . . . . 5 ⊢ (𝜑 → (𝐵 · (𝐴 + 1)) ∈ ℝ) |
9 | 2rp 12746 | . . . . . . . 8 ⊢ 2 ∈ ℝ+ | |
10 | rpmulcl 12764 | . . . . . . . 8 ⊢ ((2 ∈ ℝ+ ∧ 𝐴 ∈ ℝ+) → (2 · 𝐴) ∈ ℝ+) | |
11 | 9, 3, 10 | sylancr 587 | . . . . . . 7 ⊢ (𝜑 → (2 · 𝐴) ∈ ℝ+) |
12 | 11 | rpred 12783 | . . . . . 6 ⊢ (𝜑 → (2 · 𝐴) ∈ ℝ) |
13 | 3 | relogcld 25789 | . . . . . 6 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ) |
14 | 12, 13 | remulcld 11016 | . . . . 5 ⊢ (𝜑 → ((2 · 𝐴) · (log‘𝐴)) ∈ ℝ) |
15 | 8, 14 | readdcld 11015 | . . . 4 ⊢ (𝜑 → ((𝐵 · (𝐴 + 1)) + ((2 · 𝐴) · (log‘𝐴))) ∈ ℝ) |
16 | 7 | rpgt0d 12786 | . . . . 5 ⊢ (𝜑 → 0 < (𝐵 · (𝐴 + 1))) |
17 | 11 | rprege0d 12790 | . . . . . 6 ⊢ (𝜑 → ((2 · 𝐴) ∈ ℝ ∧ 0 ≤ (2 · 𝐴))) |
18 | log1 25752 | . . . . . . 7 ⊢ (log‘1) = 0 | |
19 | chpdifbnd.1 | . . . . . . . 8 ⊢ (𝜑 → 1 ≤ 𝐴) | |
20 | logleb 25769 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ+ ∧ 𝐴 ∈ ℝ+) → (1 ≤ 𝐴 ↔ (log‘1) ≤ (log‘𝐴))) | |
21 | 4, 3, 20 | sylancr 587 | . . . . . . . 8 ⊢ (𝜑 → (1 ≤ 𝐴 ↔ (log‘1) ≤ (log‘𝐴))) |
22 | 19, 21 | mpbid 231 | . . . . . . 7 ⊢ (𝜑 → (log‘1) ≤ (log‘𝐴)) |
23 | 18, 22 | eqbrtrrid 5115 | . . . . . 6 ⊢ (𝜑 → 0 ≤ (log‘𝐴)) |
24 | mulge0 11504 | . . . . . 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 11560 | . . . 4 ⊢ (𝜑 → 0 < ((𝐵 · (𝐴 + 1)) + ((2 · 𝐴) · (log‘𝐴)))) |
27 | 15, 26 | elrpd 12780 | . . 3 ⊢ (𝜑 → ((𝐵 · (𝐴 + 1)) + ((2 · 𝐴) · (log‘𝐴))) ∈ ℝ+) |
28 | 1, 27 | eqeltrid 2845 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
29 | 3 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 𝑦 ∈ (𝑥[,](𝐴 · 𝑥)))) → 𝐴 ∈ ℝ+) |
30 | 19 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 𝑦 ∈ (𝑥[,](𝐴 · 𝑥)))) → 1 ≤ 𝐴) |
31 | 2 | adantr 481 | . . . 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 481 | . . . 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 768 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 𝑦 ∈ (𝑥[,](𝐴 · 𝑥)))) → 𝑥 ∈ (1(,)+∞)) | |
35 | simprr 770 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 𝑦 ∈ (𝑥[,](𝐴 · 𝑥)))) → 𝑦 ∈ (𝑥[,](𝐴 · 𝑥))) | |
36 | 29, 30, 31, 33, 1, 34, 35 | chpdifbndlem1 26712 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 𝑦 ∈ (𝑥[,](𝐴 · 𝑥)))) → ((ψ‘𝑦) − (ψ‘𝑥)) ≤ ((2 · (𝑦 − 𝑥)) + (𝐶 · (𝑥 / (log‘𝑥))))) |
37 | 36 | ralrimivva 3117 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (1(,)+∞)∀𝑦 ∈ (𝑥[,](𝐴 · 𝑥))((ψ‘𝑦) − (ψ‘𝑥)) ≤ ((2 · (𝑦 − 𝑥)) + (𝐶 · (𝑥 / (log‘𝑥))))) |
38 | oveq1 7279 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐 · (𝑥 / (log‘𝑥))) = (𝐶 · (𝑥 / (log‘𝑥)))) | |
39 | 38 | oveq2d 7288 | . . . . 5 ⊢ (𝑐 = 𝐶 → ((2 · (𝑦 − 𝑥)) + (𝑐 · (𝑥 / (log‘𝑥)))) = ((2 · (𝑦 − 𝑥)) + (𝐶 · (𝑥 / (log‘𝑥))))) |
40 | 39 | breq2d 5091 | . . . 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 3125 | . . 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 3561 | . 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 584 | 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 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∀wral 3066 ∃wrex 3067 class class class wbr 5079 ‘cfv 6432 (class class class)co 7272 ℝcr 10881 0cc0 10882 1c1 10883 + caddc 10885 · cmul 10887 +∞cpnf 11017 ≤ cle 11021 − cmin 11216 / cdiv 11643 2c2 12039 ℝ+crp 12741 (,)cioo 13090 [,)cico 13092 [,]cicc 13093 ...cfz 13250 ⌊cfl 13521 abscabs 14956 Σcsu 15408 logclog 25721 Λcvma 26252 ψcchp 26253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-inf2 9387 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 ax-pre-sup 10960 ax-addf 10961 ax-mulf 10962 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-of 7528 df-om 7708 df-1st 7825 df-2nd 7826 df-supp 7970 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-2o 8290 df-oadd 8293 df-er 8490 df-map 8609 df-pm 8610 df-ixp 8678 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-fsupp 9117 df-fi 9158 df-sup 9189 df-inf 9190 df-oi 9257 df-dju 9670 df-card 9708 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-div 11644 df-nn 11985 df-2 12047 df-3 12048 df-4 12049 df-5 12050 df-6 12051 df-7 12052 df-8 12053 df-9 12054 df-n0 12245 df-z 12331 df-dec 12449 df-uz 12594 df-q 12700 df-rp 12742 df-xneg 12859 df-xadd 12860 df-xmul 12861 df-ioo 13094 df-ioc 13095 df-ico 13096 df-icc 13097 df-fz 13251 df-fzo 13394 df-fl 13523 df-mod 13601 df-seq 13733 df-exp 13794 df-fac 13999 df-bc 14028 df-hash 14056 df-shft 14789 df-cj 14821 df-re 14822 df-im 14823 df-sqrt 14957 df-abs 14958 df-limsup 15191 df-clim 15208 df-rlim 15209 df-sum 15409 df-ef 15788 df-sin 15790 df-cos 15791 df-pi 15793 df-dvds 15975 df-gcd 16213 df-prm 16388 df-pc 16549 df-struct 16859 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-ress 16953 df-plusg 16986 df-mulr 16987 df-starv 16988 df-sca 16989 df-vsca 16990 df-ip 16991 df-tset 16992 df-ple 16993 df-ds 16995 df-unif 16996 df-hom 16997 df-cco 16998 df-rest 17144 df-topn 17145 df-0g 17163 df-gsum 17164 df-topgen 17165 df-pt 17166 df-prds 17169 df-xrs 17224 df-qtop 17229 df-imas 17230 df-xps 17232 df-mre 17306 df-mrc 17307 df-acs 17309 df-mgm 18337 df-sgrp 18386 df-mnd 18397 df-submnd 18442 df-mulg 18712 df-cntz 18934 df-cmn 19399 df-psmet 20600 df-xmet 20601 df-met 20602 df-bl 20603 df-mopn 20604 df-fbas 20605 df-fg 20606 df-cnfld 20609 df-top 22054 df-topon 22071 df-topsp 22093 df-bases 22107 df-cld 22181 df-ntr 22182 df-cls 22183 df-nei 22260 df-lp 22298 df-perf 22299 df-cn 22389 df-cnp 22390 df-haus 22477 df-tx 22724 df-hmeo 22917 df-fil 23008 df-fm 23100 df-flim 23101 df-flf 23102 df-xms 23484 df-ms 23485 df-tms 23486 df-cncf 24052 df-limc 25041 df-dv 25042 df-log 25723 df-vma 26258 df-chp 26259 |
This theorem is referenced by: chpdifbnd 26714 |
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