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| Mirrors > Home > MPE Home > Th. List > selberg2b | Structured version Visualization version Unicode version | ||
Description: Convert eventual
boundedness in selberg2 25240 to boundedness on any
interval      .
(We have to bound away from zero because the
log terms diverge at zero.) (Contributed by Mario Carneiro,
25-May-2016.) |
| Ref | Expression |
|---|---|
| selberg2b |
            ψ        Λ ψ
    |
,,
is distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 1re 10039 |
. . . . . . 7
 | |
| 2 | elicopnf 12269 |
. . . . . . 7
 
![/\](wedge.gif "/\"){kind=small} | |
| 3 | 1, 2 | mp1i 13 |
. . . . . 6
 |
| 4 | 3 | simprbda 653 |
. . . . 5
|
| 5 | 4 | ex 450 |
. . . 4
|
| 6 | 5 | ssrdv 3609 |
. . 3
 |
| 7 | 1 | a1i 11 |
. . 3
|
| 8 | chpcl 24850 |
. . . . . . . . 9
ψ | |
| 9 | 4, 8 | syl 17 |
. . . . . . . 8
ψ |
| 10 | 1rp 11836 |
. . . . . . . . . . 11
| |
| 11 | 10 | a1i 11 |
. . . . . . . . . 10
|
| 12 | 3 | simplbda 654 |
. . . . . . . . . 10
|
| 13 | 4, 11, 12 | rpgecld 11911 |
. . . . . . . . 9
|
| 14 | 13 | relogcld 24369 |
. . . . . . . 8
|
| 15 | 9, 14 | remulcld 10070 |
. . . . . . 7
ψ |
| 16 | fzfid 12772 |
. . . . . . . 8
 | |
| 17 | elfznn 12370 |
. . . . . . . . . . 11
 | |
| 18 | 17 | adantl 482 |
. . . . . . . . . 10
|
| 19 | vmacl 24844 |
. . . . . . . . . 10
Λ | |
| 20 | 18, 19 | syl 17 |
. . . . . . . . 9
Λ |
| 21 | 4 | adantr 481 |
. . . . . . . . . . 11
|
| 22 | 21, 18 | nndivred 11069 |
. . . . . . . . . 10
|
| 23 | chpcl 24850 |
. . . . . . . . . 10
ψ | |
| 24 | 22, 23 | syl 17 |
. . . . . . . . 9
ψ |
| 25 | 20, 24 | remulcld 10070 |
. . . . . . . 8
Λ ψ |
| 26 | 16, 25 | fsumrecl 14465 |
. . . . . . 7
Λ ψ
|
| 27 | 15, 26 | readdcld 10069 |
. . . . . 6
ψ
Λ ψ |
| 28 | 27, 13 | rerpdivcld 11903 |
. . . . 5
ψ Λ ψ
|
| 29 | 2re 11090 |
. . . . . . 7
| |
| 30 | 29 | a1i 11 |
. . . . . 6
|
| 31 | 30, 14 | remulcld 10070 |
. . . . 5
|
| 32 | 28, 31 | resubcld 10458 |
. . . 4
ψ Λ ψ
|
| 33 | 32 | recnd 10068 |
. . 3
ψ Λ ψ
 |
| 34 | 13 | ex 450 |
. . . . 5
|
| 35 | 34 | ssrdv 3609 |
. . . 4
|
| 36 | selberg2 25240 |
. . . . 5
 ψ Λ ψ
 | |
| 37 | 36 | a1i 11 |
. . . 4
ψ Λ ψ |
| 38 | 35, 37 | o1res2 14294 |
. . 3
ψ Λ ψ |
| 39 | chpcl 24850 |
. . . . . . 7
ψ | |
| 40 | 39 | ad2antrl 764 |
. . . . . 6
ψ |
| 41 | simprl 794 |
. . . . . . . 8
| |
| 42 | 10 | a1i 11 |
. . . . . . . 8
|
| 43 | simprr 796 |
. . . . . . . 8
| |
| 44 | 41, 42, 43 | rpgecld 11911 |
. . . . . . 7
|
| 45 | 44 | relogcld 24369 |
. . . . . 6
|
| 46 | 40, 45 | remulcld 10070 |
. . . . 5
ψ |
| 47 | fzfid 12772 |
. . . . . 6
| |
| 48 | elfznn 12370 |
. . . . . . . . 9
| |
| 49 | 48 | adantl 482 |
. . . . . . . 8
|
| 50 | 49, 19 | syl 17 |
. . . . . . 7
Λ |
| 51 | 41 | adantr 481 |
. . . . . . . . 9
|
| 52 | 51, 49 | nndivred 11069 |
. . . . . . . 8
|
| 53 | chpcl 24850 |
. . . . . . . 8
ψ | |
| 54 | 52, 53 | syl 17 |
. . . . . . 7
ψ |
| 55 | 50, 54 | remulcld 10070 |
. . . . . 6
Λ ψ |
| 56 | 47, 55 | fsumrecl 14465 |
. . . . 5
Λ ψ |
| 57 | 46, 56 | readdcld 10069 |
. . . 4
ψ Λ ψ |
| 58 | 29 | a1i 11 |
. . . . 5
|
| 59 | 58, 45 | remulcld 10070 |
. . . 4
|
| 60 | 57, 59 | readdcld 10069 |
. . 3
ψ Λ ψ |
| 61 | 32 | adantr 481 |
. . . . . 6
ψ Λ ψ
|
| 62 | 61 | recnd 10068 |
. . . . 5
ψ Λ ψ
|
| 63 | 62 | abscld 14175 |
. . . 4
ψ Λ ψ |
| 64 | 28 | adantr 481 |
. . . . . . 7
ψ
Λ ψ |
| 65 | 64 | recnd 10068 |
. . . . . 6
ψ
Λ ψ |
| 66 | 65 | abscld 14175 |
. . . . 5
ψ Λ ψ
|
| 67 | 31 | adantr 481 |
. . . . . . 7
|
| 68 | 67 | recnd 10068 |
. . . . . 6
|
| 69 | 68 | abscld 14175 |
. . . . 5
|
| 70 | 66, 69 | readdcld 10069 |
. . . 4
ψ Λ ψ |
| 71 | 60 | ad2ant2r 783 |
. . . 4
ψ
Λ ψ |
| 72 | 65, 68 | abs2dif2d 14197 |
. . . 4
ψ Λ ψ ψ Λ ψ
|
| 73 | simprll 802 |
. . . . . . . 8
| |
| 74 | 73, 39 | syl 17 |
. . . . . . 7
ψ |
| 75 | 13 | adantr 481 |
. . . . . . . . 9
|
| 76 | 4 | adantr 481 |
. . . . . . . . . 10
|
| 77 | simprr 796 |
. . . . . . . . . 10
| |
| 78 | 76, 73, 77 | ltled 10185 |
. . . . . . . . 9
|
| 79 | 73, 75, 78 | rpgecld 11911 |
. . . . . . . 8
|
| 80 | 79 | relogcld 24369 |
. . . . . . 7
|
| 81 | 74, 80 | remulcld 10070 |
. . . . . 6
ψ |
| 82 | 56 | ad2ant2r 783 |
. . . . . 6
Λ ψ |
| 83 | 81, 82 | readdcld 10069 |
. . . . 5
ψ Λ ψ |
| 84 | 29 | a1i 11 |
. . . . . 6
|
| 85 | 84, 80 | remulcld 10070 |
. . . . 5
|
| 86 | 76, 8 | syl 17 |
. . . . . . . . . 10
ψ |
| 87 | 75 | relogcld 24369 |
. . . . . . . . . 10
|
| 88 | 86, 87 | remulcld 10070 |
. . . . . . . . 9
ψ |
| 89 | 26 | adantr 481 |
. . . . . . . . 9
Λ ψ
|
| 90 | 88, 89 | readdcld 10069 |
. . . . . . . 8
ψ Λ ψ |
| 91 | chpge0 24852 |
. . . . . . . . . . 11
 ψ | |
| 92 | 76, 91 | syl 17 |
. . . . . . . . . 10
ψ |
| 93 | 12 | adantr 481 |
. . . . . . . . . . 11
|
| 94 | 76, 93 | logge0d 24376 |
. . . . . . . . . 10
|
| 95 | 86, 87, 92, 94 | mulge0d 10604 |
. . . . . . . . 9
ψ |
| 96 | vmage0 24847 |
. . . . . . . . . . . . 13
Λ | |
| 97 | 18, 96 | syl 17 |
. . . . . . . . . . . 12
Λ |
| 98 | chpge0 24852 |
. . . . . . . . . . . . 13
ψ | |
| 99 | 22, 98 | syl 17 |
. . . . . . . . . . . 12
ψ |
| 100 | 20, 24, 97, 99 | mulge0d 10604 |
. . . . . . . . . . 11
Λ ψ |
| 101 | 16, 25, 100 | fsumge0 14527 |
. . . . . . . . . 10
Λ ψ |
| 102 | 101 | adantr 481 |
. . . . . . . . 9
Λ ψ |
| 103 | 88, 89, 95, 102 | addge0d 10603 |
. . . . . . . 8
ψ
Λ ψ |
| 104 | 90, 75, 103 | divge0d 11912 |
. . . . . . 7
ψ Λ ψ
|
| 105 | 64, 104 | absidd 14161 |
. . . . . 6
ψ Λ ψ
 ψ Λ ψ |
| 106 | 10 | a1i 11 |
. . . . . . . 8
|
| 107 | chpwordi 24883 |
. . . . . . . . . . 11
ψ ψ | |
| 108 | 76, 73, 78, 107 | syl3anc 1326 |
. . . . . . . . . 10
ψ
ψ |
| 109 | 75, 79 | logled 24373 |
. . . . . . . . . . 11
|
| 110 | 78, 109 | mpbid 222 |
. . . . . . . . . 10
|
| 111 | 86, 74, 87, 80, 92, 94, 108, 110 | lemul12ad 10966 |
. . . . . . . . 9
ψ ψ |
| 112 | fzfid 12772 |
. . . . . . . . . . 11
| |
| 113 | 48 | adantl 482 |
. . . . . . . . . . . . 13
|
| 114 | 113, 19 | syl 17 |
. . . . . . . . . . . 12
Λ |
| 115 | 76 | adantr 481 |
. . . . . . . . . . . . . 14
|
| 116 | 115, 113 | nndivred 11069 |
. . . . . . . . . . . . 13
|
| 117 | 116, 23 | syl 17 |
. . . . . . . . . . . 12
ψ |
| 118 | 114, 117 | remulcld 10070 |
. . . . . . . . . . 11
Λ ψ
|
| 119 | 112, 118 | fsumrecl 14465 |
. . . . . . . . . 10
Λ ψ
|
| 120 | 113, 96 | syl 17 |
. . . . . . . . . . . 12
Λ |
| 121 | 116, 98 | syl 17 |
. . . . . . . . . . . 12
ψ |
| 122 | 114, 117, 120, 121 | mulge0d 10604 |
. . . . . . . . . . 11
Λ ψ |
| 123 | flword2 12614 |
. . . . . . . . . . . . 13
 | |
| 124 | 76, 73, 78, 123 | syl3anc 1326 |
. . . . . . . . . . . 12
|
| 125 | fzss2 12381 |
. . . . . . . . . . . 12
| |
| 126 | 124, 125 | syl 17 |
. . . . . . . . . . 11
|
| 127 | 112, 118, 122, 126 | fsumless 14528 |
. . . . . . . . . 10
Λ ψ
Λ ψ |
| 128 | 73 | adantr 481 |
. . . . . . . . . . . . . 14
|
| 129 | 128, 113 | nndivred 11069 |
. . . . . . . . . . . . 13
|
| 130 | 129, 53 | syl 17 |
. . . . . . . . . . . 12
ψ |
| 131 | 114, 130 | remulcld 10070 |
. . . . . . . . . . 11
Λ ψ |
| 132 | 113 | nnrpd 11870 |
. . . . . . . . . . . . . 14
|
| 133 | 78 | adantr 481 |
. . . . . . . . . . . . . 14
|
| 134 | 115, 128, 132, 133 | lediv1dd 11930 |
. . . . . . . . . . . . 13
|
| 135 | chpwordi 24883 |
. . . . . . . . . . . . 13
ψ ψ | |
| 136 | 116, 129, 134, 135 | syl3anc 1326 |
. . . . . . . . . . . 12
ψ
ψ |
| 137 | 117, 130, 114, 120, 136 | lemul2ad 10964 |
. . . . . . . . . . 11
Λ ψ
Λ ψ |
| 138 | 112, 118, 131, 137 | fsumle 14531 |
. . . . . . . . . 10
Λ ψ
Λ ψ |
| 139 | 89, 119, 82, 127, 138 | letrd 10194 |
. . . . . . . . 9
Λ ψ
Λ ψ |
| 140 | 88, 89, 81, 82, 111, 139 | le2addd 10646 |
. . . . . . . 8
ψ Λ ψ ψ Λ ψ |
| 141 | 90, 83, 106, 76, 103, 140, 93 | lediv12ad 11931 |
. . . . . . 7
ψ
Λ ψ ψ Λ ψ |
| 142 | 83 | recnd 10068 |
. . . . . . . 8
ψ Λ ψ |
| 143 | 142 | div1d 10793 |
. . . . . . 7
ψ
Λ ψ ψ Λ ψ |
| 144 | 141, 143 | breqtrd 4679 |
. . . . . 6
ψ
Λ ψ ψ Λ ψ |
| 145 | 105, 144 | eqbrtrd 4675 |
. . . . 5
ψ Λ ψ
ψ Λ ψ |
| 146 | 2rp 11837 |
. . . . . . . . 9
| |
| 147 | rpge0 11845 |
. . . . . . . . 9
| |
| 148 | 146, 147 | mp1i 13 |
. . . . . . . 8
|
| 149 | 84, 87, 148, 94 | mulge0d 10604 |
. . . . . . 7
|
| 150 | 67, 149 | absidd 14161 |
. . . . . 6
|
| 151 | 87, 80, 84, 148, 110 | lemul2ad 10964 |
. . . . . 6
|
| 152 | 150, 151 | eqbrtrd 4675 |
. . . . 5
|
| 153 | 66, 69, 83, 85, 145, 152 | le2addd 10646 |
. . . 4
ψ Λ ψ
ψ
Λ ψ |
| 154 | 63, 70, 71, 72, 153 | letrd 10194 |
. . 3
ψ Λ ψ ψ Λ ψ |
| 155 | 6, 7, 33, 38, 60, 154 | o1bddrp 14273 |
. 2
ψ Λ ψ
|
| 156 | 155 | trud 1493 |
1
ψ Λ ψ
|
| Colors of variables: wff setvar class |
| Syntax hints:
wb 196
wa 384
wtru 1484
wcel 1990
wral 2912
wrex 2913
wss 3574
class class class wbr 4653 cmpt 4729 cfv 5888 (class class class)co 6650
cr 9935
cc0 9936
c1 9937
caddc 9939 cmul 9941 cpnf 10071 clt 10074 cle 10075 cmin 10266 cdiv 10684 cn 11020 c2 11070 cuz 11687 crp 11832 cico 12177 cfz 12326 cfl 12591 cabs 13974 co1 14217 csu 14416 clog 24301 Λcvma 24818
ψcchp 24819 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
| This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-o1 14221 df-lo1 14222 df-sum 14417 df-ef 14798 df-e 14799 df-sin 14800 df-cos 14801 df-pi 14803 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 df-cxp 24304 df-em 24719 df-cht 24823 df-vma 24824 df-chp 24825 df-ppi 24826 df-mu 24827 |
| This theorem is referenced by: chpdifbnd 25244 |
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