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Mirrors > Home > MPE Home > Th. List > pntsf | Structured version Visualization version GIF version |
Description: Functionality of the Selberg function. (Contributed by Mario Carneiro, 31-May-2016.) |
Ref | Expression |
---|---|
pntsval.1 | β’ π = (π β β β¦ Ξ£π β (1...(ββπ))((Ξβπ) Β· ((logβπ) + (Οβ(π / π))))) |
Ref | Expression |
---|---|
pntsf | β’ π:ββΆβ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pntsval.1 | . 2 β’ π = (π β β β¦ Ξ£π β (1...(ββπ))((Ξβπ) Β· ((logβπ) + (Οβ(π / π))))) | |
2 | fzfid 13945 | . . 3 β’ (π β β β (1...(ββπ)) β Fin) | |
3 | elfznn 13537 | . . . . . 6 β’ (π β (1...(ββπ)) β π β β) | |
4 | 3 | adantl 481 | . . . . 5 β’ ((π β β β§ π β (1...(ββπ))) β π β β) |
5 | vmacl 26963 | . . . . 5 β’ (π β β β (Ξβπ) β β) | |
6 | 4, 5 | syl 17 | . . . 4 β’ ((π β β β§ π β (1...(ββπ))) β (Ξβπ) β β) |
7 | 4 | nnrpd 13021 | . . . . . 6 β’ ((π β β β§ π β (1...(ββπ))) β π β β+) |
8 | 7 | relogcld 26471 | . . . . 5 β’ ((π β β β§ π β (1...(ββπ))) β (logβπ) β β) |
9 | simpl 482 | . . . . . . 7 β’ ((π β β β§ π β (1...(ββπ))) β π β β) | |
10 | 9, 4 | nndivred 12273 | . . . . . 6 β’ ((π β β β§ π β (1...(ββπ))) β (π / π) β β) |
11 | chpcl 26969 | . . . . . 6 β’ ((π / π) β β β (Οβ(π / π)) β β) | |
12 | 10, 11 | syl 17 | . . . . 5 β’ ((π β β β§ π β (1...(ββπ))) β (Οβ(π / π)) β β) |
13 | 8, 12 | readdcld 11250 | . . . 4 β’ ((π β β β§ π β (1...(ββπ))) β ((logβπ) + (Οβ(π / π))) β β) |
14 | 6, 13 | remulcld 11251 | . . 3 β’ ((π β β β§ π β (1...(ββπ))) β ((Ξβπ) Β· ((logβπ) + (Οβ(π / π)))) β β) |
15 | 2, 14 | fsumrecl 15687 | . 2 β’ (π β β β Ξ£π β (1...(ββπ))((Ξβπ) Β· ((logβπ) + (Οβ(π / π)))) β β) |
16 | 1, 15 | fmpti 7113 | 1 β’ π:ββΆβ |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 395 = wceq 1540 β wcel 2105 β¦ cmpt 5231 βΆwf 6539 βcfv 6543 (class class class)co 7412 βcr 11115 1c1 11117 + caddc 11119 Β· cmul 11121 / cdiv 11878 βcn 12219 ...cfz 13491 βcfl 13762 Ξ£csu 15639 logclog 26403 Ξcvma 26937 Οcchp 26938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-oadd 8476 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-fi 9412 df-sup 9443 df-inf 9444 df-oi 9511 df-dju 9902 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ioo 13335 df-ioc 13336 df-ico 13337 df-icc 13338 df-fz 13492 df-fzo 13635 df-fl 13764 df-mod 13842 df-seq 13974 df-exp 14035 df-fac 14241 df-bc 14270 df-hash 14298 df-shft 15021 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-limsup 15422 df-clim 15439 df-rlim 15440 df-sum 15640 df-ef 16018 df-sin 16020 df-cos 16021 df-pi 16023 df-dvds 16205 df-gcd 16443 df-prm 16616 df-pc 16777 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-mulg 18994 df-cntz 19229 df-cmn 19698 df-psmet 21225 df-xmet 21226 df-met 21227 df-bl 21228 df-mopn 21229 df-fbas 21230 df-fg 21231 df-cnfld 21234 df-top 22716 df-topon 22733 df-topsp 22755 df-bases 22769 df-cld 22843 df-ntr 22844 df-cls 22845 df-nei 22922 df-lp 22960 df-perf 22961 df-cn 23051 df-cnp 23052 df-haus 23139 df-tx 23386 df-hmeo 23579 df-fil 23670 df-fm 23762 df-flim 23763 df-flf 23764 df-xms 24146 df-ms 24147 df-tms 24148 df-cncf 24718 df-limc 25715 df-dv 25716 df-log 26405 df-vma 26943 df-chp 26944 |
This theorem is referenced by: pntrlog2bndlem1 27423 pntrlog2bndlem3 27425 pntrlog2bndlem4 27426 |
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