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Mirrors > Home > MPE Home > Th. List > chtval | Structured version Visualization version GIF version |
Description: Value of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.) |
Ref | Expression |
---|---|
chtval | β’ (π΄ β β β (ΞΈβπ΄) = Ξ£π β ((0[,]π΄) β© β)(logβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7417 | . . . 4 β’ (π₯ = π΄ β (0[,]π₯) = (0[,]π΄)) | |
2 | 1 | ineq1d 4212 | . . 3 β’ (π₯ = π΄ β ((0[,]π₯) β© β) = ((0[,]π΄) β© β)) |
3 | 2 | sumeq1d 15647 | . 2 β’ (π₯ = π΄ β Ξ£π β ((0[,]π₯) β© β)(logβπ) = Ξ£π β ((0[,]π΄) β© β)(logβπ)) |
4 | df-cht 26601 | . 2 β’ ΞΈ = (π₯ β β β¦ Ξ£π β ((0[,]π₯) β© β)(logβπ)) | |
5 | sumex 15634 | . 2 β’ Ξ£π β ((0[,]π΄) β© β)(logβπ) β V | |
6 | 3, 4, 5 | fvmpt 6999 | 1 β’ (π΄ β β β (ΞΈβπ΄) = Ξ£π β ((0[,]π΄) β© β)(logβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β© cin 3948 βcfv 6544 (class class class)co 7409 βcr 11109 0cc0 11110 [,]cicc 13327 Ξ£csu 15632 βcprime 16608 logclog 26063 ΞΈccht 26595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-iota 6496 df-fun 6546 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-seq 13967 df-sum 15633 df-cht 26601 |
This theorem is referenced by: efchtcl 26615 chtge0 26616 chtfl 26653 chtprm 26657 chtnprm 26658 chtwordi 26660 chtdif 26662 cht1 26669 prmorcht 26682 chtlepsi 26709 chtleppi 26713 chpchtsum 26722 chpub 26723 chtppilimlem1 26976 chtvalz 33641 |
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