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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 7366 | . . . 4 β’ (π₯ = π΄ β (0[,]π₯) = (0[,]π΄)) | |
2 | 1 | ineq1d 4172 | . . 3 β’ (π₯ = π΄ β ((0[,]π₯) β© β) = ((0[,]π΄) β© β)) |
3 | 2 | sumeq1d 15591 | . 2 β’ (π₯ = π΄ β Ξ£π β ((0[,]π₯) β© β)(logβπ) = Ξ£π β ((0[,]π΄) β© β)(logβπ)) |
4 | df-cht 26462 | . 2 β’ ΞΈ = (π₯ β β β¦ Ξ£π β ((0[,]π₯) β© β)(logβπ)) | |
5 | sumex 15578 | . 2 β’ Ξ£π β ((0[,]π΄) β© β)(logβπ) β V | |
6 | 3, 4, 5 | fvmpt 6949 | 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 3910 βcfv 6497 (class class class)co 7358 βcr 11055 0cc0 11056 [,]cicc 13273 Ξ£csu 15576 βcprime 16552 logclog 25926 ΞΈccht 26456 |
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 5257 ax-nul 5264 ax-pr 5385 |
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 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-iota 6449 df-fun 6499 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-seq 13913 df-sum 15577 df-cht 26462 |
This theorem is referenced by: efchtcl 26476 chtge0 26477 chtfl 26514 chtprm 26518 chtnprm 26519 chtwordi 26521 chtdif 26523 cht1 26530 prmorcht 26543 chtlepsi 26570 chtleppi 26574 chpchtsum 26583 chpub 26584 chtppilimlem1 26837 chtvalz 33299 |
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