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Mirrors > Home > MPE Home > Th. List > dchrisum0fval | Structured version Visualization version GIF version |
Description: Value of the function πΉ, the divisor sum of a Dirichlet character. (Contributed by Mario Carneiro, 5-May-2016.) |
Ref | Expression |
---|---|
rpvmasum.z | β’ π = (β€/nβ€βπ) |
rpvmasum.l | β’ πΏ = (β€RHomβπ) |
rpvmasum.a | β’ (π β π β β) |
rpvmasum2.g | β’ πΊ = (DChrβπ) |
rpvmasum2.d | β’ π· = (BaseβπΊ) |
rpvmasum2.1 | β’ 1 = (0gβπΊ) |
dchrisum0f.f | β’ πΉ = (π β β β¦ Ξ£π£ β {π β β β£ π β₯ π} (πβ(πΏβπ£))) |
Ref | Expression |
---|---|
dchrisum0fval | β’ (π΄ β β β (πΉβπ΄) = Ξ£π‘ β {π β β β£ π β₯ π΄} (πβ(πΏβπ‘))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5114 | . . . . 5 β’ (π = π΄ β (π β₯ π β π β₯ π΄)) | |
2 | 1 | rabbidv 3418 | . . . 4 β’ (π = π΄ β {π β β β£ π β₯ π} = {π β β β£ π β₯ π΄}) |
3 | 2 | sumeq1d 15593 | . . 3 β’ (π = π΄ β Ξ£π£ β {π β β β£ π β₯ π} (πβ(πΏβπ£)) = Ξ£π£ β {π β β β£ π β₯ π΄} (πβ(πΏβπ£))) |
4 | 2fveq3 6852 | . . . 4 β’ (π£ = π‘ β (πβ(πΏβπ£)) = (πβ(πΏβπ‘))) | |
5 | 4 | cbvsumv 15588 | . . 3 β’ Ξ£π£ β {π β β β£ π β₯ π΄} (πβ(πΏβπ£)) = Ξ£π‘ β {π β β β£ π β₯ π΄} (πβ(πΏβπ‘)) |
6 | 3, 5 | eqtrdi 2793 | . 2 β’ (π = π΄ β Ξ£π£ β {π β β β£ π β₯ π} (πβ(πΏβπ£)) = Ξ£π‘ β {π β β β£ π β₯ π΄} (πβ(πΏβπ‘))) |
7 | dchrisum0f.f | . 2 β’ πΉ = (π β β β¦ Ξ£π£ β {π β β β£ π β₯ π} (πβ(πΏβπ£))) | |
8 | sumex 15579 | . 2 β’ Ξ£π‘ β {π β β β£ π β₯ π΄} (πβ(πΏβπ‘)) β V | |
9 | 6, 7, 8 | fvmpt 6953 | 1 β’ (π΄ β β β (πΉβπ΄) = Ξ£π‘ β {π β β β£ π β₯ π΄} (πβ(πΏβπ‘))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 {crab 3410 class class class wbr 5110 β¦ cmpt 5193 βcfv 6501 βcn 12160 Ξ£csu 15577 β₯ cdvds 16143 Basecbs 17090 0gc0g 17328 β€RHomczrh 20916 β€/nβ€czn 20919 DChrcdchr 26596 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-iota 6453 df-fun 6503 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-seq 13914 df-sum 15578 |
This theorem is referenced by: dchrisum0fmul 26870 dchrisum0flblem1 26872 dchrisum0 26884 |
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