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Mirrors > Home > MPE Home > Th. List > dchr1 | Structured version Visualization version GIF version |
Description: Value of the principal Dirichlet character. (Contributed by Mario Carneiro, 28-Apr-2016.) |
Ref | Expression |
---|---|
dchr1.g | β’ πΊ = (DChrβπ) |
dchr1.z | β’ π = (β€/nβ€βπ) |
dchr1.o | β’ 1 = (0gβπΊ) |
dchr1.u | β’ π = (Unitβπ) |
dchr1.n | β’ (π β π β β) |
dchr1.a | β’ (π β π΄ β π) |
Ref | Expression |
---|---|
dchr1 | β’ (π β ( 1 βπ΄) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchr1.g | . . . 4 β’ πΊ = (DChrβπ) | |
2 | dchr1.z | . . . 4 β’ π = (β€/nβ€βπ) | |
3 | eqid 2724 | . . . 4 β’ (BaseβπΊ) = (BaseβπΊ) | |
4 | eqid 2724 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
5 | dchr1.u | . . . 4 β’ π = (Unitβπ) | |
6 | eqid 2724 | . . . 4 β’ (π β (Baseβπ) β¦ if(π β π, 1, 0)) = (π β (Baseβπ) β¦ if(π β π, 1, 0)) | |
7 | dchr1.n | . . . 4 β’ (π β π β β) | |
8 | 1, 2, 3, 4, 5, 6, 7 | dchr1cl 27124 | . . 3 β’ (π β (π β (Baseβπ) β¦ if(π β π, 1, 0)) β (BaseβπΊ)) |
9 | eleq1w 2808 | . . . . . 6 β’ (π = π₯ β (π β π β π₯ β π)) | |
10 | 9 | ifbid 4544 | . . . . 5 β’ (π = π₯ β if(π β π, 1, 0) = if(π₯ β π, 1, 0)) |
11 | 10 | cbvmptv 5252 | . . . 4 β’ (π β (Baseβπ) β¦ if(π β π, 1, 0)) = (π₯ β (Baseβπ) β¦ if(π₯ β π, 1, 0)) |
12 | eqid 2724 | . . . 4 β’ (+gβπΊ) = (+gβπΊ) | |
13 | 1, 2, 3, 4, 5, 11, 12, 8 | dchrmullid 27125 | . . 3 β’ (π β ((π β (Baseβπ) β¦ if(π β π, 1, 0))(+gβπΊ)(π β (Baseβπ) β¦ if(π β π, 1, 0))) = (π β (Baseβπ) β¦ if(π β π, 1, 0))) |
14 | 1 | dchrabl 27127 | . . . 4 β’ (π β β β πΊ β Abel) |
15 | ablgrp 19701 | . . . 4 β’ (πΊ β Abel β πΊ β Grp) | |
16 | dchr1.o | . . . . 5 β’ 1 = (0gβπΊ) | |
17 | 3, 12, 16 | isgrpid2 18902 | . . . 4 β’ (πΊ β Grp β (((π β (Baseβπ) β¦ if(π β π, 1, 0)) β (BaseβπΊ) β§ ((π β (Baseβπ) β¦ if(π β π, 1, 0))(+gβπΊ)(π β (Baseβπ) β¦ if(π β π, 1, 0))) = (π β (Baseβπ) β¦ if(π β π, 1, 0))) β 1 = (π β (Baseβπ) β¦ if(π β π, 1, 0)))) |
18 | 7, 14, 15, 17 | 4syl 19 | . . 3 β’ (π β (((π β (Baseβπ) β¦ if(π β π, 1, 0)) β (BaseβπΊ) β§ ((π β (Baseβπ) β¦ if(π β π, 1, 0))(+gβπΊ)(π β (Baseβπ) β¦ if(π β π, 1, 0))) = (π β (Baseβπ) β¦ if(π β π, 1, 0))) β 1 = (π β (Baseβπ) β¦ if(π β π, 1, 0)))) |
19 | 8, 13, 18 | mpbi2and 709 | . 2 β’ (π β 1 = (π β (Baseβπ) β¦ if(π β π, 1, 0))) |
20 | simpr 484 | . . . 4 β’ ((π β§ π = π΄) β π = π΄) | |
21 | dchr1.a | . . . . 5 β’ (π β π΄ β π) | |
22 | 21 | adantr 480 | . . . 4 β’ ((π β§ π = π΄) β π΄ β π) |
23 | 20, 22 | eqeltrd 2825 | . . 3 β’ ((π β§ π = π΄) β π β π) |
24 | 23 | iftrued 4529 | . 2 β’ ((π β§ π = π΄) β if(π β π, 1, 0) = 1) |
25 | 4, 5 | unitss 20274 | . . 3 β’ π β (Baseβπ) |
26 | 25, 21 | sselid 3973 | . 2 β’ (π β π΄ β (Baseβπ)) |
27 | 1cnd 11208 | . 2 β’ (π β 1 β β) | |
28 | 19, 24, 26, 27 | fvmptd 6996 | 1 β’ (π β ( 1 βπ΄) = 1) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 ifcif 4521 β¦ cmpt 5222 βcfv 6534 (class class class)co 7402 βcc 11105 0cc0 11107 1c1 11108 βcn 12211 Basecbs 17149 +gcplusg 17202 0gc0g 17390 Grpcgrp 18859 Abelcabl 19697 Unitcui 20253 β€/nβ€czn 21378 DChrcdchr 27105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-addf 11186 ax-mulf 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-ec 8702 df-qs 8706 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-0g 17392 df-imas 17459 df-qus 17460 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18709 df-grp 18862 df-minusg 18863 df-sbg 18864 df-subg 19046 df-nsg 19047 df-eqg 19048 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-subrng 20442 df-subrg 20467 df-lmod 20704 df-lss 20775 df-lsp 20815 df-sra 21017 df-rgmod 21018 df-lidl 21063 df-rsp 21064 df-2idl 21103 df-cnfld 21235 df-zring 21323 df-zn 21382 df-dchr 27106 |
This theorem is referenced by: dchrinv 27134 dchr1re 27136 dchrsum2 27141 rpvmasumlem 27360 rpvmasum2 27385 |
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