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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 2728 | . . . 4 β’ (BaseβπΊ) = (BaseβπΊ) | |
4 | eqid 2728 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
5 | dchr1.u | . . . 4 β’ π = (Unitβπ) | |
6 | eqid 2728 | . . . 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 27183 | . . 3 β’ (π β (π β (Baseβπ) β¦ if(π β π, 1, 0)) β (BaseβπΊ)) |
9 | eleq1w 2812 | . . . . . 6 β’ (π = π₯ β (π β π β π₯ β π)) | |
10 | 9 | ifbid 4552 | . . . . 5 β’ (π = π₯ β if(π β π, 1, 0) = if(π₯ β π, 1, 0)) |
11 | 10 | cbvmptv 5261 | . . . 4 β’ (π β (Baseβπ) β¦ if(π β π, 1, 0)) = (π₯ β (Baseβπ) β¦ if(π₯ β π, 1, 0)) |
12 | eqid 2728 | . . . 4 β’ (+gβπΊ) = (+gβπΊ) | |
13 | 1, 2, 3, 4, 5, 11, 12, 8 | dchrmullid 27184 | . . 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 27186 | . . . 4 β’ (π β β β πΊ β Abel) |
15 | ablgrp 19739 | . . . 4 β’ (πΊ β Abel β πΊ β Grp) | |
16 | dchr1.o | . . . . 5 β’ 1 = (0gβπΊ) | |
17 | 3, 12, 16 | isgrpid2 18932 | . . . 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 711 | . 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 2829 | . . 3 β’ ((π β§ π = π΄) β π β π) |
24 | 23 | iftrued 4537 | . 2 β’ ((π β§ π = π΄) β if(π β π, 1, 0) = 1) |
25 | 4, 5 | unitss 20314 | . . 3 β’ π β (Baseβπ) |
26 | 25, 21 | sselid 3978 | . 2 β’ (π β π΄ β (Baseβπ)) |
27 | 1cnd 11239 | . 2 β’ (π β 1 β β) | |
28 | 19, 24, 26, 27 | fvmptd 7012 | 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 1534 β wcel 2099 ifcif 4529 β¦ cmpt 5231 βcfv 6548 (class class class)co 7420 βcc 11136 0cc0 11138 1c1 11139 βcn 12242 Basecbs 17179 +gcplusg 17232 0gc0g 17420 Grpcgrp 18889 Abelcabl 19735 Unitcui 20293 β€/nβ€czn 21427 DChrcdchr 27164 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-addf 11217 ax-mulf 11218 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8231 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-ec 8726 df-qs 8730 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-0g 17422 df-imas 17489 df-qus 17490 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-grp 18892 df-minusg 18893 df-sbg 18894 df-subg 19077 df-nsg 19078 df-eqg 19079 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-oppr 20272 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-subrng 20482 df-subrg 20507 df-lmod 20744 df-lss 20815 df-lsp 20855 df-sra 21057 df-rgmod 21058 df-lidl 21103 df-rsp 21104 df-2idl 21143 df-cnfld 21279 df-zring 21372 df-zn 21431 df-dchr 27165 |
This theorem is referenced by: dchrinv 27193 dchr1re 27195 dchrsum2 27200 rpvmasumlem 27419 rpvmasum2 27444 |
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