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Mirrors > Home > MPE Home > Th. List > dchr1re | Structured version Visualization version GIF version |
Description: The principal Dirichlet character is a real character. (Contributed by Mario Carneiro, 2-May-2016.) |
Ref | Expression |
---|---|
dchr1re.g | β’ πΊ = (DChrβπ) |
dchr1re.z | β’ π = (β€/nβ€βπ) |
dchr1re.o | β’ 1 = (0gβπΊ) |
dchr1re.b | β’ π΅ = (Baseβπ) |
dchr1re.n | β’ (π β π β β) |
Ref | Expression |
---|---|
dchr1re | β’ (π β 1 :π΅βΆβ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchr1re.g | . . . 4 β’ πΊ = (DChrβπ) | |
2 | dchr1re.z | . . . 4 β’ π = (β€/nβ€βπ) | |
3 | eqid 2733 | . . . 4 β’ (BaseβπΊ) = (BaseβπΊ) | |
4 | dchr1re.b | . . . 4 β’ π΅ = (Baseβπ) | |
5 | dchr1re.n | . . . . 5 β’ (π β π β β) | |
6 | 1 | dchrabl 26757 | . . . . 5 β’ (π β β β πΊ β Abel) |
7 | ablgrp 19653 | . . . . 5 β’ (πΊ β Abel β πΊ β Grp) | |
8 | dchr1re.o | . . . . . 6 β’ 1 = (0gβπΊ) | |
9 | 3, 8 | grpidcl 18850 | . . . . 5 β’ (πΊ β Grp β 1 β (BaseβπΊ)) |
10 | 5, 6, 7, 9 | 4syl 19 | . . . 4 β’ (π β 1 β (BaseβπΊ)) |
11 | 1, 2, 3, 4, 10 | dchrf 26745 | . . 3 β’ (π β 1 :π΅βΆβ) |
12 | 11 | ffnd 6719 | . 2 β’ (π β 1 Fn π΅) |
13 | simpr 486 | . . . . 5 β’ (((π β§ π₯ β π΅) β§ ( 1 βπ₯) = 0) β ( 1 βπ₯) = 0) | |
14 | 0re 11216 | . . . . 5 β’ 0 β β | |
15 | 13, 14 | eqeltrdi 2842 | . . . 4 β’ (((π β§ π₯ β π΅) β§ ( 1 βπ₯) = 0) β ( 1 βπ₯) β β) |
16 | eqid 2733 | . . . . . 6 β’ (Unitβπ) = (Unitβπ) | |
17 | 5 | ad2antrr 725 | . . . . . 6 β’ (((π β§ π₯ β π΅) β§ ( 1 βπ₯) β 0) β π β β) |
18 | 10 | adantr 482 | . . . . . . . 8 β’ ((π β§ π₯ β π΅) β 1 β (BaseβπΊ)) |
19 | simpr 486 | . . . . . . . 8 β’ ((π β§ π₯ β π΅) β π₯ β π΅) | |
20 | 1, 2, 3, 4, 16, 18, 19 | dchrn0 26753 | . . . . . . 7 β’ ((π β§ π₯ β π΅) β (( 1 βπ₯) β 0 β π₯ β (Unitβπ))) |
21 | 20 | biimpa 478 | . . . . . 6 β’ (((π β§ π₯ β π΅) β§ ( 1 βπ₯) β 0) β π₯ β (Unitβπ)) |
22 | 1, 2, 8, 16, 17, 21 | dchr1 26760 | . . . . 5 β’ (((π β§ π₯ β π΅) β§ ( 1 βπ₯) β 0) β ( 1 βπ₯) = 1) |
23 | 1re 11214 | . . . . 5 β’ 1 β β | |
24 | 22, 23 | eqeltrdi 2842 | . . . 4 β’ (((π β§ π₯ β π΅) β§ ( 1 βπ₯) β 0) β ( 1 βπ₯) β β) |
25 | 15, 24 | pm2.61dane 3030 | . . 3 β’ ((π β§ π₯ β π΅) β ( 1 βπ₯) β β) |
26 | 25 | ralrimiva 3147 | . 2 β’ (π β βπ₯ β π΅ ( 1 βπ₯) β β) |
27 | ffnfv 7118 | . 2 β’ ( 1 :π΅βΆβ β ( 1 Fn π΅ β§ βπ₯ β π΅ ( 1 βπ₯) β β)) | |
28 | 12, 26, 27 | sylanbrc 584 | 1 β’ (π β 1 :π΅βΆβ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2941 βwral 3062 Fn wfn 6539 βΆwf 6540 βcfv 6544 βcc 11108 βcr 11109 0cc0 11110 1c1 11111 βcn 12212 Basecbs 17144 0gc0g 17385 Grpcgrp 18819 Abelcabl 19649 Unitcui 20169 β€/nβ€czn 21052 DChrcdchr 26735 |
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-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-addf 11189 ax-mulf 11190 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 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-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-tpos 8211 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-ec 8705 df-qs 8709 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-fz 13485 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-0g 17387 df-imas 17454 df-qus 17455 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-grp 18822 df-minusg 18823 df-sbg 18824 df-subg 19003 df-nsg 19004 df-eqg 19005 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-ring 20058 df-cring 20059 df-oppr 20150 df-dvdsr 20171 df-unit 20172 df-invr 20202 df-subrg 20317 df-lmod 20473 df-lss 20543 df-lsp 20583 df-sra 20785 df-rgmod 20786 df-lidl 20787 df-rsp 20788 df-2idl 20857 df-cnfld 20945 df-zring 21018 df-zn 21056 df-dchr 26736 |
This theorem is referenced by: rpvmasumlem 26990 |
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