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Mirrors > Home > MPE Home > Th. List > dchrisum0 | Structured version Visualization version Unicode version |
Description: The sum is nonzero for all non-principal Dirichlet characters (i.e. the assumption is contradictory). This is the key result that allows us to eliminate the conditionals from dchrmusum2 25183 and dchrvmasumif 25192. Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-May-2016.) |
Ref | Expression |
---|---|
rpvmasum.z | ℤ/nℤ |
rpvmasum.l | RHom |
rpvmasum.a | |
rpvmasum2.g | DChr |
rpvmasum2.d | |
rpvmasum2.1 | |
rpvmasum2.w | |
dchrisum0.b |
Ref | Expression |
---|---|
dchrisum0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpvmasum.z | . 2 ℤ/nℤ | |
2 | rpvmasum.l | . 2 RHom | |
3 | rpvmasum.a | . 2 | |
4 | rpvmasum2.g | . 2 DChr | |
5 | rpvmasum2.d | . 2 | |
6 | rpvmasum2.1 | . 2 | |
7 | eqid 2622 | . 2 | |
8 | rpvmasum2.w | . . . . 5 | |
9 | ssrab2 3687 | . . . . 5 | |
10 | 8, 9 | eqsstri 3635 | . . . 4 |
11 | difss 3737 | . . . 4 | |
12 | 10, 11 | sstri 3612 | . . 3 |
13 | dchrisum0.b | . . 3 | |
14 | 12, 13 | sseldi 3601 | . 2 |
15 | 1, 2, 3, 4, 5, 6, 8, 13 | dchrisum0re 25202 | . 2 |
16 | fveq2 6191 | . . . . . . . 8 | |
17 | 16 | oveq2d 6666 | . . . . . . 7 |
18 | rpre 11839 | . . . . . . . 8 | |
19 | 18 | adantl 482 | . . . . . . 7 |
20 | 14 | ad3antrrr 766 | . . . . . . . . . 10 |
21 | elrabi 3359 | . . . . . . . . . . . 12 | |
22 | 21 | nnzd 11481 | . . . . . . . . . . 11 |
23 | 22 | adantl 482 | . . . . . . . . . 10 |
24 | 4, 1, 5, 2, 20, 23 | dchrzrhcl 24970 | . . . . . . . . 9 |
25 | elfznn 12370 | . . . . . . . . . . . . . 14 | |
26 | 25 | adantl 482 | . . . . . . . . . . . . 13 |
27 | 26 | nnrpd 11870 | . . . . . . . . . . . 12 |
28 | 27 | rpsqrtcld 14150 | . . . . . . . . . . 11 |
29 | 28 | rpcnd 11874 | . . . . . . . . . 10 |
30 | 29 | adantr 481 | . . . . . . . . 9 |
31 | 28 | rpne0d 11877 | . . . . . . . . . 10 |
32 | 31 | adantr 481 | . . . . . . . . 9 |
33 | 24, 30, 32 | divcld 10801 | . . . . . . . 8 |
34 | 33 | anasss 679 | . . . . . . 7 |
35 | 17, 19, 34 | dvdsflsumcom 24914 | . . . . . 6 |
36 | 1, 2, 3, 4, 5, 6, 7 | dchrisum0fval 25194 | . . . . . . . . . 10 |
37 | 26, 36 | syl 17 | . . . . . . . . 9 |
38 | 37 | oveq1d 6665 | . . . . . . . 8 |
39 | fzfid 12772 | . . . . . . . . . 10 | |
40 | dvdsssfz1 15040 | . . . . . . . . . . 11 | |
41 | 26, 40 | syl 17 | . . . . . . . . . 10 |
42 | ssfi 8180 | . . . . . . . . . 10 | |
43 | 39, 41, 42 | syl2anc 693 | . . . . . . . . 9 |
44 | 43, 29, 24, 31 | fsumdivc 14518 | . . . . . . . 8 |
45 | 38, 44 | eqtrd 2656 | . . . . . . 7 |
46 | 45 | sumeq2dv 14433 | . . . . . 6 |
47 | rprege0 11847 | . . . . . . . . . . 11 | |
48 | 47 | adantl 482 | . . . . . . . . . 10 |
49 | resqrtth 13996 | . . . . . . . . . 10 | |
50 | 48, 49 | syl 17 | . . . . . . . . 9 |
51 | 50 | fveq2d 6195 | . . . . . . . 8 |
52 | 51 | oveq2d 6666 | . . . . . . 7 |
53 | 50 | oveq1d 6665 | . . . . . . . . . . 11 |
54 | 53 | fveq2d 6195 | . . . . . . . . . 10 |
55 | 54 | oveq2d 6666 | . . . . . . . . 9 |
56 | 55 | sumeq1d 14431 | . . . . . . . 8 |
57 | 56 | adantr 481 | . . . . . . 7 |
58 | 52, 57 | sumeq12dv 14437 | . . . . . 6 |
59 | 35, 46, 58 | 3eqtr4d 2666 | . . . . 5 |
60 | 59 | mpteq2dva 4744 | . . . 4 |
61 | rpsqrtcl 14005 | . . . . . 6 | |
62 | 61 | adantl 482 | . . . . 5 |
63 | eqidd 2623 | . . . . 5 | |
64 | eqidd 2623 | . . . . 5 | |
65 | oveq1 6657 | . . . . . . . 8 | |
66 | 65 | fveq2d 6195 | . . . . . . 7 |
67 | 66 | oveq2d 6666 | . . . . . 6 |
68 | 65 | oveq1d 6665 | . . . . . . . . . 10 |
69 | 68 | fveq2d 6195 | . . . . . . . . 9 |
70 | 69 | oveq2d 6666 | . . . . . . . 8 |
71 | 70 | sumeq1d 14431 | . . . . . . 7 |
72 | 71 | adantr 481 | . . . . . 6 |
73 | 67, 72 | sumeq12dv 14437 | . . . . 5 |
74 | 62, 63, 64, 73 | fmptco 6396 | . . . 4 |
75 | 60, 74 | eqtr4d 2659 | . . 3 |
76 | eqid 2622 | . . . . . . . 8 | |
77 | 1, 2, 3, 4, 5, 6, 8, 13, 76 | dchrisum0lema 25203 | . . . . . . 7 |
78 | 3 | adantr 481 | . . . . . . . . . 10 |
79 | 13 | adantr 481 | . . . . . . . . . 10 |
80 | simprl 794 | . . . . . . . . . 10 | |
81 | simprrl 804 | . . . . . . . . . 10 | |
82 | simprrr 805 | . . . . . . . . . 10 | |
83 | 1, 2, 78, 4, 5, 6, 8, 79, 76, 80, 81, 82 | dchrisum0lem3 25208 | . . . . . . . . 9 |
84 | 83 | rexlimdvaa 3032 | . . . . . . . 8 |
85 | 84 | exlimdv 1861 | . . . . . . 7 |
86 | 77, 85 | mpd 15 | . . . . . 6 |
87 | o1f 14260 | . . . . . 6 | |
88 | 86, 87 | syl 17 | . . . . 5 |
89 | sumex 14418 | . . . . . . 7 | |
90 | eqid 2622 | . . . . . . 7 | |
91 | 89, 90 | dmmpti 6023 | . . . . . 6 |
92 | 91 | feq2i 6037 | . . . . 5 |
93 | 88, 92 | sylib 208 | . . . 4 |
94 | rpssre 11843 | . . . . 5 | |
95 | 94 | a1i 11 | . . . 4 |
96 | resqcl 12931 | . . . . . 6 | |
97 | 96 | adantl 482 | . . . . 5 |
98 | 0red 10041 | . . . . . . . 8 | |
99 | simplr 792 | . . . . . . . 8 | |
100 | simplrr 801 | . . . . . . . . . 10 | |
101 | 47 | ad2antrl 764 | . . . . . . . . . . . 12 |
102 | 101 | adantr 481 | . . . . . . . . . . 11 |
103 | 102, 49 | syl 17 | . . . . . . . . . 10 |
104 | 100, 103 | breqtrrd 4681 | . . . . . . . . 9 |
105 | 99 | adantr 481 | . . . . . . . . . 10 |
106 | 62 | rpred 11872 | . . . . . . . . . . . 12 |
107 | 106 | ad2ant2r 783 | . . . . . . . . . . 11 |
108 | 107 | adantr 481 | . . . . . . . . . 10 |
109 | simpr 477 | . . . . . . . . . 10 | |
110 | sqrtge0 13998 | . . . . . . . . . . . 12 | |
111 | 101, 110 | syl 17 | . . . . . . . . . . 11 |
112 | 111 | adantr 481 | . . . . . . . . . 10 |
113 | 105, 108, 109, 112 | le2sqd 13044 | . . . . . . . . 9 |
114 | 104, 113 | mpbird 247 | . . . . . . . 8 |
115 | 99 | adantr 481 | . . . . . . . . 9 |
116 | 0red 10041 | . . . . . . . . 9 | |
117 | 107 | adantr 481 | . . . . . . . . 9 |
118 | simpr 477 | . . . . . . . . 9 | |
119 | 111 | adantr 481 | . . . . . . . . 9 |
120 | 115, 116, 117, 118, 119 | letrd 10194 | . . . . . . . 8 |
121 | 98, 99, 114, 120 | lecasei 10143 | . . . . . . 7 |
122 | 121 | expr 643 | . . . . . 6 |
123 | 122 | ralrimiva 2966 | . . . . 5 |
124 | breq1 4656 | . . . . . . . 8 | |
125 | 124 | imbi1d 331 | . . . . . . 7 |
126 | 125 | ralbidv 2986 | . . . . . 6 |
127 | 126 | rspcev 3309 | . . . . 5 |
128 | 97, 123, 127 | syl2anc 693 | . . . 4 |
129 | 93, 86, 62, 95, 128 | o1compt 14318 | . . 3 |
130 | 75, 129 | eqeltrd 2701 | . 2 |
131 | 1, 2, 3, 4, 5, 6, 7, 14, 15, 130 | dchrisum0fno1 25200 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 384 wceq 1483 wex 1704 wcel 1990 wne 2794 wral 2912 wrex 2913 crab 2916 cdif 3571 wss 3574 csn 4177 class class class wbr 4653 cmpt 4729 cdm 5114 ccom 5118 wf 5884 cfv 5888 (class class class)co 6650 cfn 7955 cc 9934 cr 9935 cc0 9936 c1 9937 caddc 9939 cmul 9941 cpnf 10071 cle 10075 cmin 10266 cdiv 10684 cn 11020 c2 11070 cz 11377 crp 11832 cico 12177 cfz 12326 cfl 12591 cseq 12801 cexp 12860 csqrt 13973 cabs 13974 cli 14215 co1 14217 csu 14416 cdvds 14983 cbs 15857 c0g 16100 RHomczrh 19848 ℤ/nℤczn 19851 DChrcdchr 24957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-rpss 6937 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-o1 14221 df-lo1 14222 df-sum 14417 df-ef 14798 df-e 14799 df-sin 14800 df-cos 14801 df-pi 14803 df-dvds 14984 df-gcd 15217 df-prm 15386 df-numer 15443 df-denom 15444 df-phi 15471 df-pc 15542 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-qus 16169 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-nsg 17592 df-eqg 17593 df-ghm 17658 df-gim 17701 df-ga 17723 df-cntz 17750 df-oppg 17776 df-od 17948 df-gex 17949 df-pgp 17950 df-lsm 18051 df-pj1 18052 df-cmn 18195 df-abl 18196 df-cyg 18280 df-dprd 18394 df-dpj 18395 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-rnghom 18715 df-drng 18749 df-subrg 18778 df-lmod 18865 df-lss 18933 df-lsp 18972 df-sra 19172 df-rgmod 19173 df-lidl 19174 df-rsp 19175 df-2idl 19232 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-zring 19819 df-zrh 19852 df-zn 19855 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-0p 23437 df-limc 23630 df-dv 23631 df-ply 23944 df-idp 23945 df-coe 23946 df-dgr 23947 df-quot 24046 df-log 24303 df-cxp 24304 df-em 24719 df-cht 24823 df-vma 24824 df-chp 24825 df-ppi 24826 df-mu 24827 df-dchr 24958 |
This theorem is referenced by: dchrisumn0 25210 rpvmasum 25215 |
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