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Mirrors > Home > MPE Home > Th. List > dchr2sum | Structured version Visualization version GIF version |
Description: An orthogonality relation for Dirichlet characters: the sum of 𝑋(𝑎) · ∗𝑌(𝑎) over all 𝑎 is nonzero only when 𝑋 = 𝑌. Part of Theorem 6.5.2 of [Shapiro] p. 232. (Contributed by Mario Carneiro, 28-Apr-2016.) |
Ref | Expression |
---|---|
dchr2sum.g | ⊢ 𝐺 = (DChr‘𝑁) |
dchr2sum.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
dchr2sum.d | ⊢ 𝐷 = (Base‘𝐺) |
dchr2sum.b | ⊢ 𝐵 = (Base‘𝑍) |
dchr2sum.x | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
dchr2sum.y | ⊢ (𝜑 → 𝑌 ∈ 𝐷) |
Ref | Expression |
---|---|
dchr2sum | ⊢ (𝜑 → Σ𝑎 ∈ 𝐵 ((𝑋‘𝑎) · (∗‘(𝑌‘𝑎))) = if(𝑋 = 𝑌, (ϕ‘𝑁), 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchr2sum.g | . . 3 ⊢ 𝐺 = (DChr‘𝑁) | |
2 | dchr2sum.z | . . 3 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
3 | dchr2sum.d | . . 3 ⊢ 𝐷 = (Base‘𝐺) | |
4 | eqid 2610 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
5 | dchr2sum.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
6 | 1, 3 | dchrrcl 24765 | . . . . . 6 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
8 | 1 | dchrabl 24779 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) |
9 | ablgrp 18021 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
10 | 7, 8, 9 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) |
11 | dchr2sum.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐷) | |
12 | eqid 2610 | . . . . 5 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
13 | 3, 12 | grpsubcl 17318 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) → (𝑋(-g‘𝐺)𝑌) ∈ 𝐷) |
14 | 10, 5, 11, 13 | syl3anc 1318 | . . 3 ⊢ (𝜑 → (𝑋(-g‘𝐺)𝑌) ∈ 𝐷) |
15 | dchr2sum.b | . . 3 ⊢ 𝐵 = (Base‘𝑍) | |
16 | 1, 2, 3, 4, 14, 15 | dchrsum 24794 | . 2 ⊢ (𝜑 → Σ𝑎 ∈ 𝐵 ((𝑋(-g‘𝐺)𝑌)‘𝑎) = if((𝑋(-g‘𝐺)𝑌) = (0g‘𝐺), (ϕ‘𝑁), 0)) |
17 | 5 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋 ∈ 𝐷) |
18 | 11 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑌 ∈ 𝐷) |
19 | eqid 2610 | . . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
20 | eqid 2610 | . . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
21 | 3, 19, 20, 12 | grpsubval 17288 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) → (𝑋(-g‘𝐺)𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
22 | 17, 18, 21 | syl2anc 691 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑋(-g‘𝐺)𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
23 | 7 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑁 ∈ ℕ) |
24 | 23, 8, 9 | 3syl 18 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐺 ∈ Grp) |
25 | 3, 20 | grpinvcl 17290 | . . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐷) → ((invg‘𝐺)‘𝑌) ∈ 𝐷) |
26 | 24, 18, 25 | syl2anc 691 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) ∈ 𝐷) |
27 | 1, 2, 3, 19, 17, 26 | dchrmul 24773 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) = (𝑋 ∘𝑓 · ((invg‘𝐺)‘𝑌))) |
28 | 22, 27 | eqtrd 2644 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑋(-g‘𝐺)𝑌) = (𝑋 ∘𝑓 · ((invg‘𝐺)‘𝑌))) |
29 | 28 | fveq1d 6105 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑋(-g‘𝐺)𝑌)‘𝑎) = ((𝑋 ∘𝑓 · ((invg‘𝐺)‘𝑌))‘𝑎)) |
30 | 1, 2, 3, 15, 17 | dchrf 24767 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋:𝐵⟶ℂ) |
31 | ffn 5958 | . . . . . 6 ⊢ (𝑋:𝐵⟶ℂ → 𝑋 Fn 𝐵) | |
32 | 30, 31 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋 Fn 𝐵) |
33 | 1, 2, 3, 15, 26 | dchrf 24767 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((invg‘𝐺)‘𝑌):𝐵⟶ℂ) |
34 | ffn 5958 | . . . . . 6 ⊢ (((invg‘𝐺)‘𝑌):𝐵⟶ℂ → ((invg‘𝐺)‘𝑌) Fn 𝐵) | |
35 | 33, 34 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) Fn 𝐵) |
36 | fvex 6113 | . . . . . . 7 ⊢ (Base‘𝑍) ∈ V | |
37 | 15, 36 | eqeltri 2684 | . . . . . 6 ⊢ 𝐵 ∈ V |
38 | 37 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐵 ∈ V) |
39 | simpr 476 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵) | |
40 | fnfvof 6809 | . . . . 5 ⊢ (((𝑋 Fn 𝐵 ∧ ((invg‘𝐺)‘𝑌) Fn 𝐵) ∧ (𝐵 ∈ V ∧ 𝑎 ∈ 𝐵)) → ((𝑋 ∘𝑓 · ((invg‘𝐺)‘𝑌))‘𝑎) = ((𝑋‘𝑎) · (((invg‘𝐺)‘𝑌)‘𝑎))) | |
41 | 32, 35, 38, 39, 40 | syl22anc 1319 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑋 ∘𝑓 · ((invg‘𝐺)‘𝑌))‘𝑎) = ((𝑋‘𝑎) · (((invg‘𝐺)‘𝑌)‘𝑎))) |
42 | 1, 3, 18, 20 | dchrinv 24786 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) = (∗ ∘ 𝑌)) |
43 | 42 | fveq1d 6105 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (((invg‘𝐺)‘𝑌)‘𝑎) = ((∗ ∘ 𝑌)‘𝑎)) |
44 | 1, 2, 3, 15, 18 | dchrf 24767 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑌:𝐵⟶ℂ) |
45 | fvco3 6185 | . . . . . . 7 ⊢ ((𝑌:𝐵⟶ℂ ∧ 𝑎 ∈ 𝐵) → ((∗ ∘ 𝑌)‘𝑎) = (∗‘(𝑌‘𝑎))) | |
46 | 44, 39, 45 | syl2anc 691 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((∗ ∘ 𝑌)‘𝑎) = (∗‘(𝑌‘𝑎))) |
47 | 43, 46 | eqtrd 2644 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (((invg‘𝐺)‘𝑌)‘𝑎) = (∗‘(𝑌‘𝑎))) |
48 | 47 | oveq2d 6565 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑋‘𝑎) · (((invg‘𝐺)‘𝑌)‘𝑎)) = ((𝑋‘𝑎) · (∗‘(𝑌‘𝑎)))) |
49 | 29, 41, 48 | 3eqtrd 2648 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑋(-g‘𝐺)𝑌)‘𝑎) = ((𝑋‘𝑎) · (∗‘(𝑌‘𝑎)))) |
50 | 49 | sumeq2dv 14281 | . 2 ⊢ (𝜑 → Σ𝑎 ∈ 𝐵 ((𝑋(-g‘𝐺)𝑌)‘𝑎) = Σ𝑎 ∈ 𝐵 ((𝑋‘𝑎) · (∗‘(𝑌‘𝑎)))) |
51 | 3, 4, 12 | grpsubeq0 17324 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) → ((𝑋(-g‘𝐺)𝑌) = (0g‘𝐺) ↔ 𝑋 = 𝑌)) |
52 | 10, 5, 11, 51 | syl3anc 1318 | . . 3 ⊢ (𝜑 → ((𝑋(-g‘𝐺)𝑌) = (0g‘𝐺) ↔ 𝑋 = 𝑌)) |
53 | 52 | ifbid 4058 | . 2 ⊢ (𝜑 → if((𝑋(-g‘𝐺)𝑌) = (0g‘𝐺), (ϕ‘𝑁), 0) = if(𝑋 = 𝑌, (ϕ‘𝑁), 0)) |
54 | 16, 50, 53 | 3eqtr3d 2652 | 1 ⊢ (𝜑 → Σ𝑎 ∈ 𝐵 ((𝑋‘𝑎) · (∗‘(𝑌‘𝑎))) = if(𝑋 = 𝑌, (ϕ‘𝑁), 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ifcif 4036 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 ℂcc 9813 0cc0 9815 · cmul 9820 ℕcn 10897 ∗ccj 13684 Σcsu 14264 ϕcphi 15307 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Grpcgrp 17245 invgcminusg 17246 -gcsg 17247 Abelcabl 18017 ℤ/nℤczn 19670 DChrcdchr 24757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-xnn0 11241 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-dvds 14822 df-gcd 15055 df-phi 15309 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-qus 15992 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-nsg 17415 df-eqg 17416 df-ghm 17481 df-cntz 17573 df-od 17771 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-rnghom 18538 df-drng 18572 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-sra 18993 df-rgmod 18994 df-lidl 18995 df-rsp 18996 df-2idl 19053 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-zring 19638 df-zrh 19671 df-zn 19674 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107 df-cxp 24108 df-dchr 24758 |
This theorem is referenced by: (None) |
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