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Mirrors > Home > MPE Home > Th. List > sum2dchr | Structured version Visualization version GIF version |
Description: An orthogonality relation for Dirichlet characters: the sum of 𝑥(𝐴) for fixed 𝐴 and all 𝑥 is 0 if 𝐴 = 1 and ϕ(𝑛) otherwise. Part of Theorem 6.5.2 of [Shapiro] p. 232. (Contributed by Mario Carneiro, 28-Apr-2016.) |
Ref | Expression |
---|---|
sum2dchr.g | ⊢ 𝐺 = (DChr‘𝑁) |
sum2dchr.d | ⊢ 𝐷 = (Base‘𝐺) |
sum2dchr.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
sum2dchr.b | ⊢ 𝐵 = (Base‘𝑍) |
sum2dchr.u | ⊢ 𝑈 = (Unit‘𝑍) |
sum2dchr.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
sum2dchr.a | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
sum2dchr.c | ⊢ (𝜑 → 𝐶 ∈ 𝑈) |
Ref | Expression |
---|---|
sum2dchr | ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶))) = if(𝐴 = 𝐶, (ϕ‘𝑁), 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sum2dchr.g | . . 3 ⊢ 𝐺 = (DChr‘𝑁) | |
2 | sum2dchr.d | . . 3 ⊢ 𝐷 = (Base‘𝐺) | |
3 | sum2dchr.z | . . 3 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
4 | eqid 2740 | . . 3 ⊢ (1r‘𝑍) = (1r‘𝑍) | |
5 | sum2dchr.b | . . 3 ⊢ 𝐵 = (Base‘𝑍) | |
6 | sum2dchr.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
7 | 6 | nnnn0d 12613 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
8 | 3 | zncrng 21586 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing) |
9 | crngring 20272 | . . . . 5 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring) | |
10 | 7, 8, 9 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝑍 ∈ Ring) |
11 | sum2dchr.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
12 | sum2dchr.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑈) | |
13 | sum2dchr.u | . . . . 5 ⊢ 𝑈 = (Unit‘𝑍) | |
14 | eqid 2740 | . . . . 5 ⊢ (/r‘𝑍) = (/r‘𝑍) | |
15 | 5, 13, 14 | dvrcl 20430 | . . . 4 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝑈) → (𝐴(/r‘𝑍)𝐶) ∈ 𝐵) |
16 | 10, 11, 12, 15 | syl3anc 1371 | . . 3 ⊢ (𝜑 → (𝐴(/r‘𝑍)𝐶) ∈ 𝐵) |
17 | 1, 2, 3, 4, 5, 6, 16 | sumdchr 27334 | . 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘(𝐴(/r‘𝑍)𝐶)) = if((𝐴(/r‘𝑍)𝐶) = (1r‘𝑍), (ϕ‘𝑁), 0)) |
18 | eqid 2740 | . . . . . . . 8 ⊢ (.r‘𝑍) = (.r‘𝑍) | |
19 | eqid 2740 | . . . . . . . 8 ⊢ (invr‘𝑍) = (invr‘𝑍) | |
20 | 5, 18, 13, 19, 14 | dvrval 20429 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝑈) → (𝐴(/r‘𝑍)𝐶) = (𝐴(.r‘𝑍)((invr‘𝑍)‘𝐶))) |
21 | 11, 12, 20 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → (𝐴(/r‘𝑍)𝐶) = (𝐴(.r‘𝑍)((invr‘𝑍)‘𝐶))) |
22 | 21 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴(/r‘𝑍)𝐶) = (𝐴(.r‘𝑍)((invr‘𝑍)‘𝐶))) |
23 | 22 | fveq2d 6924 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘(𝐴(/r‘𝑍)𝐶)) = (𝑥‘(𝐴(.r‘𝑍)((invr‘𝑍)‘𝐶)))) |
24 | 1, 3, 2 | dchrmhm 27303 | . . . . . 6 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) |
25 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷) | |
26 | 24, 25 | sselid 4006 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) |
27 | 11 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝐵) |
28 | 5, 13 | unitss 20402 | . . . . . 6 ⊢ 𝑈 ⊆ 𝐵 |
29 | 13, 19 | unitinvcl 20416 | . . . . . . . 8 ⊢ ((𝑍 ∈ Ring ∧ 𝐶 ∈ 𝑈) → ((invr‘𝑍)‘𝐶) ∈ 𝑈) |
30 | 10, 12, 29 | syl2anc 583 | . . . . . . 7 ⊢ (𝜑 → ((invr‘𝑍)‘𝐶) ∈ 𝑈) |
31 | 30 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invr‘𝑍)‘𝐶) ∈ 𝑈) |
32 | 28, 31 | sselid 4006 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invr‘𝑍)‘𝐶) ∈ 𝐵) |
33 | eqid 2740 | . . . . . . 7 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍) | |
34 | 33, 5 | mgpbas 20167 | . . . . . 6 ⊢ 𝐵 = (Base‘(mulGrp‘𝑍)) |
35 | 33, 18 | mgpplusg 20165 | . . . . . 6 ⊢ (.r‘𝑍) = (+g‘(mulGrp‘𝑍)) |
36 | eqid 2740 | . . . . . . 7 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
37 | cnfldmul 21395 | . . . . . . 7 ⊢ · = (.r‘ℂfld) | |
38 | 36, 37 | mgpplusg 20165 | . . . . . 6 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
39 | 34, 35, 38 | mhmlin 18828 | . . . . 5 ⊢ ((𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ 𝐴 ∈ 𝐵 ∧ ((invr‘𝑍)‘𝐶) ∈ 𝐵) → (𝑥‘(𝐴(.r‘𝑍)((invr‘𝑍)‘𝐶))) = ((𝑥‘𝐴) · (𝑥‘((invr‘𝑍)‘𝐶)))) |
40 | 26, 27, 32, 39 | syl3anc 1371 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘(𝐴(.r‘𝑍)((invr‘𝑍)‘𝐶))) = ((𝑥‘𝐴) · (𝑥‘((invr‘𝑍)‘𝐶)))) |
41 | eqid 2740 | . . . . . . . 8 ⊢ ((mulGrp‘𝑍) ↾s 𝑈) = ((mulGrp‘𝑍) ↾s 𝑈) | |
42 | eqid 2740 | . . . . . . . 8 ⊢ ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) | |
43 | 1, 3, 2, 13, 41, 42, 25 | dchrghm 27318 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ↾ 𝑈) ∈ (((mulGrp‘𝑍) ↾s 𝑈) GrpHom ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))) |
44 | 12 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐶 ∈ 𝑈) |
45 | 13, 41 | unitgrpbas 20408 | . . . . . . . 8 ⊢ 𝑈 = (Base‘((mulGrp‘𝑍) ↾s 𝑈)) |
46 | 13, 41, 19 | invrfval 20415 | . . . . . . . 8 ⊢ (invr‘𝑍) = (invg‘((mulGrp‘𝑍) ↾s 𝑈)) |
47 | cnfldbas 21391 | . . . . . . . . . 10 ⊢ ℂ = (Base‘ℂfld) | |
48 | cnfld0 21428 | . . . . . . . . . 10 ⊢ 0 = (0g‘ℂfld) | |
49 | cndrng 21434 | . . . . . . . . . 10 ⊢ ℂfld ∈ DivRing | |
50 | 47, 48, 49 | drngui 20757 | . . . . . . . . 9 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
51 | eqid 2740 | . . . . . . . . 9 ⊢ (invr‘ℂfld) = (invr‘ℂfld) | |
52 | 50, 42, 51 | invrfval 20415 | . . . . . . . 8 ⊢ (invr‘ℂfld) = (invg‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) |
53 | 45, 46, 52 | ghminv 19263 | . . . . . . 7 ⊢ (((𝑥 ↾ 𝑈) ∈ (((mulGrp‘𝑍) ↾s 𝑈) GrpHom ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) ∧ 𝐶 ∈ 𝑈) → ((𝑥 ↾ 𝑈)‘((invr‘𝑍)‘𝐶)) = ((invr‘ℂfld)‘((𝑥 ↾ 𝑈)‘𝐶))) |
54 | 43, 44, 53 | syl2anc 583 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥 ↾ 𝑈)‘((invr‘𝑍)‘𝐶)) = ((invr‘ℂfld)‘((𝑥 ↾ 𝑈)‘𝐶))) |
55 | 31 | fvresd 6940 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥 ↾ 𝑈)‘((invr‘𝑍)‘𝐶)) = (𝑥‘((invr‘𝑍)‘𝐶))) |
56 | 44 | fvresd 6940 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥 ↾ 𝑈)‘𝐶) = (𝑥‘𝐶)) |
57 | 56 | fveq2d 6924 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invr‘ℂfld)‘((𝑥 ↾ 𝑈)‘𝐶)) = ((invr‘ℂfld)‘(𝑥‘𝐶))) |
58 | 1, 3, 2, 5, 25 | dchrf 27304 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐵⟶ℂ) |
59 | 28, 44 | sselid 4006 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐶 ∈ 𝐵) |
60 | 58, 59 | ffvelcdmd 7119 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐶) ∈ ℂ) |
61 | 1, 3, 2, 5, 13, 25, 59 | dchrn0 27312 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥‘𝐶) ≠ 0 ↔ 𝐶 ∈ 𝑈)) |
62 | 44, 61 | mpbird 257 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐶) ≠ 0) |
63 | cnfldinv 21438 | . . . . . . . 8 ⊢ (((𝑥‘𝐶) ∈ ℂ ∧ (𝑥‘𝐶) ≠ 0) → ((invr‘ℂfld)‘(𝑥‘𝐶)) = (1 / (𝑥‘𝐶))) | |
64 | 60, 62, 63 | syl2anc 583 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invr‘ℂfld)‘(𝑥‘𝐶)) = (1 / (𝑥‘𝐶))) |
65 | recval 15371 | . . . . . . . . 9 ⊢ (((𝑥‘𝐶) ∈ ℂ ∧ (𝑥‘𝐶) ≠ 0) → (1 / (𝑥‘𝐶)) = ((∗‘(𝑥‘𝐶)) / ((abs‘(𝑥‘𝐶))↑2))) | |
66 | 60, 62, 65 | syl2anc 583 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (1 / (𝑥‘𝐶)) = ((∗‘(𝑥‘𝐶)) / ((abs‘(𝑥‘𝐶))↑2))) |
67 | 1, 2, 25, 3, 13, 44 | dchrabs 27322 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (abs‘(𝑥‘𝐶)) = 1) |
68 | 67 | oveq1d 7463 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((abs‘(𝑥‘𝐶))↑2) = (1↑2)) |
69 | sq1 14244 | . . . . . . . . . 10 ⊢ (1↑2) = 1 | |
70 | 68, 69 | eqtrdi 2796 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((abs‘(𝑥‘𝐶))↑2) = 1) |
71 | 70 | oveq2d 7464 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((∗‘(𝑥‘𝐶)) / ((abs‘(𝑥‘𝐶))↑2)) = ((∗‘(𝑥‘𝐶)) / 1)) |
72 | 60 | cjcld 15245 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (∗‘(𝑥‘𝐶)) ∈ ℂ) |
73 | 72 | div1d 12062 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((∗‘(𝑥‘𝐶)) / 1) = (∗‘(𝑥‘𝐶))) |
74 | 66, 71, 73 | 3eqtrd 2784 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (1 / (𝑥‘𝐶)) = (∗‘(𝑥‘𝐶))) |
75 | 57, 64, 74 | 3eqtrd 2784 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invr‘ℂfld)‘((𝑥 ↾ 𝑈)‘𝐶)) = (∗‘(𝑥‘𝐶))) |
76 | 54, 55, 75 | 3eqtr3d 2788 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘((invr‘𝑍)‘𝐶)) = (∗‘(𝑥‘𝐶))) |
77 | 76 | oveq2d 7464 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥‘𝐴) · (𝑥‘((invr‘𝑍)‘𝐶))) = ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶)))) |
78 | 23, 40, 77 | 3eqtrd 2784 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘(𝐴(/r‘𝑍)𝐶)) = ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶)))) |
79 | 78 | sumeq2dv 15750 | . 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘(𝐴(/r‘𝑍)𝐶)) = Σ𝑥 ∈ 𝐷 ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶)))) |
80 | 5, 13, 14, 4 | dvreq1 20437 | . . . 4 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝑈) → ((𝐴(/r‘𝑍)𝐶) = (1r‘𝑍) ↔ 𝐴 = 𝐶)) |
81 | 10, 11, 12, 80 | syl3anc 1371 | . . 3 ⊢ (𝜑 → ((𝐴(/r‘𝑍)𝐶) = (1r‘𝑍) ↔ 𝐴 = 𝐶)) |
82 | 81 | ifbid 4571 | . 2 ⊢ (𝜑 → if((𝐴(/r‘𝑍)𝐶) = (1r‘𝑍), (ϕ‘𝑁), 0) = if(𝐴 = 𝐶, (ϕ‘𝑁), 0)) |
83 | 17, 79, 82 | 3eqtr3d 2788 | 1 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶))) = if(𝐴 = 𝐶, (ϕ‘𝑁), 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∖ cdif 3973 ifcif 4548 {csn 4648 ↾ cres 5702 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 0cc0 11184 1c1 11185 · cmul 11189 / cdiv 11947 ℕcn 12293 2c2 12348 ℕ0cn0 12553 ↑cexp 14112 ∗ccj 15145 abscabs 15283 Σcsu 15734 ϕcphi 16811 Basecbs 17258 ↾s cress 17287 .rcmulr 17312 MndHom cmhm 18816 GrpHom cghm 19252 mulGrpcmgp 20161 1rcur 20208 Ringcrg 20260 CRingccrg 20261 Unitcui 20381 invrcinvr 20413 /rcdvr 20426 ℂfldccnfld 21387 ℤ/nℤczn 21536 DChrcdchr 27294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-rpss 7758 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-omul 8527 df-er 8763 df-ec 8765 df-qs 8769 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-acn 10011 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-xnn0 12626 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-word 14563 df-concat 14619 df-s1 14644 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-sin 16117 df-cos 16118 df-pi 16120 df-dvds 16303 df-gcd 16541 df-prm 16719 df-phi 16813 df-pc 16884 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-qus 17569 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-nsg 19164 df-eqg 19165 df-ghm 19253 df-gim 19299 df-ga 19330 df-cntz 19357 df-oppg 19386 df-od 19570 df-gex 19571 df-pgp 19572 df-lsm 19678 df-pj1 19679 df-cmn 19824 df-abl 19825 df-cyg 19920 df-dprd 20039 df-dpj 20040 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-rhm 20498 df-subrng 20572 df-subrg 20597 df-drng 20753 df-lmod 20882 df-lss 20953 df-lsp 20993 df-sra 21195 df-rgmod 21196 df-lidl 21241 df-rsp 21242 df-2idl 21283 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-zring 21481 df-zrh 21537 df-zn 21540 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-0p 25724 df-limc 25921 df-dv 25922 df-ply 26247 df-idp 26248 df-coe 26249 df-dgr 26250 df-quot 26351 df-log 26616 df-cxp 26617 df-dchr 27295 |
This theorem is referenced by: rpvmasum2 27574 |
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