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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 2734 | . . 3 ⊢ (1r‘𝑍) = (1r‘𝑍) | |
5 | sum2dchr.b | . . 3 ⊢ 𝐵 = (Base‘𝑍) | |
6 | sum2dchr.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
7 | 6 | nnnn0d 12609 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
8 | 3 | zncrng 21581 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing) |
9 | crngring 20267 | . . . . 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 2734 | . . . . 5 ⊢ (/r‘𝑍) = (/r‘𝑍) | |
15 | 5, 13, 14 | dvrcl 20425 | . . . 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 27325 | . 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘(𝐴(/r‘𝑍)𝐶)) = if((𝐴(/r‘𝑍)𝐶) = (1r‘𝑍), (ϕ‘𝑁), 0)) |
18 | eqid 2734 | . . . . . . . 8 ⊢ (.r‘𝑍) = (.r‘𝑍) | |
19 | eqid 2734 | . . . . . . . 8 ⊢ (invr‘𝑍) = (invr‘𝑍) | |
20 | 5, 18, 13, 19, 14 | dvrval 20424 | . . . . . . 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 6923 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘(𝐴(/r‘𝑍)𝐶)) = (𝑥‘(𝐴(.r‘𝑍)((invr‘𝑍)‘𝐶)))) |
24 | 1, 3, 2 | dchrmhm 27294 | . . . . . 6 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) |
25 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷) | |
26 | 24, 25 | sselid 4000 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) |
27 | 11 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝐵) |
28 | 5, 13 | unitss 20397 | . . . . . 6 ⊢ 𝑈 ⊆ 𝐵 |
29 | 13, 19 | unitinvcl 20411 | . . . . . . . 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 4000 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invr‘𝑍)‘𝐶) ∈ 𝐵) |
33 | eqid 2734 | . . . . . . 7 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍) | |
34 | 33, 5 | mgpbas 20162 | . . . . . 6 ⊢ 𝐵 = (Base‘(mulGrp‘𝑍)) |
35 | 33, 18 | mgpplusg 20160 | . . . . . 6 ⊢ (.r‘𝑍) = (+g‘(mulGrp‘𝑍)) |
36 | eqid 2734 | . . . . . . 7 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
37 | cnfldmul 21390 | . . . . . . 7 ⊢ · = (.r‘ℂfld) | |
38 | 36, 37 | mgpplusg 20160 | . . . . . 6 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
39 | 34, 35, 38 | mhmlin 18823 | . . . . 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 2734 | . . . . . . . 8 ⊢ ((mulGrp‘𝑍) ↾s 𝑈) = ((mulGrp‘𝑍) ↾s 𝑈) | |
42 | eqid 2734 | . . . . . . . 8 ⊢ ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) | |
43 | 1, 3, 2, 13, 41, 42, 25 | dchrghm 27309 | . . . . . . 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 20403 | . . . . . . . 8 ⊢ 𝑈 = (Base‘((mulGrp‘𝑍) ↾s 𝑈)) |
46 | 13, 41, 19 | invrfval 20410 | . . . . . . . 8 ⊢ (invr‘𝑍) = (invg‘((mulGrp‘𝑍) ↾s 𝑈)) |
47 | cnfldbas 21386 | . . . . . . . . . 10 ⊢ ℂ = (Base‘ℂfld) | |
48 | cnfld0 21423 | . . . . . . . . . 10 ⊢ 0 = (0g‘ℂfld) | |
49 | cndrng 21429 | . . . . . . . . . 10 ⊢ ℂfld ∈ DivRing | |
50 | 47, 48, 49 | drngui 20752 | . . . . . . . . 9 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
51 | eqid 2734 | . . . . . . . . 9 ⊢ (invr‘ℂfld) = (invr‘ℂfld) | |
52 | 50, 42, 51 | invrfval 20410 | . . . . . . . 8 ⊢ (invr‘ℂfld) = (invg‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) |
53 | 45, 46, 52 | ghminv 19258 | . . . . . . 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 6939 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥 ↾ 𝑈)‘((invr‘𝑍)‘𝐶)) = (𝑥‘((invr‘𝑍)‘𝐶))) |
56 | 44 | fvresd 6939 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥 ↾ 𝑈)‘𝐶) = (𝑥‘𝐶)) |
57 | 56 | fveq2d 6923 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invr‘ℂfld)‘((𝑥 ↾ 𝑈)‘𝐶)) = ((invr‘ℂfld)‘(𝑥‘𝐶))) |
58 | 1, 3, 2, 5, 25 | dchrf 27295 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐵⟶ℂ) |
59 | 28, 44 | sselid 4000 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐶 ∈ 𝐵) |
60 | 58, 59 | ffvelcdmd 7117 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐶) ∈ ℂ) |
61 | 1, 3, 2, 5, 13, 25, 59 | dchrn0 27303 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥‘𝐶) ≠ 0 ↔ 𝐶 ∈ 𝑈)) |
62 | 44, 61 | mpbird 257 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐶) ≠ 0) |
63 | cnfldinv 21433 | . . . . . . . 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 15367 | . . . . . . . . 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 27313 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (abs‘(𝑥‘𝐶)) = 1) |
68 | 67 | oveq1d 7460 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((abs‘(𝑥‘𝐶))↑2) = (1↑2)) |
69 | sq1 14240 | . . . . . . . . . 10 ⊢ (1↑2) = 1 | |
70 | 68, 69 | eqtrdi 2790 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((abs‘(𝑥‘𝐶))↑2) = 1) |
71 | 70 | oveq2d 7461 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((∗‘(𝑥‘𝐶)) / ((abs‘(𝑥‘𝐶))↑2)) = ((∗‘(𝑥‘𝐶)) / 1)) |
72 | 60 | cjcld 15241 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (∗‘(𝑥‘𝐶)) ∈ ℂ) |
73 | 72 | div1d 12058 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((∗‘(𝑥‘𝐶)) / 1) = (∗‘(𝑥‘𝐶))) |
74 | 66, 71, 73 | 3eqtrd 2778 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (1 / (𝑥‘𝐶)) = (∗‘(𝑥‘𝐶))) |
75 | 57, 64, 74 | 3eqtrd 2778 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invr‘ℂfld)‘((𝑥 ↾ 𝑈)‘𝐶)) = (∗‘(𝑥‘𝐶))) |
76 | 54, 55, 75 | 3eqtr3d 2782 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘((invr‘𝑍)‘𝐶)) = (∗‘(𝑥‘𝐶))) |
77 | 76 | oveq2d 7461 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥‘𝐴) · (𝑥‘((invr‘𝑍)‘𝐶))) = ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶)))) |
78 | 23, 40, 77 | 3eqtrd 2778 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘(𝐴(/r‘𝑍)𝐶)) = ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶)))) |
79 | 78 | sumeq2dv 15746 | . 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘(𝐴(/r‘𝑍)𝐶)) = Σ𝑥 ∈ 𝐷 ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶)))) |
80 | 5, 13, 14, 4 | dvreq1 20432 | . . . 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 2782 | 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 2103 ≠ wne 2942 ∖ cdif 3967 ifcif 4548 {csn 4648 ↾ cres 5701 ‘cfv 6572 (class class class)co 7445 ℂcc 11178 0cc0 11180 1c1 11181 · cmul 11185 / cdiv 11943 ℕcn 12289 2c2 12344 ℕ0cn0 12549 ↑cexp 14108 ∗ccj 15141 abscabs 15279 Σcsu 15730 ϕcphi 16806 Basecbs 17253 ↾s cress 17282 .rcmulr 17307 MndHom cmhm 18811 GrpHom cghm 19247 mulGrpcmgp 20156 1rcur 20203 Ringcrg 20255 CRingccrg 20256 Unitcui 20376 invrcinvr 20408 /rcdvr 20421 ℂfldccnfld 21382 ℤ/nℤczn 21531 DChrcdchr 27285 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-inf2 9706 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-pre-sup 11258 ax-addf 11259 ax-mulf 11260 |
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 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-iin 5022 df-disj 5137 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-se 5655 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-isom 6581 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-of 7710 df-rpss 7754 df-om 7900 df-1st 8026 df-2nd 8027 df-supp 8198 df-tpos 8263 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-oadd 8522 df-omul 8523 df-er 8759 df-ec 8761 df-qs 8765 df-map 8882 df-pm 8883 df-ixp 8952 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-fsupp 9428 df-fi 9476 df-sup 9507 df-inf 9508 df-oi 9575 df-dju 9966 df-card 10004 df-acn 10007 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-xnn0 12622 df-z 12636 df-dec 12755 df-uz 12900 df-q 13010 df-rp 13054 df-xneg 13171 df-xadd 13172 df-xmul 13173 df-ioo 13407 df-ioc 13408 df-ico 13409 df-icc 13410 df-fz 13564 df-fzo 13708 df-fl 13839 df-mod 13917 df-seq 14049 df-exp 14109 df-fac 14319 df-bc 14348 df-hash 14376 df-word 14559 df-concat 14615 df-s1 14640 df-shft 15112 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-limsup 15513 df-clim 15530 df-rlim 15531 df-sum 15731 df-ef 16109 df-sin 16111 df-cos 16112 df-pi 16114 df-dvds 16297 df-gcd 16535 df-prm 16713 df-phi 16808 df-pc 16879 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-starv 17321 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-hom 17330 df-cco 17331 df-rest 17477 df-topn 17478 df-0g 17496 df-gsum 17497 df-topgen 17498 df-pt 17499 df-prds 17502 df-xrs 17557 df-qtop 17562 df-imas 17563 df-qus 17564 df-xps 17565 df-mre 17639 df-mrc 17640 df-acs 17642 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-mhm 18813 df-submnd 18814 df-grp 18971 df-minusg 18972 df-sbg 18973 df-mulg 19103 df-subg 19158 df-nsg 19159 df-eqg 19160 df-ghm 19248 df-gim 19294 df-ga 19325 df-cntz 19352 df-oppg 19381 df-od 19565 df-gex 19566 df-pgp 19567 df-lsm 19673 df-pj1 19674 df-cmn 19819 df-abl 19820 df-cyg 19915 df-dprd 20034 df-dpj 20035 df-mgp 20157 df-rng 20175 df-ur 20204 df-ring 20257 df-cring 20258 df-oppr 20355 df-dvdsr 20378 df-unit 20379 df-invr 20409 df-dvr 20422 df-rhm 20493 df-subrng 20567 df-subrg 20592 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-2idl 21278 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-zring 21476 df-zrh 21532 df-zn 21535 df-top 22914 df-topon 22931 df-topsp 22953 df-bases 22967 df-cld 23041 df-ntr 23042 df-cls 23043 df-nei 23120 df-lp 23158 df-perf 23159 df-cn 23249 df-cnp 23250 df-haus 23337 df-tx 23584 df-hmeo 23777 df-fil 23868 df-fm 23960 df-flim 23961 df-flf 23962 df-xms 24344 df-ms 24345 df-tms 24346 df-cncf 24916 df-0p 25717 df-limc 25913 df-dv 25914 df-ply 26239 df-idp 26240 df-coe 26241 df-dgr 26242 df-quot 26343 df-log 26607 df-cxp 26608 df-dchr 27286 |
This theorem is referenced by: rpvmasum2 27565 |
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