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Theorem sum2dchr 27218

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.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
sum2dchr	⊢ (𝜑 → Σ𝑥 ∈ 𝐷 ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶))) = if(𝐴 = 𝐶, (ϕ‘𝑁), 0))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐶 𝑥,𝐷 𝑥,𝐺 𝑥,𝑁 𝜑,𝑥 𝑥,𝑍 Allowed substitution hints: 𝐵(𝑥) 𝑈(𝑥)

**Proof of Theorem sum2dchr**
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 2729	. . 3 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
5		sum2dchr.b	. . 3 ⊢ 𝐵 = (Base‘𝑍)
6		sum2dchr.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
7	6	nnnn0d 12479	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
8	3	zncrng 21486	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
9		crngring 20165	. . . . 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 2729	. . . . 5 ⊢ (/_r‘𝑍) = (/_r‘𝑍)
15	5, 13, 14	dvrcl 20324	. . . 4 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝑈) → (𝐴(/_r‘𝑍)𝐶) ∈ 𝐵)
16	10, 11, 12, 15	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝐴(/_r‘𝑍)𝐶) ∈ 𝐵)
17	1, 2, 3, 4, 5, 6, 16	sumdchr 27216	. 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘(𝐴(/_r‘𝑍)𝐶)) = if((𝐴(/_r‘𝑍)𝐶) = (1_r‘𝑍), (ϕ‘𝑁), 0))
18		eqid 2729	. . . . . . . 8 ⊢ (._r‘𝑍) = (._r‘𝑍)
19		eqid 2729	. . . . . . . 8 ⊢ (inv_r‘𝑍) = (inv_r‘𝑍)
20	5, 18, 13, 19, 14	dvrval 20323	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝑈) → (𝐴(/_r‘𝑍)𝐶) = (𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐶)))
21	11, 12, 20	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐴(/_r‘𝑍)𝐶) = (𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐶)))
22	21	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴(/_r‘𝑍)𝐶) = (𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐶)))
23	22	fveq2d 6844	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘(𝐴(/_r‘𝑍)𝐶)) = (𝑥‘(𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐶))))
24	1, 3, 2	dchrmhm 27185	. . . . . 6 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
25		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
26	24, 25	sselid 3941	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
27	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝐵)
28	5, 13	unitss 20296	. . . . . 6 ⊢ 𝑈 ⊆ 𝐵
29	13, 19	unitinvcl 20310	. . . . . . . 8 ⊢ ((𝑍 ∈ Ring ∧ 𝐶 ∈ 𝑈) → ((inv_r‘𝑍)‘𝐶) ∈ 𝑈)
30	10, 12, 29	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((inv_r‘𝑍)‘𝐶) ∈ 𝑈)
31	30	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((inv_r‘𝑍)‘𝐶) ∈ 𝑈)
32	28, 31	sselid 3941	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((inv_r‘𝑍)‘𝐶) ∈ 𝐵)
33		eqid 2729	. . . . . . 7 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
34	33, 5	mgpbas 20065	. . . . . 6 ⊢ 𝐵 = (Base‘(mulGrp‘𝑍))
35	33, 18	mgpplusg 20064	. . . . . 6 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
36		eqid 2729	. . . . . . 7 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
37		cnfldmul 21304	. . . . . . 7 ⊢ · = (._r‘ℂ_fld)
38	36, 37	mgpplusg 20064	. . . . . 6 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
39	34, 35, 38	mhmlin 18702	. . . . 5 ⊢ ((𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ 𝐴 ∈ 𝐵 ∧ ((inv_r‘𝑍)‘𝐶) ∈ 𝐵) → (𝑥‘(𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐶))) = ((𝑥‘𝐴) · (𝑥‘((inv_r‘𝑍)‘𝐶))))
40	26, 27, 32, 39	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘(𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐶))) = ((𝑥‘𝐴) · (𝑥‘((inv_r‘𝑍)‘𝐶))))
41		eqid 2729	. . . . . . . 8 ⊢ ((mulGrp‘𝑍) ↾_s 𝑈) = ((mulGrp‘𝑍) ↾_s 𝑈)
42		eqid 2729	. . . . . . . 8 ⊢ ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})) = ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))
43	1, 3, 2, 13, 41, 42, 25	dchrghm 27200	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ↾ 𝑈) ∈ (((mulGrp‘𝑍) ↾_s 𝑈) GrpHom ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))))
44	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐶 ∈ 𝑈)
45	13, 41	unitgrpbas 20302	. . . . . . . 8 ⊢ 𝑈 = (Base‘((mulGrp‘𝑍) ↾_s 𝑈))
46	13, 41, 19	invrfval 20309	. . . . . . . 8 ⊢ (inv_r‘𝑍) = (inv_g‘((mulGrp‘𝑍) ↾_s 𝑈))
47		cnfldbas 21300	. . . . . . . . . 10 ⊢ ℂ = (Base‘ℂ_fld)
48		cnfld0 21334	. . . . . . . . . 10 ⊢ 0 = (0_g‘ℂ_fld)
49		cndrng 21340	. . . . . . . . . 10 ⊢ ℂ_fld ∈ DivRing
50	47, 48, 49	drngui 20655	. . . . . . . . 9 ⊢ (ℂ ∖ {0}) = (Unit‘ℂ_fld)
51		eqid 2729	. . . . . . . . 9 ⊢ (inv_r‘ℂ_fld) = (inv_r‘ℂ_fld)
52	50, 42, 51	invrfval 20309	. . . . . . . 8 ⊢ (inv_r‘ℂ_fld) = (inv_g‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})))
53	45, 46, 52	ghminv 19137	. . . . . . 7 ⊢ (((𝑥 ↾ 𝑈) ∈ (((mulGrp‘𝑍) ↾_s 𝑈) GrpHom ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))) ∧ 𝐶 ∈ 𝑈) → ((𝑥 ↾ 𝑈)‘((inv_r‘𝑍)‘𝐶)) = ((inv_r‘ℂ_fld)‘((𝑥 ↾ 𝑈)‘𝐶)))
54	43, 44, 53	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥 ↾ 𝑈)‘((inv_r‘𝑍)‘𝐶)) = ((inv_r‘ℂ_fld)‘((𝑥 ↾ 𝑈)‘𝐶)))
55	31	fvresd 6860	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥 ↾ 𝑈)‘((inv_r‘𝑍)‘𝐶)) = (𝑥‘((inv_r‘𝑍)‘𝐶)))
56	44	fvresd 6860	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥 ↾ 𝑈)‘𝐶) = (𝑥‘𝐶))
57	56	fveq2d 6844	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((inv_r‘ℂ_fld)‘((𝑥 ↾ 𝑈)‘𝐶)) = ((inv_r‘ℂ_fld)‘(𝑥‘𝐶)))
58	1, 3, 2, 5, 25	dchrf 27186	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐵⟶ℂ)
59	28, 44	sselid 3941	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐶 ∈ 𝐵)
60	58, 59	ffvelcdmd 7039	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐶) ∈ ℂ)
61	1, 3, 2, 5, 13, 25, 59	dchrn0 27194	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥‘𝐶) ≠ 0 ↔ 𝐶 ∈ 𝑈))
62	44, 61	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐶) ≠ 0)
63		cnfldinv 21344	. . . . . . . 8 ⊢ (((𝑥‘𝐶) ∈ ℂ ∧ (𝑥‘𝐶) ≠ 0) → ((inv_r‘ℂ_fld)‘(𝑥‘𝐶)) = (1 / (𝑥‘𝐶)))
64	60, 62, 63	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((inv_r‘ℂ_fld)‘(𝑥‘𝐶)) = (1 / (𝑥‘𝐶)))
65		recval 15265	. . . . . . . . 9 ⊢ (((𝑥‘𝐶) ∈ ℂ ∧ (𝑥‘𝐶) ≠ 0) → (1 / (𝑥‘𝐶)) = ((∗‘(𝑥‘𝐶)) / ((abs‘(𝑥‘𝐶))↑2)))
66	60, 62, 65	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (1 / (𝑥‘𝐶)) = ((∗‘(𝑥‘𝐶)) / ((abs‘(𝑥‘𝐶))↑2)))
67	1, 2, 25, 3, 13, 44	dchrabs 27204	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (abs‘(𝑥‘𝐶)) = 1)
68	67	oveq1d 7384	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((abs‘(𝑥‘𝐶))↑2) = (1↑2))
69		sq1 14136	. . . . . . . . . 10 ⊢ (1↑2) = 1
70	68, 69	eqtrdi 2780	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((abs‘(𝑥‘𝐶))↑2) = 1)
71	70	oveq2d 7385	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((∗‘(𝑥‘𝐶)) / ((abs‘(𝑥‘𝐶))↑2)) = ((∗‘(𝑥‘𝐶)) / 1))
72	60	cjcld 15138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (∗‘(𝑥‘𝐶)) ∈ ℂ)
73	72	div1d 11926	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((∗‘(𝑥‘𝐶)) / 1) = (∗‘(𝑥‘𝐶)))
74	66, 71, 73	3eqtrd 2768	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (1 / (𝑥‘𝐶)) = (∗‘(𝑥‘𝐶)))
75	57, 64, 74	3eqtrd 2768	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((inv_r‘ℂ_fld)‘((𝑥 ↾ 𝑈)‘𝐶)) = (∗‘(𝑥‘𝐶)))
76	54, 55, 75	3eqtr3d 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘((inv_r‘𝑍)‘𝐶)) = (∗‘(𝑥‘𝐶)))
77	76	oveq2d 7385	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥‘𝐴) · (𝑥‘((inv_r‘𝑍)‘𝐶))) = ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶))))
78	23, 40, 77	3eqtrd 2768	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘(𝐴(/_r‘𝑍)𝐶)) = ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶))))
79	78	sumeq2dv 15644	. 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘(𝐴(/_r‘𝑍)𝐶)) = Σ𝑥 ∈ 𝐷 ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶))))
80	5, 13, 14, 4	dvreq1 20331	. . . 4 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝑈) → ((𝐴(/_r‘𝑍)𝐶) = (1_r‘𝑍) ↔ 𝐴 = 𝐶))
81	10, 11, 12, 80	syl3anc 1373	. . 3 ⊢ (𝜑 → ((𝐴(/_r‘𝑍)𝐶) = (1_r‘𝑍) ↔ 𝐴 = 𝐶))
82	81	ifbid 4508	. 2 ⊢ (𝜑 → if((𝐴(/_r‘𝑍)𝐶) = (1_r‘𝑍), (ϕ‘𝑁), 0) = if(𝐴 = 𝐶, (ϕ‘𝑁), 0))
83	17, 79, 82	3eqtr3d 2772	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 1540 ∈ wcel 2109 ≠ wne 2925 ∖ cdif 3908 ifcif 4484 {csn 4585 ↾ cres 5633 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 0cc0 11044 1c1 11045 · cmul 11049 / cdiv 11811 ℕcn 12162 2c2 12217 ℕ₀cn0 12418 ↑cexp 14002 ∗ccj 15038 abscabs 15176 Σcsu 15628 ϕcphi 16710 Basecbs 17155 ↾_s cress 17176 ._rcmulr 17197 MndHom cmhm 18690 GrpHom cghm 19126 mulGrpcmgp 20060 1_rcur 20101 Ringcrg 20153 CRingccrg 20154 Unitcui 20275 inv_rcinvr 20307 /_rcdvr 20320 ℂ_fldccnfld 21296 ℤ/nℤczn 21444 DChrcdchr 27176

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-disj 5070 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-rpss 7679 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-tpos 8182 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-omul 8416 df-er 8648 df-ec 8650 df-qs 8654 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-dju 9830 df-card 9868 df-acn 9871 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-xnn0 12492 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ioc 13287 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-fac 14215 df-bc 14244 df-hash 14272 df-word 14455 df-concat 14512 df-s1 14537 df-shft 15009 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15629 df-ef 16009 df-sin 16011 df-cos 16012 df-pi 16014 df-dvds 16199 df-gcd 16441 df-prm 16618 df-phi 16712 df-pc 16784 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-qus 17448 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-mhm 18692 df-submnd 18693 df-grp 18850 df-minusg 18851 df-sbg 18852 df-mulg 18982 df-subg 19037 df-nsg 19038 df-eqg 19039 df-ghm 19127 df-gim 19173 df-ga 19204 df-cntz 19231 df-oppg 19260 df-od 19442 df-gex 19443 df-pgp 19444 df-lsm 19550 df-pj1 19551 df-cmn 19696 df-abl 19697 df-cyg 19792 df-dprd 19911 df-dpj 19912 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-cring 20156 df-oppr 20257 df-dvdsr 20277 df-unit 20278 df-invr 20308 df-dvr 20321 df-rhm 20392 df-subrng 20466 df-subrg 20490 df-drng 20651 df-lmod 20800 df-lss 20870 df-lsp 20910 df-sra 21112 df-rgmod 21113 df-lidl 21150 df-rsp 21151 df-2idl 21192 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-zring 21389 df-zrh 21445 df-zn 21448 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22866 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-lp 23056 df-perf 23057 df-cn 23147 df-cnp 23148 df-haus 23235 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-xms 24241 df-ms 24242 df-tms 24243 df-cncf 24804 df-0p 25604 df-limc 25800 df-dv 25801 df-ply 26126 df-idp 26127 df-coe 26128 df-dgr 26129 df-quot 26232 df-log 26498 df-cxp 26499 df-dchr 27177

This theorem is referenced by: rpvmasum2 27456

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