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Theorem sum2dchr 27191

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 2730	. . 3 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
5		sum2dchr.b	. . 3 ⊢ 𝐵 = (Base‘𝑍)
6		sum2dchr.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
7	6	nnnn0d 12509	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
8	3	zncrng 21460	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
9		crngring 20160	. . . . 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 2730	. . . . 5 ⊢ (/_r‘𝑍) = (/_r‘𝑍)
15	5, 13, 14	dvrcl 20319	. . . 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 27189	. 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘(𝐴(/_r‘𝑍)𝐶)) = if((𝐴(/_r‘𝑍)𝐶) = (1_r‘𝑍), (ϕ‘𝑁), 0))
18		eqid 2730	. . . . . . . 8 ⊢ (._r‘𝑍) = (._r‘𝑍)
19		eqid 2730	. . . . . . . 8 ⊢ (inv_r‘𝑍) = (inv_r‘𝑍)
20	5, 18, 13, 19, 14	dvrval 20318	. . . . . . 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 6864	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘(𝐴(/_r‘𝑍)𝐶)) = (𝑥‘(𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐶))))
24	1, 3, 2	dchrmhm 27158	. . . . . 6 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
25		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
26	24, 25	sselid 3946	. . . . 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 20291	. . . . . 6 ⊢ 𝑈 ⊆ 𝐵
29	13, 19	unitinvcl 20305	. . . . . . . 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 3946	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((inv_r‘𝑍)‘𝐶) ∈ 𝐵)
33		eqid 2730	. . . . . . 7 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
34	33, 5	mgpbas 20060	. . . . . 6 ⊢ 𝐵 = (Base‘(mulGrp‘𝑍))
35	33, 18	mgpplusg 20059	. . . . . 6 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
36		eqid 2730	. . . . . . 7 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
37		cnfldmul 21278	. . . . . . 7 ⊢ · = (._r‘ℂ_fld)
38	36, 37	mgpplusg 20059	. . . . . 6 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
39	34, 35, 38	mhmlin 18726	. . . . 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 2730	. . . . . . . 8 ⊢ ((mulGrp‘𝑍) ↾_s 𝑈) = ((mulGrp‘𝑍) ↾_s 𝑈)
42		eqid 2730	. . . . . . . 8 ⊢ ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})) = ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))
43	1, 3, 2, 13, 41, 42, 25	dchrghm 27173	. . . . . . 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 20297	. . . . . . . 8 ⊢ 𝑈 = (Base‘((mulGrp‘𝑍) ↾_s 𝑈))
46	13, 41, 19	invrfval 20304	. . . . . . . 8 ⊢ (inv_r‘𝑍) = (inv_g‘((mulGrp‘𝑍) ↾_s 𝑈))
47		cnfldbas 21274	. . . . . . . . . 10 ⊢ ℂ = (Base‘ℂ_fld)
48		cnfld0 21310	. . . . . . . . . 10 ⊢ 0 = (0_g‘ℂ_fld)
49		cndrng 21316	. . . . . . . . . 10 ⊢ ℂ_fld ∈ DivRing
50	47, 48, 49	drngui 20650	. . . . . . . . 9 ⊢ (ℂ ∖ {0}) = (Unit‘ℂ_fld)
51		eqid 2730	. . . . . . . . 9 ⊢ (inv_r‘ℂ_fld) = (inv_r‘ℂ_fld)
52	50, 42, 51	invrfval 20304	. . . . . . . 8 ⊢ (inv_r‘ℂ_fld) = (inv_g‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})))
53	45, 46, 52	ghminv 19161	. . . . . . 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 6880	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥 ↾ 𝑈)‘((inv_r‘𝑍)‘𝐶)) = (𝑥‘((inv_r‘𝑍)‘𝐶)))
56	44	fvresd 6880	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥 ↾ 𝑈)‘𝐶) = (𝑥‘𝐶))
57	56	fveq2d 6864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((inv_r‘ℂ_fld)‘((𝑥 ↾ 𝑈)‘𝐶)) = ((inv_r‘ℂ_fld)‘(𝑥‘𝐶)))
58	1, 3, 2, 5, 25	dchrf 27159	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐵⟶ℂ)
59	28, 44	sselid 3946	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐶 ∈ 𝐵)
60	58, 59	ffvelcdmd 7059	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐶) ∈ ℂ)
61	1, 3, 2, 5, 13, 25, 59	dchrn0 27167	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥‘𝐶) ≠ 0 ↔ 𝐶 ∈ 𝑈))
62	44, 61	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐶) ≠ 0)
63		cnfldinv 21320	. . . . . . . 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 15295	. . . . . . . . 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 27177	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (abs‘(𝑥‘𝐶)) = 1)
68	67	oveq1d 7404	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((abs‘(𝑥‘𝐶))↑2) = (1↑2))
69		sq1 14166	. . . . . . . . . 10 ⊢ (1↑2) = 1
70	68, 69	eqtrdi 2781	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((abs‘(𝑥‘𝐶))↑2) = 1)
71	70	oveq2d 7405	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((∗‘(𝑥‘𝐶)) / ((abs‘(𝑥‘𝐶))↑2)) = ((∗‘(𝑥‘𝐶)) / 1))
72	60	cjcld 15168	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (∗‘(𝑥‘𝐶)) ∈ ℂ)
73	72	div1d 11956	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((∗‘(𝑥‘𝐶)) / 1) = (∗‘(𝑥‘𝐶)))
74	66, 71, 73	3eqtrd 2769	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (1 / (𝑥‘𝐶)) = (∗‘(𝑥‘𝐶)))
75	57, 64, 74	3eqtrd 2769	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((inv_r‘ℂ_fld)‘((𝑥 ↾ 𝑈)‘𝐶)) = (∗‘(𝑥‘𝐶)))
76	54, 55, 75	3eqtr3d 2773	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘((inv_r‘𝑍)‘𝐶)) = (∗‘(𝑥‘𝐶)))
77	76	oveq2d 7405	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥‘𝐴) · (𝑥‘((inv_r‘𝑍)‘𝐶))) = ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶))))
78	23, 40, 77	3eqtrd 2769	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘(𝐴(/_r‘𝑍)𝐶)) = ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶))))
79	78	sumeq2dv 15674	. 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘(𝐴(/_r‘𝑍)𝐶)) = Σ𝑥 ∈ 𝐷 ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶))))
80	5, 13, 14, 4	dvreq1 20326	. . . 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 4514	. 2 ⊢ (𝜑 → if((𝐴(/_r‘𝑍)𝐶) = (1_r‘𝑍), (ϕ‘𝑁), 0) = if(𝐴 = 𝐶, (ϕ‘𝑁), 0))
83	17, 79, 82	3eqtr3d 2773	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 2926 ∖ cdif 3913 ifcif 4490 {csn 4591 ↾ cres 5642 ‘cfv 6513 (class class class)co 7389 ℂcc 11072 0cc0 11074 1c1 11075 · cmul 11079 / cdiv 11841 ℕcn 12187 2c2 12242 ℕ₀cn0 12448 ↑cexp 14032 ∗ccj 15068 abscabs 15206 Σcsu 15658 ϕcphi 16740 Basecbs 17185 ↾_s cress 17206 ._rcmulr 17227 MndHom cmhm 18714 GrpHom cghm 19150 mulGrpcmgp 20055 1_rcur 20096 Ringcrg 20148 CRingccrg 20149 Unitcui 20270 inv_rcinvr 20302 /_rcdvr 20315 ℂ_fldccnfld 21270 ℤ/nℤczn 21418 DChrcdchr 27149

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 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-inf2 9600 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152 ax-addf 11153 ax-mulf 11154

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-iin 4960 df-disj 5077 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-isom 6522 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-of 7655 df-rpss 7701 df-om 7845 df-1st 7970 df-2nd 7971 df-supp 8142 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-2o 8437 df-oadd 8440 df-omul 8441 df-er 8673 df-ec 8675 df-qs 8679 df-map 8803 df-pm 8804 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9319 df-fi 9368 df-sup 9399 df-inf 9400 df-oi 9469 df-dju 9860 df-card 9898 df-acn 9901 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-xnn0 12522 df-z 12536 df-dec 12656 df-uz 12800 df-q 12914 df-rp 12958 df-xneg 13078 df-xadd 13079 df-xmul 13080 df-ioo 13316 df-ioc 13317 df-ico 13318 df-icc 13319 df-fz 13475 df-fzo 13622 df-fl 13760 df-mod 13838 df-seq 13973 df-exp 14033 df-fac 14245 df-bc 14274 df-hash 14302 df-word 14485 df-concat 14542 df-s1 14567 df-shft 15039 df-cj 15071 df-re 15072 df-im 15073 df-sqrt 15207 df-abs 15208 df-limsup 15443 df-clim 15460 df-rlim 15461 df-sum 15659 df-ef 16039 df-sin 16041 df-cos 16042 df-pi 16044 df-dvds 16229 df-gcd 16471 df-prm 16648 df-phi 16742 df-pc 16814 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17186 df-ress 17207 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-rest 17391 df-topn 17392 df-0g 17410 df-gsum 17411 df-topgen 17412 df-pt 17413 df-prds 17416 df-xrs 17471 df-qtop 17476 df-imas 17477 df-qus 17478 df-xps 17479 df-mre 17553 df-mrc 17554 df-acs 17556 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18716 df-submnd 18717 df-grp 18874 df-minusg 18875 df-sbg 18876 df-mulg 19006 df-subg 19061 df-nsg 19062 df-eqg 19063 df-ghm 19151 df-gim 19197 df-ga 19228 df-cntz 19255 df-oppg 19284 df-od 19464 df-gex 19465 df-pgp 19466 df-lsm 19572 df-pj1 19573 df-cmn 19718 df-abl 19719 df-cyg 19814 df-dprd 19933 df-dpj 19934 df-mgp 20056 df-rng 20068 df-ur 20097 df-ring 20150 df-cring 20151 df-oppr 20252 df-dvdsr 20272 df-unit 20273 df-invr 20303 df-dvr 20316 df-rhm 20387 df-subrng 20461 df-subrg 20485 df-drng 20646 df-lmod 20774 df-lss 20844 df-lsp 20884 df-sra 21086 df-rgmod 21087 df-lidl 21124 df-rsp 21125 df-2idl 21166 df-psmet 21262 df-xmet 21263 df-met 21264 df-bl 21265 df-mopn 21266 df-fbas 21267 df-fg 21268 df-cnfld 21271 df-zring 21363 df-zrh 21419 df-zn 21422 df-top 22787 df-topon 22804 df-topsp 22826 df-bases 22839 df-cld 22912 df-ntr 22913 df-cls 22914 df-nei 22991 df-lp 23029 df-perf 23030 df-cn 23120 df-cnp 23121 df-haus 23208 df-tx 23455 df-hmeo 23648 df-fil 23739 df-fm 23831 df-flim 23832 df-flf 23833 df-xms 24214 df-ms 24215 df-tms 24216 df-cncf 24777 df-0p 25577 df-limc 25773 df-dv 25774 df-ply 26099 df-idp 26100 df-coe 26101 df-dgr 26102 df-quot 26205 df-log 26471 df-cxp 26472 df-dchr 27150

This theorem is referenced by: rpvmasum2 27429

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