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Theorem sum2dchr 27338

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 2762	. . 3 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
5		sum2dchr.b	. . 3 ⊢ 𝐵 = (Base‘𝑍)
6		sum2dchr.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
7	6	nnnn0d 12542	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
8	3	zncrng 21596	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
9		crngring 20295	. . . . 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 2762	. . . . 5 ⊢ (/_r‘𝑍) = (/_r‘𝑍)
15	5, 13, 14	dvrcl 20453	. . . 4 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝑈) → (𝐴(/_r‘𝑍)𝐶) ∈ 𝐵)
16	10, 11, 12, 15	syl3anc 1390	. . 3 ⊢ (𝜑 → (𝐴(/_r‘𝑍)𝐶) ∈ 𝐵)
17	1, 2, 3, 4, 5, 6, 16	sumdchr 27336	. 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘(𝐴(/_r‘𝑍)𝐶)) = if((𝐴(/_r‘𝑍)𝐶) = (1_r‘𝑍), (ϕ‘𝑁), 0))
18		eqid 2762	. . . . . . . 8 ⊢ (._r‘𝑍) = (._r‘𝑍)
19		eqid 2762	. . . . . . . 8 ⊢ (inv_r‘𝑍) = (inv_r‘𝑍)
20	5, 18, 13, 19, 14	dvrval 20452	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝑈) → (𝐴(/_r‘𝑍)𝐶) = (𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐶)))
21	11, 12, 20	syl2anc 593	. . . . . 6 ⊢ (𝜑 → (𝐴(/_r‘𝑍)𝐶) = (𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐶)))
22	21	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴(/_r‘𝑍)𝐶) = (𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐶)))
23	22	fveq2d 6871	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘(𝐴(/_r‘𝑍)𝐶)) = (𝑥‘(𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐶))))
24	1, 3, 2	dchrmhm 27305	. . . . . 6 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
25		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
26	24, 25	sselid 3934	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
27	11	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝐵)
28	5, 13	unitss 20425	. . . . . 6 ⊢ 𝑈 ⊆ 𝐵
29	13, 19	unitinvcl 20439	. . . . . . . 8 ⊢ ((𝑍 ∈ Ring ∧ 𝐶 ∈ 𝑈) → ((inv_r‘𝑍)‘𝐶) ∈ 𝑈)
30	10, 12, 29	syl2anc 593	. . . . . . 7 ⊢ (𝜑 → ((inv_r‘𝑍)‘𝐶) ∈ 𝑈)
31	30	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((inv_r‘𝑍)‘𝐶) ∈ 𝑈)
32	28, 31	sselid 3934	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((inv_r‘𝑍)‘𝐶) ∈ 𝐵)
33		eqid 2762	. . . . . . 7 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
34	33, 5	mgpbas 20191	. . . . . 6 ⊢ 𝐵 = (Base‘(mulGrp‘𝑍))
35	33, 18	mgpplusg 20190	. . . . . 6 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
36		eqid 2762	. . . . . . 7 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
37		cnfldmul 21432	. . . . . . 7 ⊢ · = (._r‘ℂ_fld)
38	36, 37	mgpplusg 20190	. . . . . 6 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
39	34, 35, 38	mhmlin 18827	. . . . 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 1390	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘(𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐶))) = ((𝑥‘𝐴) · (𝑥‘((inv_r‘𝑍)‘𝐶))))
41		eqid 2762	. . . . . . . 8 ⊢ ((mulGrp‘𝑍) ↾_s 𝑈) = ((mulGrp‘𝑍) ↾_s 𝑈)
42		eqid 2762	. . . . . . . 8 ⊢ ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})) = ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))
43	1, 3, 2, 13, 41, 42, 25	dchrghm 27320	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ↾ 𝑈) ∈ (((mulGrp‘𝑍) ↾_s 𝑈) GrpHom ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))))
44	12	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐶 ∈ 𝑈)
45	13, 41	unitgrpbas 20431	. . . . . . . 8 ⊢ 𝑈 = (Base‘((mulGrp‘𝑍) ↾_s 𝑈))
46	13, 41, 19	invrfval 20438	. . . . . . . 8 ⊢ (inv_r‘𝑍) = (inv_g‘((mulGrp‘𝑍) ↾_s 𝑈))
47		cnfldbas 21428	. . . . . . . . . 10 ⊢ ℂ = (Base‘ℂ_fld)
48		cnfld0 21448	. . . . . . . . . 10 ⊢ 0 = (0_g‘ℂ_fld)
49		cndrng 21453	. . . . . . . . . 10 ⊢ ℂ_fld ∈ DivRing
50	47, 48, 49	drngui 20785	. . . . . . . . 9 ⊢ (ℂ ∖ {0}) = (Unit‘ℂ_fld)
51		eqid 2762	. . . . . . . . 9 ⊢ (inv_r‘ℂ_fld) = (inv_r‘ℂ_fld)
52	50, 42, 51	invrfval 20438	. . . . . . . 8 ⊢ (inv_r‘ℂ_fld) = (inv_g‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})))
53	45, 46, 52	ghminv 19263	. . . . . . 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 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥 ↾ 𝑈)‘((inv_r‘𝑍)‘𝐶)) = ((inv_r‘ℂ_fld)‘((𝑥 ↾ 𝑈)‘𝐶)))
55	31	fvresd 6887	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥 ↾ 𝑈)‘((inv_r‘𝑍)‘𝐶)) = (𝑥‘((inv_r‘𝑍)‘𝐶)))
56	44	fvresd 6887	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥 ↾ 𝑈)‘𝐶) = (𝑥‘𝐶))
57	56	fveq2d 6871	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((inv_r‘ℂ_fld)‘((𝑥 ↾ 𝑈)‘𝐶)) = ((inv_r‘ℂ_fld)‘(𝑥‘𝐶)))
58	1, 3, 2, 5, 25	dchrf 27306	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐵⟶ℂ)
59	28, 44	sselid 3934	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐶 ∈ 𝐵)
60	58, 59	ffvelcdmd 7066	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐶) ∈ ℂ)
61	1, 3, 2, 5, 13, 25, 59	dchrn0 27314	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥‘𝐶) ≠ 0 ↔ 𝐶 ∈ 𝑈))
62	44, 61	mpbird 259	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐶) ≠ 0)
63		cnfldinv 21455	. . . . . . . 8 ⊢ (((𝑥‘𝐶) ∈ ℂ ∧ (𝑥‘𝐶) ≠ 0) → ((inv_r‘ℂ_fld)‘(𝑥‘𝐶)) = (1 / (𝑥‘𝐶)))
64	60, 62, 63	syl2anc 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((inv_r‘ℂ_fld)‘(𝑥‘𝐶)) = (1 / (𝑥‘𝐶)))
65		recval 15350	. . . . . . . . 9 ⊢ (((𝑥‘𝐶) ∈ ℂ ∧ (𝑥‘𝐶) ≠ 0) → (1 / (𝑥‘𝐶)) = ((∗‘(𝑥‘𝐶)) / ((abs‘(𝑥‘𝐶))↑2)))
66	60, 62, 65	syl2anc 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (1 / (𝑥‘𝐶)) = ((∗‘(𝑥‘𝐶)) / ((abs‘(𝑥‘𝐶))↑2)))
67	1, 2, 25, 3, 13, 44	dchrabs 27324	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (abs‘(𝑥‘𝐶)) = 1)
68	67	oveq1d 7411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((abs‘(𝑥‘𝐶))↑2) = (1↑2))
69		sq1 14208	. . . . . . . . . 10 ⊢ (1↑2) = 1
70	68, 69	eqtrdi 2813	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((abs‘(𝑥‘𝐶))↑2) = 1)
71	70	oveq2d 7412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((∗‘(𝑥‘𝐶)) / ((abs‘(𝑥‘𝐶))↑2)) = ((∗‘(𝑥‘𝐶)) / 1))
72	60	cjcld 15223	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (∗‘(𝑥‘𝐶)) ∈ ℂ)
73	72	div1d 11959	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((∗‘(𝑥‘𝐶)) / 1) = (∗‘(𝑥‘𝐶)))
74	66, 71, 73	3eqtrd 2801	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (1 / (𝑥‘𝐶)) = (∗‘(𝑥‘𝐶)))
75	57, 64, 74	3eqtrd 2801	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((inv_r‘ℂ_fld)‘((𝑥 ↾ 𝑈)‘𝐶)) = (∗‘(𝑥‘𝐶)))
76	54, 55, 75	3eqtr3d 2805	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘((inv_r‘𝑍)‘𝐶)) = (∗‘(𝑥‘𝐶)))
77	76	oveq2d 7412	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥‘𝐴) · (𝑥‘((inv_r‘𝑍)‘𝐶))) = ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶))))
78	23, 40, 77	3eqtrd 2801	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘(𝐴(/_r‘𝑍)𝐶)) = ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶))))
79	78	sumeq2dv 15729	. 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘(𝐴(/_r‘𝑍)𝐶)) = Σ𝑥 ∈ 𝐷 ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶))))
80	5, 13, 14, 4	dvreq1 20460	. . . 4 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝑈) → ((𝐴(/_r‘𝑍)𝐶) = (1_r‘𝑍) ↔ 𝐴 = 𝐶))
81	10, 11, 12, 80	syl3anc 1390	. . 3 ⊢ (𝜑 → ((𝐴(/_r‘𝑍)𝐶) = (1_r‘𝑍) ↔ 𝐴 = 𝐶))
82	81	ifbid 4504	. 2 ⊢ (𝜑 → if((𝐴(/_r‘𝑍)𝐶) = (1_r‘𝑍), (ϕ‘𝑁), 0) = if(𝐴 = 𝐶, (ϕ‘𝑁), 0))
83	17, 79, 82	3eqtr3d 2805	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 208 ∧ wa 399 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 ∖ cdif 3901 ifcif 4480 {csn 4582 ↾ cres 5649 ‘cfv 6521 (class class class)co 7396 ℂcc 11071 0cc0 11073 1c1 11074 · cmul 11078 / cdiv 11844 ℕcn 12210 2c2 12272 ℕ₀cn0 12481 ↑cexp 14074 ∗ccj 15123 abscabs 15261 Σcsu 15713 ϕcphi 16799 Basecbs 17245 ↾_s cress 17266 ._rcmulr 17287 MndHom cmhm 18815 GrpHom cghm 19253 mulGrpcmgp 20186 1_rcur 20231 Ringcrg 20283 CRingccrg 20284 Unitcui 20404 inv_rcinvr 20436 /_rcdvr 20449 ℂ_fldccnfld 21424 ℤ/nℤczn 21554 DChrcdchr 27296

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-inf2 9596 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151 ax-addf 11152 ax-mulf 11153

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-iin 4952 df-disj 5068 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-rpss 7706 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-tpos 8206 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-omul 8442 df-er 8678 df-ec 8680 df-qs 8684 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9308 df-fi 9357 df-sup 9388 df-inf 9389 df-oi 9458 df-dju 9859 df-card 9897 df-acn 9900 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-xnn0 12555 df-z 12569 df-dec 12689 df-uz 12840 df-q 12950 df-rp 12994 df-xneg 13114 df-xadd 13115 df-xmul 13116 df-ioo 13353 df-ioc 13354 df-ico 13355 df-icc 13356 df-fz 13513 df-fzo 13660 df-fl 13802 df-mod 13880 df-seq 14015 df-exp 14075 df-fac 14287 df-bc 14316 df-hash 14344 df-word 14527 df-concat 14584 df-s1 14610 df-shft 15080 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-limsup 15498 df-clim 15515 df-rlim 15516 df-sum 15714 df-ef 16097 df-sin 16099 df-cos 16100 df-pi 16102 df-dvds 16287 df-gcd 16529 df-prm 16706 df-phi 16801 df-pc 16873 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-starv 17301 df-sca 17302 df-vsca 17303 df-ip 17304 df-tset 17305 df-ple 17306 df-ds 17308 df-unif 17309 df-hom 17310 df-cco 17311 df-rest 17451 df-topn 17452 df-0g 17470 df-gsum 17471 df-topgen 17472 df-pt 17473 df-prds 17476 df-xrs 17532 df-qtop 17537 df-imas 17538 df-qus 17539 df-xps 17540 df-mre 17614 df-mrc 17615 df-acs 17617 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-mhm 18817 df-submnd 18818 df-grp 18978 df-minusg 18979 df-sbg 18980 df-mulg 19110 df-subg 19165 df-nsg 19166 df-eqg 19167 df-ghm 19254 df-gim 19299 df-ga 19330 df-cntz 19357 df-oppg 19386 df-od 19568 df-gex 19569 df-pgp 19570 df-lsm 19676 df-pj1 19677 df-cmn 19822 df-abl 19823 df-cyg 19918 df-dprd 20037 df-dpj 20038 df-mgp 20187 df-rng 20199 df-ur 20232 df-ring 20285 df-cring 20286 df-oppr 20386 df-dvdsr 20406 df-unit 20407 df-invr 20437 df-dvr 20450 df-rhm 20521 df-subrng 20596 df-subrg 20620 df-drng 20781 df-lmod 20929 df-lss 20999 df-lsp 21039 df-sra 21240 df-rgmod 21241 df-lidl 21278 df-rsp 21279 df-2idl 21320 df-psmet 21416 df-xmet 21417 df-met 21418 df-bl 21419 df-mopn 21420 df-fbas 21421 df-fg 21422 df-cnfld 21425 df-zring 21499 df-zrh 21555 df-zn 21558 df-top 22954 df-topon 22971 df-topsp 22993 df-bases 23006 df-cld 23079 df-ntr 23080 df-cls 23081 df-nei 23158 df-lp 23196 df-perf 23197 df-cn 23287 df-cnp 23288 df-haus 23375 df-tx 23622 df-hmeo 23815 df-fil 23906 df-fm 23998 df-flim 23999 df-flf 24000 df-xms 24380 df-ms 24381 df-tms 24382 df-cncf 24940 df-0p 25732 df-limc 25928 df-dv 25929 df-ply 26248 df-idp 26249 df-coe 26250 df-dgr 26251 df-quot 26355 df-log 26621 df-cxp 26622 df-dchr 27297

This theorem is referenced by: rpvmasum2 27576

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