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Theorem sum2dchr 27242

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 2736	. . 3 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
5		sum2dchr.b	. . 3 ⊢ 𝐵 = (Base‘𝑍)
6		sum2dchr.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
7	6	nnnn0d 12567	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
8	3	zncrng 21510	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
9		crngring 20210	. . . . 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 2736	. . . . 5 ⊢ (/_r‘𝑍) = (/_r‘𝑍)
15	5, 13, 14	dvrcl 20369	. . . 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 27240	. 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘(𝐴(/_r‘𝑍)𝐶)) = if((𝐴(/_r‘𝑍)𝐶) = (1_r‘𝑍), (ϕ‘𝑁), 0))
18		eqid 2736	. . . . . . . 8 ⊢ (._r‘𝑍) = (._r‘𝑍)
19		eqid 2736	. . . . . . . 8 ⊢ (inv_r‘𝑍) = (inv_r‘𝑍)
20	5, 18, 13, 19, 14	dvrval 20368	. . . . . . 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 6885	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘(𝐴(/_r‘𝑍)𝐶)) = (𝑥‘(𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐶))))
24	1, 3, 2	dchrmhm 27209	. . . . . 6 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
25		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
26	24, 25	sselid 3961	. . . . 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 20341	. . . . . 6 ⊢ 𝑈 ⊆ 𝐵
29	13, 19	unitinvcl 20355	. . . . . . . 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 3961	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((inv_r‘𝑍)‘𝐶) ∈ 𝐵)
33		eqid 2736	. . . . . . 7 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
34	33, 5	mgpbas 20110	. . . . . 6 ⊢ 𝐵 = (Base‘(mulGrp‘𝑍))
35	33, 18	mgpplusg 20109	. . . . . 6 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
36		eqid 2736	. . . . . . 7 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
37		cnfldmul 21328	. . . . . . 7 ⊢ · = (._r‘ℂ_fld)
38	36, 37	mgpplusg 20109	. . . . . 6 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
39	34, 35, 38	mhmlin 18776	. . . . 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 2736	. . . . . . . 8 ⊢ ((mulGrp‘𝑍) ↾_s 𝑈) = ((mulGrp‘𝑍) ↾_s 𝑈)
42		eqid 2736	. . . . . . . 8 ⊢ ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})) = ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))
43	1, 3, 2, 13, 41, 42, 25	dchrghm 27224	. . . . . . 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 20347	. . . . . . . 8 ⊢ 𝑈 = (Base‘((mulGrp‘𝑍) ↾_s 𝑈))
46	13, 41, 19	invrfval 20354	. . . . . . . 8 ⊢ (inv_r‘𝑍) = (inv_g‘((mulGrp‘𝑍) ↾_s 𝑈))
47		cnfldbas 21324	. . . . . . . . . 10 ⊢ ℂ = (Base‘ℂ_fld)
48		cnfld0 21360	. . . . . . . . . 10 ⊢ 0 = (0_g‘ℂ_fld)
49		cndrng 21366	. . . . . . . . . 10 ⊢ ℂ_fld ∈ DivRing
50	47, 48, 49	drngui 20700	. . . . . . . . 9 ⊢ (ℂ ∖ {0}) = (Unit‘ℂ_fld)
51		eqid 2736	. . . . . . . . 9 ⊢ (inv_r‘ℂ_fld) = (inv_r‘ℂ_fld)
52	50, 42, 51	invrfval 20354	. . . . . . . 8 ⊢ (inv_r‘ℂ_fld) = (inv_g‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})))
53	45, 46, 52	ghminv 19211	. . . . . . 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 6901	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥 ↾ 𝑈)‘((inv_r‘𝑍)‘𝐶)) = (𝑥‘((inv_r‘𝑍)‘𝐶)))
56	44	fvresd 6901	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥 ↾ 𝑈)‘𝐶) = (𝑥‘𝐶))
57	56	fveq2d 6885	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((inv_r‘ℂ_fld)‘((𝑥 ↾ 𝑈)‘𝐶)) = ((inv_r‘ℂ_fld)‘(𝑥‘𝐶)))
58	1, 3, 2, 5, 25	dchrf 27210	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐵⟶ℂ)
59	28, 44	sselid 3961	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐶 ∈ 𝐵)
60	58, 59	ffvelcdmd 7080	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐶) ∈ ℂ)
61	1, 3, 2, 5, 13, 25, 59	dchrn0 27218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥‘𝐶) ≠ 0 ↔ 𝐶 ∈ 𝑈))
62	44, 61	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐶) ≠ 0)
63		cnfldinv 21370	. . . . . . . 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 15346	. . . . . . . . 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 27228	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (abs‘(𝑥‘𝐶)) = 1)
68	67	oveq1d 7425	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((abs‘(𝑥‘𝐶))↑2) = (1↑2))
69		sq1 14218	. . . . . . . . . 10 ⊢ (1↑2) = 1
70	68, 69	eqtrdi 2787	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((abs‘(𝑥‘𝐶))↑2) = 1)
71	70	oveq2d 7426	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((∗‘(𝑥‘𝐶)) / ((abs‘(𝑥‘𝐶))↑2)) = ((∗‘(𝑥‘𝐶)) / 1))
72	60	cjcld 15220	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (∗‘(𝑥‘𝐶)) ∈ ℂ)
73	72	div1d 12014	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((∗‘(𝑥‘𝐶)) / 1) = (∗‘(𝑥‘𝐶)))
74	66, 71, 73	3eqtrd 2775	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (1 / (𝑥‘𝐶)) = (∗‘(𝑥‘𝐶)))
75	57, 64, 74	3eqtrd 2775	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((inv_r‘ℂ_fld)‘((𝑥 ↾ 𝑈)‘𝐶)) = (∗‘(𝑥‘𝐶)))
76	54, 55, 75	3eqtr3d 2779	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘((inv_r‘𝑍)‘𝐶)) = (∗‘(𝑥‘𝐶)))
77	76	oveq2d 7426	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥‘𝐴) · (𝑥‘((inv_r‘𝑍)‘𝐶))) = ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶))))
78	23, 40, 77	3eqtrd 2775	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘(𝐴(/_r‘𝑍)𝐶)) = ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶))))
79	78	sumeq2dv 15723	. 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘(𝐴(/_r‘𝑍)𝐶)) = Σ𝑥 ∈ 𝐷 ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶))))
80	5, 13, 14, 4	dvreq1 20376	. . . 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 4529	. 2 ⊢ (𝜑 → if((𝐴(/_r‘𝑍)𝐶) = (1_r‘𝑍), (ϕ‘𝑁), 0) = if(𝐴 = 𝐶, (ϕ‘𝑁), 0))
83	17, 79, 82	3eqtr3d 2779	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 2933 ∖ cdif 3928 ifcif 4505 {csn 4606 ↾ cres 5661 ‘cfv 6536 (class class class)co 7410 ℂcc 11132 0cc0 11134 1c1 11135 · cmul 11139 / cdiv 11899 ℕcn 12245 2c2 12300 ℕ₀cn0 12506 ↑cexp 14084 ∗ccj 15120 abscabs 15258 Σcsu 15707 ϕcphi 16788 Basecbs 17233 ↾_s cress 17256 ._rcmulr 17277 MndHom cmhm 18764 GrpHom cghm 19200 mulGrpcmgp 20105 1_rcur 20146 Ringcrg 20198 CRingccrg 20199 Unitcui 20320 inv_rcinvr 20352 /_rcdvr 20365 ℂ_fldccnfld 21320 ℤ/nℤczn 21468 DChrcdchr 27200

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213 ax-mulf 11214

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-disj 5092 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-rpss 7722 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-omul 8490 df-er 8724 df-ec 8726 df-qs 8730 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-dju 9920 df-card 9958 df-acn 9961 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-xnn0 12580 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-fac 14297 df-bc 14326 df-hash 14354 df-word 14537 df-concat 14594 df-s1 14619 df-shft 15091 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-limsup 15492 df-clim 15509 df-rlim 15510 df-sum 15708 df-ef 16088 df-sin 16090 df-cos 16091 df-pi 16093 df-dvds 16278 df-gcd 16519 df-prm 16696 df-phi 16790 df-pc 16862 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17521 df-qtop 17526 df-imas 17527 df-qus 17528 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-mhm 18766 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mulg 19056 df-subg 19111 df-nsg 19112 df-eqg 19113 df-ghm 19201 df-gim 19247 df-ga 19278 df-cntz 19305 df-oppg 19334 df-od 19514 df-gex 19515 df-pgp 19516 df-lsm 19622 df-pj1 19623 df-cmn 19768 df-abl 19769 df-cyg 19864 df-dprd 19983 df-dpj 19984 df-mgp 20106 df-rng 20118 df-ur 20147 df-ring 20200 df-cring 20201 df-oppr 20302 df-dvdsr 20322 df-unit 20323 df-invr 20353 df-dvr 20366 df-rhm 20437 df-subrng 20511 df-subrg 20535 df-drng 20696 df-lmod 20824 df-lss 20894 df-lsp 20934 df-sra 21136 df-rgmod 21137 df-lidl 21174 df-rsp 21175 df-2idl 21216 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-fbas 21317 df-fg 21318 df-cnfld 21321 df-zring 21413 df-zrh 21469 df-zn 21472 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-haus 23258 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-0p 25628 df-limc 25824 df-dv 25825 df-ply 26150 df-idp 26151 df-coe 26152 df-dgr 26153 df-quot 26256 df-log 26522 df-cxp 26523 df-dchr 27201

This theorem is referenced by: rpvmasum2 27480

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