sum2dchr - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > sum2dchr		Structured version Visualization version GIF version

Theorem sum2dchr 27243

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 12464	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
8	3	zncrng 21501	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
9		crngring 20182	. . . . 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 20342	. . . 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 27241	. 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 20341	. . . . . . 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 6838	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘(𝐴(/_r‘𝑍)𝐶)) = (𝑥‘(𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐶))))
24	1, 3, 2	dchrmhm 27210	. . . . . 6 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
25		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
26	24, 25	sselid 3931	. . . . 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 20314	. . . . . 6 ⊢ 𝑈 ⊆ 𝐵
29	13, 19	unitinvcl 20328	. . . . . . . 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 3931	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((inv_r‘𝑍)‘𝐶) ∈ 𝐵)
33		eqid 2736	. . . . . . 7 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
34	33, 5	mgpbas 20082	. . . . . 6 ⊢ 𝐵 = (Base‘(mulGrp‘𝑍))
35	33, 18	mgpplusg 20081	. . . . . 6 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
36		eqid 2736	. . . . . . 7 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
37		cnfldmul 21319	. . . . . . 7 ⊢ · = (._r‘ℂ_fld)
38	36, 37	mgpplusg 20081	. . . . . 6 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
39	34, 35, 38	mhmlin 18720	. . . . 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 27225	. . . . . . 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 20320	. . . . . . . 8 ⊢ 𝑈 = (Base‘((mulGrp‘𝑍) ↾_s 𝑈))
46	13, 41, 19	invrfval 20327	. . . . . . . 8 ⊢ (inv_r‘𝑍) = (inv_g‘((mulGrp‘𝑍) ↾_s 𝑈))
47		cnfldbas 21315	. . . . . . . . . 10 ⊢ ℂ = (Base‘ℂ_fld)
48		cnfld0 21349	. . . . . . . . . 10 ⊢ 0 = (0_g‘ℂ_fld)
49		cndrng 21355	. . . . . . . . . 10 ⊢ ℂ_fld ∈ DivRing
50	47, 48, 49	drngui 20670	. . . . . . . . 9 ⊢ (ℂ ∖ {0}) = (Unit‘ℂ_fld)
51		eqid 2736	. . . . . . . . 9 ⊢ (inv_r‘ℂ_fld) = (inv_r‘ℂ_fld)
52	50, 42, 51	invrfval 20327	. . . . . . . 8 ⊢ (inv_r‘ℂ_fld) = (inv_g‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0})))
53	45, 46, 52	ghminv 19154	. . . . . . 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 6854	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥 ↾ 𝑈)‘((inv_r‘𝑍)‘𝐶)) = (𝑥‘((inv_r‘𝑍)‘𝐶)))
56	44	fvresd 6854	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥 ↾ 𝑈)‘𝐶) = (𝑥‘𝐶))
57	56	fveq2d 6838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((inv_r‘ℂ_fld)‘((𝑥 ↾ 𝑈)‘𝐶)) = ((inv_r‘ℂ_fld)‘(𝑥‘𝐶)))
58	1, 3, 2, 5, 25	dchrf 27211	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐵⟶ℂ)
59	28, 44	sselid 3931	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐶 ∈ 𝐵)
60	58, 59	ffvelcdmd 7030	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐶) ∈ ℂ)
61	1, 3, 2, 5, 13, 25, 59	dchrn0 27219	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑥‘𝐶) ≠ 0 ↔ 𝐶 ∈ 𝑈))
62	44, 61	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐶) ≠ 0)
63		cnfldinv 21359	. . . . . . . 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 15248	. . . . . . . . 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 27229	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (abs‘(𝑥‘𝐶)) = 1)
68	67	oveq1d 7373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((abs‘(𝑥‘𝐶))↑2) = (1↑2))
69		sq1 14120	. . . . . . . . . 10 ⊢ (1↑2) = 1
70	68, 69	eqtrdi 2787	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((abs‘(𝑥‘𝐶))↑2) = 1)
71	70	oveq2d 7374	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((∗‘(𝑥‘𝐶)) / ((abs‘(𝑥‘𝐶))↑2)) = ((∗‘(𝑥‘𝐶)) / 1))
72	60	cjcld 15121	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (∗‘(𝑥‘𝐶)) ∈ ℂ)
73	72	div1d 11911	. . . . . . . 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 7374	. . . 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 15627	. 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘(𝐴(/_r‘𝑍)𝐶)) = Σ𝑥 ∈ 𝐷 ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶))))
80	5, 13, 14, 4	dvreq1 20349	. . . 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 4503	. 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 1541 ∈ wcel 2113 ≠ wne 2932 ∖ cdif 3898 ifcif 4479 {csn 4580 ↾ cres 5626 ‘cfv 6492 (class class class)co 7358 ℂcc 11026 0cc0 11028 1c1 11029 · cmul 11033 / cdiv 11796 ℕcn 12147 2c2 12202 ℕ₀cn0 12403 ↑cexp 13986 ∗ccj 15021 abscabs 15159 Σcsu 15611 ϕcphi 16693 Basecbs 17138 ↾_s cress 17159 ._rcmulr 17180 MndHom cmhm 18708 GrpHom cghm 19143 mulGrpcmgp 20077 1_rcur 20118 Ringcrg 20170 CRingccrg 20171 Unitcui 20293 inv_rcinvr 20325 /_rcdvr 20338 ℂ_fldccnfld 21311 ℤ/nℤczn 21459 DChrcdchr 27201

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9552 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107 ax-mulf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-disj 5066 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-rpss 7668 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-tpos 8168 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-omul 8402 df-er 8635 df-ec 8637 df-qs 8641 df-map 8767 df-pm 8768 df-ixp 8838 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fsupp 9267 df-fi 9316 df-sup 9347 df-inf 9348 df-oi 9417 df-dju 9815 df-card 9853 df-acn 9856 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-xnn0 12477 df-z 12491 df-dec 12610 df-uz 12754 df-q 12864 df-rp 12908 df-xneg 13028 df-xadd 13029 df-xmul 13030 df-ioo 13267 df-ioc 13268 df-ico 13269 df-icc 13270 df-fz 13426 df-fzo 13573 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-fac 14199 df-bc 14228 df-hash 14256 df-word 14439 df-concat 14496 df-s1 14522 df-shft 14992 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-limsup 15396 df-clim 15413 df-rlim 15414 df-sum 15612 df-ef 15992 df-sin 15994 df-cos 15995 df-pi 15997 df-dvds 16182 df-gcd 16424 df-prm 16601 df-phi 16695 df-pc 16767 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17425 df-qtop 17430 df-imas 17431 df-qus 17432 df-xps 17433 df-mre 17507 df-mrc 17508 df-acs 17510 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-nsg 19056 df-eqg 19057 df-ghm 19144 df-gim 19190 df-ga 19221 df-cntz 19248 df-oppg 19277 df-od 19459 df-gex 19460 df-pgp 19461 df-lsm 19567 df-pj1 19568 df-cmn 19713 df-abl 19714 df-cyg 19809 df-dprd 19928 df-dpj 19929 df-mgp 20078 df-rng 20090 df-ur 20119 df-ring 20172 df-cring 20173 df-oppr 20275 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-dvr 20339 df-rhm 20410 df-subrng 20481 df-subrg 20505 df-drng 20666 df-lmod 20815 df-lss 20885 df-lsp 20925 df-sra 21127 df-rgmod 21128 df-lidl 21165 df-rsp 21166 df-2idl 21207 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-fbas 21308 df-fg 21309 df-cnfld 21312 df-zring 21404 df-zrh 21460 df-zn 21463 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22892 df-cld 22965 df-ntr 22966 df-cls 22967 df-nei 23044 df-lp 23082 df-perf 23083 df-cn 23173 df-cnp 23174 df-haus 23261 df-tx 23508 df-hmeo 23701 df-fil 23792 df-fm 23884 df-flim 23885 df-flf 23886 df-xms 24266 df-ms 24267 df-tms 24268 df-cncf 24829 df-0p 25629 df-limc 25825 df-dv 25826 df-ply 26151 df-idp 26152 df-coe 26153 df-dgr 26154 df-quot 26257 df-log 26523 df-cxp 26524 df-dchr 27202

This theorem is referenced by: rpvmasum2 27481

W3C validator