Theorem tgptsmscls 24129

Description: A sum in a topological group is uniquely determined up to a coset of cls({0}), which is a normal subgroup by clsnsg 24089, 0nsg 19139. (Contributed by Mario Carneiro, 22-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)

Hypotheses

Ref	Expression
tgptsmscls.b	⊢ 𝐵 = (Base‘𝐺)
tgptsmscls.j	⊢ 𝐽 = (TopOpen‘𝐺)
tgptsmscls.1	⊢ (𝜑 → 𝐺 ∈ CMnd)
tgptsmscls.2	⊢ (𝜑 → 𝐺 ∈ TopGrp)
tgptsmscls.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
tgptsmscls.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
tgptsmscls.x	⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹))

Assertion

Ref	Expression
tgptsmscls	⊢ (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑋}))

Proof of Theorem tgptsmscls

Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgptsmscls.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ TopGrp)
2	1	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ TopGrp)
3		tgpgrp 24057	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
4	2, 3	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Grp)
5		eqid 2737	. . . . . . . . . . 11 ⊢ (0g‘𝐺) = (0g‘𝐺)
6	5	0subg 19122	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → {(0g‘𝐺)} ∈ (SubGrp‘𝐺))
7	4, 6	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → {(0g‘𝐺)} ∈ (SubGrp‘𝐺))
8		tgptsmscls.j	. . . . . . . . . 10 ⊢ 𝐽 = (TopOpen‘𝐺)
9	8	clssubg 24088	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ {(0g‘𝐺)} ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘{(0g‘𝐺)}) ∈ (SubGrp‘𝐺))
10	2, 7, 9	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{(0g‘𝐺)}) ∈ (SubGrp‘𝐺))
11		tgptsmscls.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
12		eqid 2737	. . . . . . . . 9 ⊢ (𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)})) = (𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))
13	11, 12	eqger 19148	. . . . . . . 8 ⊢ (((cls‘𝐽)‘{(0g‘𝐺)}) ∈ (SubGrp‘𝐺) → (𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)})) Er 𝐵)
14	10, 13	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)})) Er 𝐵)
15		tgptsmscls.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ CMnd)
16		tgptps 24059	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
17	1, 16	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ TopSp)
18		tgptsmscls.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
19		tgptsmscls.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
20	11, 15, 17, 18, 19	tsmscl 24114	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵)
21	20	sselda 3922	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ 𝐵)
22		tgptsmscls.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹))
23	20, 22	sseldd 3923	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
24	23	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋 ∈ 𝐵)
25		eqid 2737	. . . . . . . . . 10 ⊢ (-g‘𝐺) = (-g‘𝐺)
26	15	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ CMnd)
27	18	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐴 ∈ 𝑉)
28	19	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐹:𝐴⟶𝐵)
29	22	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋 ∈ (𝐺 tsums 𝐹))
30		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ (𝐺 tsums 𝐹))
31	11, 25, 26, 2, 27, 28, 28, 29, 30	tsmssub 24128	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑋(-g‘𝐺)𝑥) ∈ (𝐺 tsums (𝐹 ∘f (-g‘𝐺)𝐹)))
32	28	ffvelcdmda 7032	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ 𝐵)
33	28	feqmptd 6904	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
34	27, 32, 32, 33, 33	offval2 7646	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐹 ∘f (-g‘𝐺)𝐹) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(-g‘𝐺)(𝐹‘𝑘))))
35	4	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘 ∈ 𝐴) → 𝐺 ∈ Grp)
36	11, 5, 25	grpsubid 18995	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑘) ∈ 𝐵) → ((𝐹‘𝑘)(-g‘𝐺)(𝐹‘𝑘)) = (0g‘𝐺))
37	35, 32, 36	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘 ∈ 𝐴) → ((𝐹‘𝑘)(-g‘𝐺)(𝐹‘𝑘)) = (0g‘𝐺))
38	37	mpteq2dva 5179	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(-g‘𝐺)(𝐹‘𝑘))) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))
39	34, 38	eqtrd 2772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐹 ∘f (-g‘𝐺)𝐹) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))
40	39	oveq2d 7378	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝐹 ∘f (-g‘𝐺)𝐹)) = (𝐺 tsums (𝑘 ∈ 𝐴 ↦ (0g‘𝐺))))
41	2, 16	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ TopSp)
42	11, 5	grpidcl 18936	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵)
43	4, 42	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (0g‘𝐺) ∈ 𝐵)
44	43	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘 ∈ 𝐴) → (0g‘𝐺) ∈ 𝐵)
45	44	fmpttd 7063	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)):𝐴⟶𝐵)
46		fconstmpt 5688	. . . . . . . . . . . 12 ⊢ (𝐴 × {(0g‘𝐺)}) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐺))
47		fvexd 6851	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝐺) ∈ V)
48	18, 47	fczfsuppd 9294	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 × {(0g‘𝐺)}) finSupp (0g‘𝐺))
49	48	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐴 × {(0g‘𝐺)}) finSupp (0g‘𝐺))
50	46, 49	eqbrtrrid 5122	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)) finSupp (0g‘𝐺))
51	11, 5, 26, 41, 27, 45, 50, 8	tsmsgsum 24118	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝑘 ∈ 𝐴 ↦ (0g‘𝐺))) = ((cls‘𝐽)‘{(𝐺 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))}))
52		cmnmnd 19767	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
53	26, 52	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Mnd)
54	5	gsumz 18799	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐺))) = (0g‘𝐺))
55	53, 27, 54	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐺))) = (0g‘𝐺))
56	55	sneqd 4580	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → {(𝐺 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))} = {(0g‘𝐺)})
57	56	fveq2d 6840	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{(𝐺 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))}) = ((cls‘𝐽)‘{(0g‘𝐺)}))
58	40, 51, 57	3eqtrd 2776	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝐹 ∘f (-g‘𝐺)𝐹)) = ((cls‘𝐽)‘{(0g‘𝐺)}))
59	31, 58	eleqtrd 2839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑋(-g‘𝐺)𝑥) ∈ ((cls‘𝐽)‘{(0g‘𝐺)}))
60		isabl 19754	. . . . . . . . . 10 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
61	4, 26, 60	sylanbrc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Abel)
62	11	subgss 19098	. . . . . . . . . 10 ⊢ (((cls‘𝐽)‘{(0g‘𝐺)}) ∈ (SubGrp‘𝐺) → ((cls‘𝐽)‘{(0g‘𝐺)}) ⊆ 𝐵)
63	10, 62	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{(0g‘𝐺)}) ⊆ 𝐵)
64	11, 25, 12	eqgabl 19804	. . . . . . . . 9 ⊢ ((𝐺 ∈ Abel ∧ ((cls‘𝐽)‘{(0g‘𝐺)}) ⊆ 𝐵) → (𝑥(𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))𝑋 ↔ (𝑥 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ (𝑋(-g‘𝐺)𝑥) ∈ ((cls‘𝐽)‘{(0g‘𝐺)}))))
65	61, 63, 64	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑥(𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))𝑋 ↔ (𝑥 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ (𝑋(-g‘𝐺)𝑥) ∈ ((cls‘𝐽)‘{(0g‘𝐺)}))))
66	21, 24, 59, 65	mpbir3and 1344	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥(𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))𝑋)
67	14, 66	ersym 8651	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋(𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))𝑥)
68	12	releqg 19145	. . . . . . 7 ⊢ Rel (𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))
69		relelec 8686	. . . . . . 7 ⊢ (Rel (𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)})) → (𝑥 ∈ [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)})) ↔ 𝑋(𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))𝑥))
70	68, 69	ax-mp 5	. . . . . 6 ⊢ (𝑥 ∈ [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)})) ↔ 𝑋(𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))𝑥)
71	67, 70	sylibr 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)})))
72		eqid 2737	. . . . . . 7 ⊢ ((cls‘𝐽)‘{(0g‘𝐺)}) = ((cls‘𝐽)‘{(0g‘𝐺)})
73	11, 8, 5, 12, 72	snclseqg 24095	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑋 ∈ 𝐵) → [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)})) = ((cls‘𝐽)‘{𝑋}))
74	2, 24, 73	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)})) = ((cls‘𝐽)‘{𝑋}))
75	71, 74	eleqtrd 2839	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ ((cls‘𝐽)‘{𝑋}))
76	75	ex 412	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘{𝑋})))
77	76	ssrdv 3928	. 2 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ ((cls‘𝐽)‘{𝑋}))
78	11, 8, 15, 17, 18, 19, 22	tsmscls 24117	. 2 ⊢ (𝜑 → ((cls‘𝐽)‘{𝑋}) ⊆ (𝐺 tsums 𝐹))
79	77, 78	eqssd 3940	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑋}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ⊆ wss 3890  {csn 4568   class class class wbr 5086   ↦ cmpt 5167   × cxp 5624  Rel wrel 5631  ⟶wf 6490  ‘cfv 6494  (class class class)co 7362   ∘_f cof 7624   Er wer 8635  [cec 8636   finSupp cfsupp 9269  Basecbs 17174  TopOpenctopn 17379  0_gc0g 17397   Σ_g cgsu 17398  Mndcmnd 18697  Grpcgrp 18904  -_gcsg 18906  SubGrpcsubg 19091   ~_QG cqg 19093  CMndccmn 19750  Abelcabl 19751  TopSpctps 22911  clsccl 22997  TopGrpctgp 24050   tsums ctsu 24105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-se 5580  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-isom 6503  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7626  df-om 7813  df-1st 7937  df-2nd 7938  df-supp 8106  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-er 8638  df-ec 8640  df-map 8770  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-fsupp 9270  df-oi 9420  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-n0 12433  df-z 12520  df-uz 12784  df-fz 13457  df-fzo 13604  df-seq 13959  df-hash 14288  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-plusg 17228  df-0g 17399  df-gsum 17400  df-topgen 17401  df-plusf 18602  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-mhm 18746  df-submnd 18747  df-grp 18907  df-minusg 18908  df-sbg 18909  df-subg 19094  df-eqg 19096  df-ghm 19183  df-cntz 19287  df-cmn 19752  df-abl 19753  df-fbas 21345  df-fg 21346  df-top 22873  df-topon 22890  df-topsp 22912  df-bases 22925  df-cld 22998  df-ntr 22999  df-cls 23000  df-nei 23077  df-cn 23206  df-cnp 23207  df-tx 23541  df-hmeo 23734  df-fil 23825  df-fm 23917  df-flim 23918  df-flf 23919  df-tmd 24051  df-tgp 24052  df-tsms 24106

This theorem is referenced by:  tgptsmscld  24130