Theorem tgptsmscls 24276

Description: A sum in a topological group is uniquely determined up to a coset of cls({0}), which is a normal subgroup by clsnsg 24236, 0nsg 19235. (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 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ TopGrp)
3		tgpgrp 24204	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
4	2, 3	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Grp)
5		eqid 2769	. . . . . . . . . . 11 ⊢ (0g‘𝐺) = (0g‘𝐺)
6	5	0subg 19218	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → {(0g‘𝐺)} ∈ (SubGrp‘𝐺))
7	4, 6	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → {(0g‘𝐺)} ∈ (SubGrp‘𝐺))
8		tgptsmscls.j	. . . . . . . . . 10 ⊢ 𝐽 = (TopOpen‘𝐺)
9	8	clssubg 24235	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ {(0g‘𝐺)} ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘{(0g‘𝐺)}) ∈ (SubGrp‘𝐺))
10	2, 7, 9	syl2anc 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{(0g‘𝐺)}) ∈ (SubGrp‘𝐺))
11		tgptsmscls.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
12		eqid 2769	. . . . . . . . 9 ⊢ (𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)})) = (𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))
13	11, 12	eqger 19246	. . . . . . . 8 ⊢ (((cls‘𝐽)‘{(0g‘𝐺)}) ∈ (SubGrp‘𝐺) → (𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)})) Er 𝐵)
14	10, 13	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)})) Er 𝐵)
15		tgptsmscls.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ CMnd)
16		tgptps 24206	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
17	1, 16	syl 18	. . . . . . . . . 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 24261	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵)
21	20	sselda 3945	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ 𝐵)
22		tgptsmscls.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹))
23	20, 22	sseldd 3946	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
24	23	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋 ∈ 𝐵)
25		eqid 2769	. . . . . . . . . 10 ⊢ (-g‘𝐺) = (-g‘𝐺)
26	15	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ CMnd)
27	18	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐴 ∈ 𝑉)
28	19	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐹:𝐴⟶𝐵)
29	22	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋 ∈ (𝐺 tsums 𝐹))
30		simpr 489	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ (𝐺 tsums 𝐹))
31	11, 25, 26, 2, 27, 28, 28, 29, 30	tsmssub 24275	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑋(-g‘𝐺)𝑥) ∈ (𝐺 tsums (𝐹 ∘f (-g‘𝐺)𝐹)))
32	28	ffvelcdmda 7080	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ 𝐵)
33	28	feqmptd 6950	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
34	27, 32, 32, 33, 33	offval2 7695	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐹 ∘f (-g‘𝐺)𝐹) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(-g‘𝐺)(𝐹‘𝑘))))
35	4	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘 ∈ 𝐴) → 𝐺 ∈ Grp)
36	11, 5, 25	grpsubid 19090	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑘) ∈ 𝐵) → ((𝐹‘𝑘)(-g‘𝐺)(𝐹‘𝑘)) = (0g‘𝐺))
37	35, 32, 36	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘 ∈ 𝐴) → ((𝐹‘𝑘)(-g‘𝐺)(𝐹‘𝑘)) = (0g‘𝐺))
38	37	mpteq2dva 5208	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(-g‘𝐺)(𝐹‘𝑘))) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))
39	34, 38	eqtrd 2804	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐹 ∘f (-g‘𝐺)𝐹) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))
40	39	oveq2d 7427	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝐹 ∘f (-g‘𝐺)𝐹)) = (𝐺 tsums (𝑘 ∈ 𝐴 ↦ (0g‘𝐺))))
41	2, 16	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ TopSp)
42	11, 5	grpidcl 19032	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵)
43	4, 42	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (0g‘𝐺) ∈ 𝐵)
44	43	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘 ∈ 𝐴) → (0g‘𝐺) ∈ 𝐵)
45	44	fmpttd 7111	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)):𝐴⟶𝐵)
46		fconstmpt 5724	. . . . . . . . . . . 12 ⊢ (𝐴 × {(0g‘𝐺)}) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐺))
47		fvexd 6897	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝐺) ∈ V)
48	18, 47	fczfsuppd 9346	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 × {(0g‘𝐺)}) finSupp (0g‘𝐺))
49	48	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐴 × {(0g‘𝐺)}) finSupp (0g‘𝐺))
50	46, 49	eqbrtrrid 5151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)) finSupp (0g‘𝐺))
51	11, 5, 26, 41, 27, 45, 50, 8	tsmsgsum 24265	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝑘 ∈ 𝐴 ↦ (0g‘𝐺))) = ((cls‘𝐽)‘{(𝐺 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))}))
52		cmnmnd 19867	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
53	26, 52	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Mnd)
54	5	gsumz 18895	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐺))) = (0g‘𝐺))
55	53, 27, 54	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐺))) = (0g‘𝐺))
56	55	sneqd 4606	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → {(𝐺 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))} = {(0g‘𝐺)})
57	56	fveq2d 6886	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{(𝐺 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))}) = ((cls‘𝐽)‘{(0g‘𝐺)}))
58	40, 51, 57	3eqtrd 2808	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝐹 ∘f (-g‘𝐺)𝐹)) = ((cls‘𝐽)‘{(0g‘𝐺)}))
59	31, 58	eleqtrd 2871	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑋(-g‘𝐺)𝑥) ∈ ((cls‘𝐽)‘{(0g‘𝐺)}))
60		isabl 19854	. . . . . . . . . 10 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
61	4, 26, 60	sylanbrc 594	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Abel)
62	11	subgss 19193	. . . . . . . . . 10 ⊢ (((cls‘𝐽)‘{(0g‘𝐺)}) ∈ (SubGrp‘𝐺) → ((cls‘𝐽)‘{(0g‘𝐺)}) ⊆ 𝐵)
63	10, 62	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{(0g‘𝐺)}) ⊆ 𝐵)
64	11, 25, 12	eqgabl 19904	. . . . . . . . 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 595	. . . . . . . 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 1359	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥(𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))𝑋)
67	14, 66	ersym 8707	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋(𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))𝑥)
68	12	releqg 19241	. . . . . . 7 ⊢ Rel (𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))
69		relelec 8742	. . . . . . 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 237	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)})))
72		eqid 2769	. . . . . . 7 ⊢ ((cls‘𝐽)‘{(0g‘𝐺)}) = ((cls‘𝐽)‘{(0g‘𝐺)})
73	11, 8, 5, 12, 72	snclseqg 24242	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑋 ∈ 𝐵) → [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)})) = ((cls‘𝐽)‘{𝑋}))
74	2, 24, 73	syl2anc 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)})) = ((cls‘𝐽)‘{𝑋}))
75	71, 74	eleqtrd 2871	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ ((cls‘𝐽)‘{𝑋}))
76	75	ex 417	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘{𝑋})))
77	76	ssrdv 3951	. 2 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ ((cls‘𝐽)‘{𝑋}))
78	11, 8, 15, 17, 18, 19, 22	tsmscls 24264	. 2 ⊢ (𝜑 → ((cls‘𝐽)‘{𝑋}) ⊆ (𝐺 tsums 𝐹))
79	77, 78	eqssd 3962	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑋}))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ⊆ wss 3913  {csn 4594   class class class wbr 5113   ↦ cmpt 5196   × cxp 5660  Rel wrel 5667  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ∘_f cof 7673   Er wer 8691  [cec 8692   finSupp cfsupp 9321  Basecbs 17269  TopOpenctopn 17474  0_gc0g 17492   Σ_g cgsu 17493  Mndcmnd 18792  Grpcgrp 19000  -_gcsg 19002  SubGrpcsubg 19186   ~_QG cqg 19188  CMndccmn 19850  Abelcabl 19851  TopSpctps 23058  clsccl 23144  TopGrpctgp 24197   tsums ctsu 24252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7675  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8157  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-ec 8696  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fsupp 9322  df-oi 9472  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-n0 12505  df-z 12592  df-uz 12863  df-fz 13536  df-fzo 13683  df-seq 14038  df-hash 14367  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-ress 17291  df-plusg 17323  df-0g 17494  df-gsum 17495  df-topgen 17496  df-plusf 18697  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-mhm 18841  df-submnd 18842  df-grp 19003  df-minusg 19004  df-sbg 19005  df-subg 19189  df-eqg 19191  df-ghm 19284  df-cntz 19387  df-cmn 19852  df-abl 19853  df-fbas 21488  df-fg 21489  df-top 23020  df-topon 23037  df-topsp 23059  df-bases 23072  df-cld 23145  df-ntr 23146  df-cls 23147  df-nei 23224  df-cn 23353  df-cnp 23354  df-tx 23688  df-hmeo 23881  df-fil 23972  df-fm 24064  df-flim 24065  df-flf 24066  df-tmd 24198  df-tgp 24199  df-tsms 24253

This theorem is referenced by:  tgptsmscld  24277