Theorem tgptsmscls 24070

Description: A sum in a topological group is uniquely determined up to a coset of cls({0}), which is a normal subgroup by clsnsg 24030, 0nsg 19083. (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 23998	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
4	2, 3	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Grp)
5		eqid 2729	. . . . . . . . . . 11 ⊢ (0g‘𝐺) = (0g‘𝐺)
6	5	0subg 19065	. . . . . . . . . 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 24029	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ {(0g‘𝐺)} ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘{(0g‘𝐺)}) ∈ (SubGrp‘𝐺))
10	2, 7, 9	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{(0g‘𝐺)}) ∈ (SubGrp‘𝐺))
11		tgptsmscls.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
12		eqid 2729	. . . . . . . . 9 ⊢ (𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)})) = (𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))
13	11, 12	eqger 19092	. . . . . . . 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 24000	. . . . . . . . . . 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 24055	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵)
21	20	sselda 3943	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ 𝐵)
22		tgptsmscls.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹))
23	20, 22	sseldd 3944	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
24	23	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋 ∈ 𝐵)
25		eqid 2729	. . . . . . . . . 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 24069	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑋(-g‘𝐺)𝑥) ∈ (𝐺 tsums (𝐹 ∘f (-g‘𝐺)𝐹)))
32	28	ffvelcdmda 7038	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ 𝐵)
33	28	feqmptd 6911	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
34	27, 32, 32, 33, 33	offval2 7653	. . . . . . . . . . . 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 18938	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑘) ∈ 𝐵) → ((𝐹‘𝑘)(-g‘𝐺)(𝐹‘𝑘)) = (0g‘𝐺))
37	35, 32, 36	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘 ∈ 𝐴) → ((𝐹‘𝑘)(-g‘𝐺)(𝐹‘𝑘)) = (0g‘𝐺))
38	37	mpteq2dva 5195	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(-g‘𝐺)(𝐹‘𝑘))) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))
39	34, 38	eqtrd 2764	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐹 ∘f (-g‘𝐺)𝐹) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))
40	39	oveq2d 7385	. . . . . . . . . 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 18879	. . . . . . . . . . . . . 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 7069	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)):𝐴⟶𝐵)
46		fconstmpt 5693	. . . . . . . . . . . 12 ⊢ (𝐴 × {(0g‘𝐺)}) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐺))
47		fvexd 6855	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝐺) ∈ V)
48	18, 47	fczfsuppd 9313	. . . . . . . . . . . . 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 5138	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)) finSupp (0g‘𝐺))
51	11, 5, 26, 41, 27, 45, 50, 8	tsmsgsum 24059	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝑘 ∈ 𝐴 ↦ (0g‘𝐺))) = ((cls‘𝐽)‘{(𝐺 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))}))
52		cmnmnd 19711	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
53	26, 52	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Mnd)
54	5	gsumz 18745	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐺))) = (0g‘𝐺))
55	53, 27, 54	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐺))) = (0g‘𝐺))
56	55	sneqd 4597	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → {(𝐺 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))} = {(0g‘𝐺)})
57	56	fveq2d 6844	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{(𝐺 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))}) = ((cls‘𝐽)‘{(0g‘𝐺)}))
58	40, 51, 57	3eqtrd 2768	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝐹 ∘f (-g‘𝐺)𝐹)) = ((cls‘𝐽)‘{(0g‘𝐺)}))
59	31, 58	eleqtrd 2830	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑋(-g‘𝐺)𝑥) ∈ ((cls‘𝐽)‘{(0g‘𝐺)}))
60		isabl 19698	. . . . . . . . . 10 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
61	4, 26, 60	sylanbrc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Abel)
62	11	subgss 19041	. . . . . . . . . 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 19748	. . . . . . . . 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 584	. . . . . . . 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 1343	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥(𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))𝑋)
67	14, 66	ersym 8660	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋(𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))𝑥)
68	12	releqg 19089	. . . . . . 7 ⊢ Rel (𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))
69		relelec 8695	. . . . . . 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 2729	. . . . . . 7 ⊢ ((cls‘𝐽)‘{(0g‘𝐺)}) = ((cls‘𝐽)‘{(0g‘𝐺)})
73	11, 8, 5, 12, 72	snclseqg 24036	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑋 ∈ 𝐵) → [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)})) = ((cls‘𝐽)‘{𝑋}))
74	2, 24, 73	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)})) = ((cls‘𝐽)‘{𝑋}))
75	71, 74	eleqtrd 2830	. . . 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 3949	. 2 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ ((cls‘𝐽)‘{𝑋}))
78	11, 8, 15, 17, 18, 19, 22	tsmscls 24058	. 2 ⊢ (𝜑 → ((cls‘𝐽)‘{𝑋}) ⊆ (𝐺 tsums 𝐹))
79	77, 78	eqssd 3961	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑋}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ⊆ wss 3911  {csn 4585   class class class wbr 5102   ↦ cmpt 5183   × cxp 5629  Rel wrel 5636  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369   ∘_f cof 7631   Er wer 8645  [cec 8646   finSupp cfsupp 9288  Basecbs 17155  TopOpenctopn 17360  0_gc0g 17378   Σ_g cgsu 17379  Mndcmnd 18643  Grpcgrp 18847  -_gcsg 18849  SubGrpcsubg 19034   ~_QG cqg 19036  CMndccmn 19694  Abelcabl 19695  TopSpctps 22852  clsccl 22938  TopGrpctgp 23991   tsums ctsu 24046

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-om 7823  df-1st 7947  df-2nd 7948  df-supp 8117  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-ec 8650  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9289  df-oi 9439  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-n0 12419  df-z 12506  df-uz 12770  df-fz 13445  df-fzo 13592  df-seq 13943  df-hash 14272  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-0g 17380  df-gsum 17381  df-topgen 17382  df-plusf 18548  df-mgm 18549  df-sgrp 18628  df-mnd 18644  df-mhm 18692  df-submnd 18693  df-grp 18850  df-minusg 18851  df-sbg 18852  df-subg 19037  df-eqg 19039  df-ghm 19127  df-cntz 19231  df-cmn 19696  df-abl 19697  df-fbas 21293  df-fg 21294  df-top 22814  df-topon 22831  df-topsp 22853  df-bases 22866  df-cld 22939  df-ntr 22940  df-cls 22941  df-nei 23018  df-cn 23147  df-cnp 23148  df-tx 23482  df-hmeo 23675  df-fil 23766  df-fm 23858  df-flim 23859  df-flf 23860  df-tmd 23992  df-tgp 23993  df-tsms 24047

This theorem is referenced by:  tgptsmscld  24071