Theorem tgptsmscls 24111

Description: A sum in a topological group is uniquely determined up to a coset of cls({0}), which is a normal subgroup by clsnsg 24071, 0nsg 19115. (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 24039	. . . . . . . . . . 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 19098	. . . . . . . . . 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 24070	. . . . . . . . 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 19124	. . . . . . . 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 24041	. . . . . . . . . . 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 24096	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵)
21	20	sselda 3935	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ 𝐵)
22		tgptsmscls.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹))
23	20, 22	sseldd 3936	. . . . . . . . 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 24110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑋(-g‘𝐺)𝑥) ∈ (𝐺 tsums (𝐹 ∘f (-g‘𝐺)𝐹)))
32	28	ffvelcdmda 7040	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ 𝐵)
33	28	feqmptd 6912	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
34	27, 32, 32, 33, 33	offval2 7654	. . . . . . . . . . . 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 18971	. . . . . . . . . . . . . 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 5193	. . . . . . . . . . . 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 7386	. . . . . . . . . 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 18912	. . . . . . . . . . . . . 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 7071	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)):𝐴⟶𝐵)
46		fconstmpt 5696	. . . . . . . . . . . 12 ⊢ (𝐴 × {(0g‘𝐺)}) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐺))
47		fvexd 6859	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝐺) ∈ V)
48	18, 47	fczfsuppd 9303	. . . . . . . . . . . . 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 5136	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)) finSupp (0g‘𝐺))
51	11, 5, 26, 41, 27, 45, 50, 8	tsmsgsum 24100	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝑘 ∈ 𝐴 ↦ (0g‘𝐺))) = ((cls‘𝐽)‘{(𝐺 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))}))
52		cmnmnd 19743	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
53	26, 52	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Mnd)
54	5	gsumz 18775	. . . . . . . . . . . . 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 4594	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → {(𝐺 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))} = {(0g‘𝐺)})
57	56	fveq2d 6848	. . . . . . . . . 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 19730	. . . . . . . . . 10 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
61	4, 26, 60	sylanbrc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Abel)
62	11	subgss 19074	. . . . . . . . . 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 19780	. . . . . . . . 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 8660	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋(𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))𝑥)
68	12	releqg 19121	. . . . . . 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 2737	. . . . . . 7 ⊢ ((cls‘𝐽)‘{(0g‘𝐺)}) = ((cls‘𝐽)‘{(0g‘𝐺)})
73	11, 8, 5, 12, 72	snclseqg 24077	. . . . . 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 3941	. 2 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ ((cls‘𝐽)‘{𝑋}))
78	11, 8, 15, 17, 18, 19, 22	tsmscls 24099	. 2 ⊢ (𝜑 → ((cls‘𝐽)‘{𝑋}) ⊆ (𝐺 tsums 𝐹))
79	77, 78	eqssd 3953	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑋}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ⊆ wss 3903  {csn 4582   class class class wbr 5100   ↦ cmpt 5181   × cxp 5632  Rel wrel 5639  ⟶wf 6498  ‘cfv 6502  (class class class)co 7370   ∘_f cof 7632   Er wer 8644  [cec 8645   finSupp cfsupp 9278  Basecbs 17150  TopOpenctopn 17355  0_gc0g 17373   Σ_g cgsu 17374  Mndcmnd 18673  Grpcgrp 18880  -_gcsg 18882  SubGrpcsubg 19067   ~_QG cqg 19069  CMndccmn 19726  Abelcabl 19727  TopSpctps 22893  clsccl 22979  TopGrpctgp 24032   tsums ctsu 24087

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 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

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 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-se 5588  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-isom 6511  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-of 7634  df-om 7821  df-1st 7945  df-2nd 7946  df-supp 8115  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-er 8647  df-ec 8649  df-map 8779  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-fsupp 9279  df-oi 9429  df-card 9865  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-2 12222  df-n0 12416  df-z 12503  df-uz 12766  df-fz 13438  df-fzo 13585  df-seq 13939  df-hash 14268  df-sets 17105  df-slot 17123  df-ndx 17135  df-base 17151  df-ress 17172  df-plusg 17204  df-0g 17375  df-gsum 17376  df-topgen 17377  df-plusf 18578  df-mgm 18579  df-sgrp 18658  df-mnd 18674  df-mhm 18722  df-submnd 18723  df-grp 18883  df-minusg 18884  df-sbg 18885  df-subg 19070  df-eqg 19072  df-ghm 19159  df-cntz 19263  df-cmn 19728  df-abl 19729  df-fbas 21323  df-fg 21324  df-top 22855  df-topon 22872  df-topsp 22894  df-bases 22907  df-cld 22980  df-ntr 22981  df-cls 22982  df-nei 23059  df-cn 23188  df-cnp 23189  df-tx 23523  df-hmeo 23716  df-fil 23807  df-fm 23899  df-flim 23900  df-flf 23901  df-tmd 24033  df-tgp 24034  df-tsms 24088

This theorem is referenced by:  tgptsmscld  24112