Theorem tgptsmscls 23301

Description: A sum in a topological group is uniquely determined up to a coset of cls({0}), which is a normal subgroup by clsnsg 23261, 0nsg 18797. (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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ TopGrp)
3		tgpgrp 23229	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
4	2, 3	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Grp)
5		eqid 2738	. . . . . . . . . . 11 ⊢ (0g‘𝐺) = (0g‘𝐺)
6	5	0subg 18780	. . . . . . . . . 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 23260	. . . . . . . . 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 2738	. . . . . . . . 9 ⊢ (𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)})) = (𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))
13	11, 12	eqger 18806	. . . . . . . 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 23231	. . . . . . . . . . 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 23286	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵)
21	20	sselda 3921	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ 𝐵)
22		tgptsmscls.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹))
23	20, 22	sseldd 3922	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
24	23	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋 ∈ 𝐵)
25		eqid 2738	. . . . . . . . . 10 ⊢ (-g‘𝐺) = (-g‘𝐺)
26	15	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ CMnd)
27	18	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐴 ∈ 𝑉)
28	19	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐹:𝐴⟶𝐵)
29	22	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋 ∈ (𝐺 tsums 𝐹))
30		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ (𝐺 tsums 𝐹))
31	11, 25, 26, 2, 27, 28, 28, 29, 30	tsmssub 23300	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑋(-g‘𝐺)𝑥) ∈ (𝐺 tsums (𝐹 ∘f (-g‘𝐺)𝐹)))
32	28	ffvelrnda 6961	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ 𝐵)
33	28	feqmptd 6837	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
34	27, 32, 32, 33, 33	offval2 7553	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐹 ∘f (-g‘𝐺)𝐹) = (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(-g‘𝐺)(𝐹‘𝑘))))
35	4	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘 ∈ 𝐴) → 𝐺 ∈ Grp)
36	11, 5, 25	grpsubid 18659	. . . . . . . . . . . . . 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 5174	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)(-g‘𝐺)(𝐹‘𝑘))) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))
39	34, 38	eqtrd 2778	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐹 ∘f (-g‘𝐺)𝐹) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))
40	39	oveq2d 7291	. . . . . . . . . 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 18607	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵)
43	4, 42	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (0g‘𝐺) ∈ 𝐵)
44	43	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘 ∈ 𝐴) → (0g‘𝐺) ∈ 𝐵)
45	44	fmpttd 6989	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)):𝐴⟶𝐵)
46		fconstmpt 5649	. . . . . . . . . . . 12 ⊢ (𝐴 × {(0g‘𝐺)}) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐺))
47		fvexd 6789	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝐺) ∈ V)
48	18, 47	fczfsuppd 9146	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 × {(0g‘𝐺)}) finSupp (0g‘𝐺))
49	48	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐴 × {(0g‘𝐺)}) finSupp (0g‘𝐺))
50	46, 49	eqbrtrrid 5110	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)) finSupp (0g‘𝐺))
51	11, 5, 26, 41, 27, 45, 50, 8	tsmsgsum 23290	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝑘 ∈ 𝐴 ↦ (0g‘𝐺))) = ((cls‘𝐽)‘{(𝐺 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))}))
52		cmnmnd 19402	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
53	26, 52	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Mnd)
54	5	gsumz 18474	. . . . . . . . . . . . 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 4573	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → {(𝐺 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))} = {(0g‘𝐺)})
57	56	fveq2d 6778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{(𝐺 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐺)))}) = ((cls‘𝐽)‘{(0g‘𝐺)}))
58	40, 51, 57	3eqtrd 2782	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝐹 ∘f (-g‘𝐺)𝐹)) = ((cls‘𝐽)‘{(0g‘𝐺)}))
59	31, 58	eleqtrd 2841	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑋(-g‘𝐺)𝑥) ∈ ((cls‘𝐽)‘{(0g‘𝐺)}))
60		isabl 19390	. . . . . . . . . 10 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
61	4, 26, 60	sylanbrc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Abel)
62	11	subgss 18756	. . . . . . . . . 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 19436	. . . . . . . . 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 1341	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥(𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))𝑋)
67	14, 66	ersym 8510	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋(𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))𝑥)
68	12	releqg 18803	. . . . . . 7 ⊢ Rel (𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)}))
69		relelec 8543	. . . . . . 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 233	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g‘𝐺)})))
72		eqid 2738	. . . . . . 7 ⊢ ((cls‘𝐽)‘{(0g‘𝐺)}) = ((cls‘𝐽)‘{(0g‘𝐺)})
73	11, 8, 5, 12, 72	snclseqg 23267	. . . . . 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 2841	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ ((cls‘𝐽)‘{𝑋}))
76	75	ex 413	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘{𝑋})))
77	76	ssrdv 3927	. 2 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ ((cls‘𝐽)‘{𝑋}))
78	11, 8, 15, 17, 18, 19, 22	tsmscls 23289	. 2 ⊢ (𝜑 → ((cls‘𝐽)‘{𝑋}) ⊆ (𝐺 tsums 𝐹))
79	77, 78	eqssd 3938	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑋}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ⊆ wss 3887  {csn 4561   class class class wbr 5074   ↦ cmpt 5157   × cxp 5587  Rel wrel 5594  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∘_f cof 7531   Er wer 8495  [cec 8496   finSupp cfsupp 9128  Basecbs 16912  TopOpenctopn 17132  0_gc0g 17150   Σ_g cgsu 17151  Mndcmnd 18385  Grpcgrp 18577  -_gcsg 18579  SubGrpcsubg 18749   ~_QG cqg 18751  CMndccmn 19386  Abelcabl 19387  TopSpctps 22081  clsccl 22169  TopGrpctgp 23222   tsums ctsu 23277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-ec 8500  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-seq 13722  df-hash 14045  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-0g 17152  df-gsum 17153  df-topgen 17154  df-plusf 18325  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-submnd 18431  df-grp 18580  df-minusg 18581  df-sbg 18582  df-subg 18752  df-eqg 18754  df-ghm 18832  df-cntz 18923  df-cmn 19388  df-abl 19389  df-fbas 20594  df-fg 20595  df-top 22043  df-topon 22060  df-topsp 22082  df-bases 22096  df-cld 22170  df-ntr 22171  df-cls 22172  df-nei 22249  df-cn 22378  df-cnp 22379  df-tx 22713  df-hmeo 22906  df-fil 22997  df-fm 23089  df-flim 23090  df-flf 23091  df-tmd 23223  df-tgp 23224  df-tsms 23278

This theorem is referenced by:  tgptsmscld  23302