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Metamath Proof Explorer

Theorem tgptsmscld

Description: The set of limit points to an infinite sum in a topological group is closed. (Contributed by Mario Carneiro, 22-Sep-2015)

Ref Expression
Hypotheses 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 )
Assertion tgptsmscld ( 𝜑 → ( 𝐺 tsums 𝐹 ) ∈ ( Clsd ‘ 𝐽 ) )

Proof

Step Hyp Ref Expression
1 tgptsmscls.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 tgptsmscls.j ⊢ 𝐽 = ( TopOpen ‘ 𝐺 )
3 tgptsmscls.1 ⊢ ( 𝜑 → 𝐺 ∈ CMnd )
4 tgptsmscls.2 ⊢ ( 𝜑 → 𝐺 ∈ TopGrp )
5 tgptsmscls.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
6 tgptsmscls.f ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
7 2 1 tgptopon ⊢ ( 𝐺 ∈ TopGrp → 𝐽 ∈ ( TopOn ‘ 𝐵 ) )
8 4 7 syl ⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝐵 ) )
9 topontop ⊢ ( 𝐽 ∈ ( TopOn ‘ 𝐵 ) → 𝐽 ∈ Top )
10 8 9 syl ⊢ ( 𝜑 → 𝐽 ∈ Top )
11 0cld ⊢ ( 𝐽 ∈ Top → ∅ ∈ ( Clsd ‘ 𝐽 ) )
12 10 11 syl ⊢ ( 𝜑 → ∅ ∈ ( Clsd ‘ 𝐽 ) )
13 eleq1 ⊢ ( ( 𝐺 tsums 𝐹 ) = ∅ → ( ( 𝐺 tsums 𝐹 ) ∈ ( Clsd ‘ 𝐽 ) ↔ ∅ ∈ ( Clsd ‘ 𝐽 ) ) )
14 12 13 syl5ibrcom ⊢ ( 𝜑 → ( ( 𝐺 tsums 𝐹 ) = ∅ → ( 𝐺 tsums 𝐹 ) ∈ ( Clsd ‘ 𝐽 ) ) )
15 n0 ⊢ ( ( 𝐺 tsums 𝐹 ) ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ ( 𝐺 tsums 𝐹 ) )
16 3 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐺 tsums 𝐹 ) ) → 𝐺 ∈ CMnd )
17 4 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐺 tsums 𝐹 ) ) → 𝐺 ∈ TopGrp )
18 5 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐺 tsums 𝐹 ) ) → 𝐴 ∈ 𝑉 )
19 6 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐺 tsums 𝐹 ) ) → 𝐹 : 𝐴 ⟶ 𝐵 )
20 simpr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐺 tsums 𝐹 ) ) → 𝑥 ∈ ( 𝐺 tsums 𝐹 ) )
21 1 2 16 17 18 19 20 tgptsmscls ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐺 tsums 𝐹 ) ) → ( 𝐺 tsums 𝐹 ) = ( ( cls ‘ 𝐽 ) ‘ { 𝑥 } ) )
22 tgptps ⊢ ( 𝐺 ∈ TopGrp → 𝐺 ∈ TopSp )
23 4 22 syl ⊢ ( 𝜑 → 𝐺 ∈ TopSp )
24 1 3 23 5 6 tsmscl ⊢ ( 𝜑 → ( 𝐺 tsums 𝐹 ) ⊆ 𝐵 )
25 toponuni ⊢ ( 𝐽 ∈ ( TopOn ‘ 𝐵 ) → 𝐵 = ∪ 𝐽 )
26 8 25 syl ⊢ ( 𝜑 → 𝐵 = ∪ 𝐽 )
27 24 26 sseqtrd ⊢ ( 𝜑 → ( 𝐺 tsums 𝐹 ) ⊆ ∪ 𝐽 )
28 27 sselda ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐺 tsums 𝐹 ) ) → 𝑥 ∈ ∪ 𝐽 )
29 28 snssd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐺 tsums 𝐹 ) ) → { 𝑥 } ⊆ ∪ 𝐽 )
30 eqid ⊢ ∪ 𝐽 = ∪ 𝐽
31 30 clscld ⊢ ( ( 𝐽 ∈ Top ∧ { 𝑥 } ⊆ ∪ 𝐽 ) → ( ( cls ‘ 𝐽 ) ‘ { 𝑥 } ) ∈ ( Clsd ‘ 𝐽 ) )
32 10 29 31 syl2an2r ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐺 tsums 𝐹 ) ) → ( ( cls ‘ 𝐽 ) ‘ { 𝑥 } ) ∈ ( Clsd ‘ 𝐽 ) )
33 21 32 eqeltrd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐺 tsums 𝐹 ) ) → ( 𝐺 tsums 𝐹 ) ∈ ( Clsd ‘ 𝐽 ) )
34 33 ex ⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐺 tsums 𝐹 ) → ( 𝐺 tsums 𝐹 ) ∈ ( Clsd ‘ 𝐽 ) ) )
35 34 exlimdv ⊢ ( 𝜑 → ( ∃ 𝑥 𝑥 ∈ ( 𝐺 tsums 𝐹 ) → ( 𝐺 tsums 𝐹 ) ∈ ( Clsd ‘ 𝐽 ) ) )
36 15 35 biimtrid ⊢ ( 𝜑 → ( ( 𝐺 tsums 𝐹 ) ≠ ∅ → ( 𝐺 tsums 𝐹 ) ∈ ( Clsd ‘ 𝐽 ) ) )
37 14 36 pm2.61dne ⊢ ( 𝜑 → ( 𝐺 tsums 𝐹 ) ∈ ( Clsd ‘ 𝐽 ) )