Description: A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | tgptmd | ⊢ ( 𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid | ⊢ ( TopOpen ‘ 𝐺 ) = ( TopOpen ‘ 𝐺 ) | |
2 | eqid | ⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 ) | |
3 | 1 2 | istgp | ⊢ ( 𝐺 ∈ TopGrp ↔ ( 𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ ( invg ‘ 𝐺 ) ∈ ( ( TopOpen ‘ 𝐺 ) Cn ( TopOpen ‘ 𝐺 ) ) ) ) |
4 | 3 | simp2bi | ⊢ ( 𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd ) |