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Metamath Proof Explorer

Theorem tmdlactcn

Description: The left group action of element A in a topological monoid G is a continuous function. (Contributed by FL, 18-Mar-2008) (Revised by Mario Carneiro, 14-Aug-2015)

Ref Expression
Hypotheses tgplacthmeo.1 ⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ ( 𝐴 + 𝑥 ) )
tgplacthmeo.2 ⊢ 𝑋 = ( Base ‘ 𝐺 )
tgplacthmeo.3 ⊢ + = ( +g ‘ 𝐺 )
tgplacthmeo.4 ⊢ 𝐽 = ( TopOpen ‘ 𝐺 )
Assertion tmdlactcn ( ( 𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋 ) → 𝐹 ∈ ( 𝐽 Cn 𝐽 ) )

Proof

Step Hyp Ref Expression
1 tgplacthmeo.1 ⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ ( 𝐴 + 𝑥 ) )
2 tgplacthmeo.2 ⊢ 𝑋 = ( Base ‘ 𝐺 )
3 tgplacthmeo.3 ⊢ + = ( +g ‘ 𝐺 )
4 tgplacthmeo.4 ⊢ 𝐽 = ( TopOpen ‘ 𝐺 )
5 simpl ⊢ ( ( 𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋 ) → 𝐺 ∈ TopMnd )
6 4 2 tmdtopon ⊢ ( 𝐺 ∈ TopMnd → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
7 6 adantr ⊢ ( ( 𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
8 simpr ⊢ ( ( 𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋 ) → 𝐴 ∈ 𝑋 )
9 7 7 8 cnmptc ⊢ ( ( 𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋 ) → ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ∈ ( 𝐽 Cn 𝐽 ) )
10 7 cnmptid ⊢ ( ( 𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋 ) → ( 𝑥 ∈ 𝑋 ↦ 𝑥 ) ∈ ( 𝐽 Cn 𝐽 ) )
11 4 3 5 7 9 10 cnmpt1plusg ⊢ ( ( 𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋 ) → ( 𝑥 ∈ 𝑋 ↦ ( 𝐴 + 𝑥 ) ) ∈ ( 𝐽 Cn 𝐽 ) )
12 1 11 eqeltrid ⊢ ( ( 𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋 ) → 𝐹 ∈ ( 𝐽 Cn 𝐽 ) )