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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 e. X |-> ( A .+ x ) )
tgplacthmeo.2 
|- X = ( Base ` G )
tgplacthmeo.3 
|- .+ = ( +g ` G )
tgplacthmeo.4 
|- J = ( TopOpen ` G )
Assertion tmdlactcn 
|- ( ( G e. TopMnd /\ A e. X ) -> F e. ( J Cn J ) )

Proof

Step Hyp Ref Expression
1 tgplacthmeo.1 
 |-  F = ( x e. X |-> ( A .+ x ) )
2 tgplacthmeo.2 
 |-  X = ( Base ` G )
3 tgplacthmeo.3 
 |-  .+ = ( +g ` G )
4 tgplacthmeo.4 
 |-  J = ( TopOpen ` G )
5 simpl 
 |-  ( ( G e. TopMnd /\ A e. X ) -> G e. TopMnd )
6 4 2 tmdtopon 
 |-  ( G e. TopMnd -> J e. ( TopOn ` X ) )
7 6 adantr 
 |-  ( ( G e. TopMnd /\ A e. X ) -> J e. ( TopOn ` X ) )
8 simpr 
 |-  ( ( G e. TopMnd /\ A e. X ) -> A e. X )
9 7 7 8 cnmptc 
 |-  ( ( G e. TopMnd /\ A e. X ) -> ( x e. X |-> A ) e. ( J Cn J ) )
10 7 cnmptid 
 |-  ( ( G e. TopMnd /\ A e. X ) -> ( x e. X |-> x ) e. ( J Cn J ) )
11 4 3 5 7 9 10 cnmpt1plusg 
 |-  ( ( G e. TopMnd /\ A e. X ) -> ( x e. X |-> ( A .+ x ) ) e. ( J Cn J ) )
12 1 11 eqeltrid 
 |-  ( ( G e. TopMnd /\ A e. X ) -> F e. ( J Cn J ) )