Metamath Proof Explorer


Theorem topgrpplusg

Description: The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015)

Ref Expression
Hypothesis topgrpfn.w
|- W = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( TopSet ` ndx ) , J >. }
Assertion topgrpplusg
|- ( .+ e. X -> .+ = ( +g ` W ) )

Proof

Step Hyp Ref Expression
1 topgrpfn.w
 |-  W = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( TopSet ` ndx ) , J >. }
2 1 topgrpstr
 |-  W Struct <. 1 , 9 >.
3 plusgid
 |-  +g = Slot ( +g ` ndx )
4 snsstp2
 |-  { <. ( +g ` ndx ) , .+ >. } C_ { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( TopSet ` ndx ) , J >. }
5 4 1 sseqtrri
 |-  { <. ( +g ` ndx ) , .+ >. } C_ W
6 2 3 5 strfv
 |-  ( .+ e. X -> .+ = ( +g ` W ) )