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Definition df-tlm 18179
Description: Define a topological left module, which is just what its name suggests: instead of a group over a ring with a scalar product connecting them, it is a topological group over a topological ring with a continuous scalar product. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
df-tlm  |- TopMod  =  {
w  e.  (TopMnd  i^i  LMod )  |  ( (Scalar `  w )  e.  TopRing  /\  ( .s f `  w
)  e.  ( ( ( TopOpen `  (Scalar `  w
) )  tX  ( TopOpen
`  w ) )  Cn  ( TopOpen `  w
) ) ) }

Detailed syntax breakdown of Definition df-tlm
StepHypRef Expression
1 ctlm 18175 . 2  class TopMod
2 vw . . . . . . 7  set  w
32cv 1651 . . . . . 6  class  w
4 csca 13520 . . . . . 6  class Scalar
53, 4cfv 5445 . . . . 5  class  (Scalar `  w )
6 ctrg 18173 . . . . 5  class  TopRing
75, 6wcel 1725 . . . 4  wff  (Scalar `  w )  e.  TopRing
8 cscaf 15939 . . . . . 6  class  .s f
93, 8cfv 5445 . . . . 5  class  ( .s f `  w )
10 ctopn 13637 . . . . . . . 8  class  TopOpen
115, 10cfv 5445 . . . . . . 7  class  ( TopOpen `  (Scalar `  w ) )
123, 10cfv 5445 . . . . . . 7  class  ( TopOpen `  w )
13 ctx 17580 . . . . . . 7  class  tX
1411, 12, 13co 6072 . . . . . 6  class  ( (
TopOpen `  (Scalar `  w
) )  tX  ( TopOpen
`  w ) )
15 ccn 17276 . . . . . 6  class  Cn
1614, 12, 15co 6072 . . . . 5  class  ( ( ( TopOpen `  (Scalar `  w
) )  tX  ( TopOpen
`  w ) )  Cn  ( TopOpen `  w
) )
179, 16wcel 1725 . . . 4  wff  ( .s f `  w )  e.  ( ( (
TopOpen `  (Scalar `  w
) )  tX  ( TopOpen
`  w ) )  Cn  ( TopOpen `  w
) )
187, 17wa 359 . . 3  wff  ( (Scalar `  w )  e.  TopRing  /\  ( .s f `  w
)  e.  ( ( ( TopOpen `  (Scalar `  w
) )  tX  ( TopOpen
`  w ) )  Cn  ( TopOpen `  w
) ) )
19 ctmd 18088 . . . 4  class TopMnd
20 clmod 15938 . . . 4  class  LMod
2119, 20cin 3311 . . 3  class  (TopMnd  i^i  LMod )
2218, 2, 21crab 2701 . 2  class  { w  e.  (TopMnd  i^i  LMod )  |  ( (Scalar `  w
)  e.  TopRing  /\  ( .s f `  w )  e.  ( ( (
TopOpen `  (Scalar `  w
) )  tX  ( TopOpen
`  w ) )  Cn  ( TopOpen `  w
) ) ) }
231, 22wceq 1652 1  wff TopMod  =  {
w  e.  (TopMnd  i^i  LMod )  |  ( (Scalar `  w )  e.  TopRing  /\  ( .s f `  w
)  e.  ( ( ( TopOpen `  (Scalar `  w
) )  tX  ( TopOpen
`  w ) )  Cn  ( TopOpen `  w
) ) ) }
Colors of variables: wff set class
This definition is referenced by:  istlm  18202
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