Definition df-tlm 21965

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 /\ ( .sf ` w ) e. ( ( ( TopOpen ` (Scalar ` w ) ) tX ( TopOpen ` w ) ) Cn ( TopOpen ` w ) ) ) }

Detailed syntax breakdown of Definition df-tlm

Step	Hyp	Ref	Expression
1		ctlm 21961	. 2 class TopMod
2		vw	. . . . . . 7 setvar w
3	2	cv 1482	. . . . . 6 class w
4		csca 15944	. . . . . 6 class Scalar
5	3, 4	cfv 5888	. . . . 5 class (Scalar ` w )
6		ctrg 21959	. . . . 5 class TopRing
7	5, 6	wcel 1990	. . . 4 wff (Scalar ` w ) e. TopRing
8		cscaf 18864	. . . . . 6 class .sf
9	3, 8	cfv 5888	. . . . 5 class ( .sf ` w )
10		ctopn 16082	. . . . . . . 8 class TopOpen
11	5, 10	cfv 5888	. . . . . . 7 class ( TopOpen ` (Scalar ` w ) )
12	3, 10	cfv 5888	. . . . . . 7 class ( TopOpen ` w )
13		ctx 21363	. . . . . . 7 class tX
14	11, 12, 13	co 6650	. . . . . 6 class ( ( TopOpen ` (Scalar ` w ) ) tX ( TopOpen ` w ) )
15		ccn 21028	. . . . . 6 class Cn
16	14, 12, 15	co 6650	. . . . 5 class ( ( ( TopOpen ` (Scalar ` w ) ) tX ( TopOpen ` w ) ) Cn ( TopOpen ` w ) )
17	9, 16	wcel 1990	. . . 4 wff ( .sf ` w ) e. ( ( ( TopOpen ` (Scalar ` w ) ) tX ( TopOpen ` w ) ) Cn ( TopOpen ` w ) )
18	7, 17	wa 384	. . 3 wff ( (Scalar ` w ) e. TopRing /\ ( .sf ` w ) e. ( ( ( TopOpen ` (Scalar ` w ) ) tX ( TopOpen ` w ) ) Cn ( TopOpen ` w ) ) )
19		ctmd 21874	. . . 4 class TopMnd
20		clmod 18863	. . . 4 class LMod
21	19, 20	cin 3573	. . 3 class (TopMnd i^i LMod )
22	18, 2, 21	crab 2916	. 2 class { w e. (TopMnd i^i LMod ) | ( (Scalar ` w ) e. TopRing /\ ( .sf ` w ) e. ( ( ( TopOpen ` (Scalar ` w ) ) tX ( TopOpen ` w ) ) Cn ( TopOpen ` w ) ) ) }
23	1, 22	wceq 1483	1 wff TopMod = { w e. (TopMnd i^i LMod ) | ( (Scalar ` w ) e. TopRing /\ ( .sf ` w ) e. ( ( ( TopOpen ` (Scalar ` w ) ) tX ( TopOpen ` w ) ) Cn ( TopOpen ` w ) ) ) }

This definition is referenced by:  istlm  21988