Definition df-tlm 23313

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 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}

Detailed syntax breakdown of Definition df-tlm

Step	Hyp	Ref	Expression
1		ctlm 23309	. 2 class TopMod
2		vw	. . . . . . 7 setvar 𝑤
3	2	cv 1538	. . . . . 6 class 𝑤
4		csca 16965	. . . . . 6 class Scalar
5	3, 4	cfv 6433	. . . . 5 class (Scalar‘𝑤)
6		ctrg 23307	. . . . 5 class TopRing
7	5, 6	wcel 2106	. . . 4 wff (Scalar‘𝑤) ∈ TopRing
8		cscaf 20124	. . . . . 6 class ·sf
9	3, 8	cfv 6433	. . . . 5 class ( ·sf ‘𝑤)
10		ctopn 17132	. . . . . . . 8 class TopOpen
11	5, 10	cfv 6433	. . . . . . 7 class (TopOpen‘(Scalar‘𝑤))
12	3, 10	cfv 6433	. . . . . . 7 class (TopOpen‘𝑤)
13		ctx 22711	. . . . . . 7 class ×t
14	11, 12, 13	co 7275	. . . . . 6 class ((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤))
15		ccn 22375	. . . . . 6 class Cn
16	14, 12, 15	co 7275	. . . . 5 class (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤))
17	9, 16	wcel 2106	. . . 4 wff ( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤))
18	7, 17	wa 396	. . 3 wff ((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))
19		ctmd 23221	. . . 4 class TopMnd
20		clmod 20123	. . . 4 class LMod
21	19, 20	cin 3886	. . 3 class (TopMnd ∩ LMod)
22	18, 2, 21	crab 3068	. 2 class {𝑤 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}
23	1, 22	wceq 1539	1 wff TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}

This definition is referenced by:  istlm  23336