Theorem tlmlmod 22904

Description: A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)

Assertion

Ref	Expression
tlmlmod	⊢ (𝑊 ∈ TopMod → 𝑊 ∈ LMod)

Proof of Theorem tlmlmod

Step	Hyp	Ref	Expression
1		eqid 2759	. . . 4 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊)
2		eqid 2759	. . . 4 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊)
3		eqid 2759	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
4		eqid 2759	. . . 4 ⊢ (TopOpen‘(Scalar‘𝑊)) = (TopOpen‘(Scalar‘𝑊))
5	1, 2, 3, 4	istlm 22900	. . 3 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing) ∧ ( ·sf ‘𝑊) ∈ (((TopOpen‘(Scalar‘𝑊)) ×t (TopOpen‘𝑊)) Cn (TopOpen‘𝑊))))
6	5	simplbi 501	. 2 ⊢ (𝑊 ∈ TopMod → (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing))
7	6	simp2d 1141	1 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ LMod)

Syntax hints:   → wi 4   ∧ w3a 1085   ∈ wcel 2112  ‘cfv 6341  (class class class)co 7157  Scalarcsca 16641  TopOpenctopn 16768  LModclmod 19717   ·_sf cscaf 19718   Cn ccn 21939   ×_t ctx 22275  TopMndctmd 22785  TopRingctrg 22871  TopModctlm 22873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-rab 3080  df-v 3412  df-un 3866  df-in 3868  df-ss 3878  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-br 5038  df-iota 6300  df-fv 6349  df-ov 7160  df-tlm 22877

This theorem is referenced by:  tlmtgp  22911  tvclmod  22913