Theorem tlmlmod 24237

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 2761	. . . 4 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊)
2		eqid 2761	. . . 4 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊)
3		eqid 2761	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
4		eqid 2761	. . . 4 ⊢ (TopOpen‘(Scalar‘𝑊)) = (TopOpen‘(Scalar‘𝑊))
5	1, 2, 3, 4	istlm 24233	. . 3 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing) ∧ ( ·sf ‘𝑊) ∈ (((TopOpen‘(Scalar‘𝑊)) ×t (TopOpen‘𝑊)) Cn (TopOpen‘𝑊))))
6	5	simplbi 500	. 2 ⊢ (𝑊 ∈ TopMod → (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing))
7	6	simp2d 1155	1 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ LMod)

Syntax hints:   → wi 4   ∧ w3a 1097   ∈ wcel 2141  ‘cfv 6516  (class class class)co 7391  Scalarcsca 17280  TopOpenctopn 17441  LModclmod 20915   ·_sf cscaf 20916   Cn ccn 23272   ×_t ctx 23608  TopMndctmd 24118  TopRingctrg 24204  TopModctlm 24206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6472  df-fv 6524  df-ov 7394  df-tlm 24210

This theorem is referenced by:  tlmtgp  24244  tvclmod  24246