Theorem tlmlmod 24213

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 2735	. . . 4 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊)
2		eqid 2735	. . . 4 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊)
3		eqid 2735	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
4		eqid 2735	. . . 4 ⊢ (TopOpen‘(Scalar‘𝑊)) = (TopOpen‘(Scalar‘𝑊))
5	1, 2, 3, 4	istlm 24209	. . 3 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing) ∧ ( ·sf ‘𝑊) ∈ (((TopOpen‘(Scalar‘𝑊)) ×t (TopOpen‘𝑊)) Cn (TopOpen‘𝑊))))
6	5	simplbi 497	. 2 ⊢ (𝑊 ∈ TopMod → (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing))
7	6	simp2d 1142	1 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ LMod)

Syntax hints:   → wi 4   ∧ w3a 1086   ∈ wcel 2106  ‘cfv 6563  (class class class)co 7431  Scalarcsca 17301  TopOpenctopn 17468  LModclmod 20875   ·_sf cscaf 20876   Cn ccn 23248   ×_t ctx 23584  TopMndctmd 24094  TopRingctrg 24180  TopModctlm 24182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-tlm 24186

This theorem is referenced by:  tlmtgp  24220  tvclmod  24222