Theorem tvctlm 24226

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

Assertion

Ref	Expression
tvctlm	⊢ (𝑊 ∈ TopVec → 𝑊 ∈ TopMod)

Proof of Theorem tvctlm

Step	Hyp	Ref	Expression
1		eqid 2740	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
2	1	istvc 24221	. 2 ⊢ (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ (Scalar‘𝑊) ∈ TopDRing))
3	2	simplbi 497	1 ⊢ (𝑊 ∈ TopVec → 𝑊 ∈ TopMod)

Syntax hints:   → wi 4   ∈ wcel 2108  ‘cfv 6573  Scalarcsca 17314  TopDRingctdrg 24186  TopModctlm 24187  TopVecctvc 24188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-tvc 24192

This theorem is referenced by:  tvclmod  24227