Theorem tvctlm 23094

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 2737	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
2	1	istvc 23089	. 2 ⊢ (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ (Scalar‘𝑊) ∈ TopDRing))
3	2	simplbi 501	1 ⊢ (𝑊 ∈ TopVec → 𝑊 ∈ TopMod)

Syntax hints:   → wi 4   ∈ wcel 2110  ‘cfv 6380  Scalarcsca 16805  TopDRingctdrg 23054  TopModctlm 23055  TopVecctvc 23056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-tvc 23060

This theorem is referenced by:  tvclmod  23095