Theorem tvctlm 24052

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 2726	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
2	1	istvc 24047	. 2 ⊢ (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ (Scalar‘𝑊) ∈ TopDRing))
3	2	simplbi 497	1 ⊢ (𝑊 ∈ TopVec → 𝑊 ∈ TopMod)

Syntax hints:   → wi 4   ∈ wcel 2098  ‘cfv 6536  Scalarcsca 17207  TopDRingctdrg 24012  TopModctlm 24013  TopVecctvc 24014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-tvc 24018

This theorem is referenced by:  tvclmod  24053