Theorem tvctlm 24365

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 2762	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
2	1	istvc 24360	. 2 ⊢ (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ (Scalar‘𝑊) ∈ TopDRing))
3	2	simplbi 501	1 ⊢ (𝑊 ∈ TopVec → 𝑊 ∈ TopMod)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142  ‘cfv 6536  Scalarcsca 17319  TopDRingctdrg 24325  TopModctlm 24326  TopVecctvc 24327

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-tvc 24331

This theorem is used by:  tvclmod  24366