Theorem tlmscatps 24313

Description: The scalar ring of a topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypothesis

Ref	Expression
tlmtrg.f	⊢ 𝐹 = (Scalar‘𝑊)

Assertion

Ref	Expression
tlmscatps	⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)

Proof of Theorem tlmscatps

Step	Hyp	Ref	Expression
1		tlmtrg.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
2	1	tlmtrg 24312	. 2 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopRing)
3		trgtps 24292	. 2 ⊢ (𝐹 ∈ TopRing → 𝐹 ∈ TopSp)
4	2, 3	syl 18	1 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ‘cfv 6534  Scalarcsca 17309  TopSpctps 23054  TopRingctrg 24278  TopModctlm 24280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5268

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-iota 6490  df-fv 6542  df-ov 7411  df-tmd 24194  df-tgp 24195  df-trg 24282  df-tlm 24284

This theorem is referenced by:  cnmpt1vsca  24316  cnmpt2vsca  24317  tlmtgp  24318