Theorem tlmscatps 24174

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 24173	. 2 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopRing)
3		trgtps 24153	. 2 ⊢ (𝐹 ∈ TopRing → 𝐹 ∈ TopSp)
4	2, 3	syl 17	1 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ‘cfv 6485  Scalarcsca 17214  TopSpctps 22915  TopRingctrg 24139  TopModctlm 24141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-nul 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-ov 7359  df-tmd 24055  df-tgp 24056  df-trg 24143  df-tlm 24145

This theorem is referenced by:  cnmpt1vsca  24177  cnmpt2vsca  24178  tlmtgp  24179