Theorem tlmscatps 24565

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 24564	. 2 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopRing)
3		trgtps 24544	. 2 ⊢ (𝐹 ∈ TopRing → 𝐹 ∈ TopSp)
4	2, 3	syl 18	1 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ‘cfv 6566  Scalarcsca 17475  TopSpctps 23305  TopRingctrg 24530  TopModctlm 24532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5266

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-iota 6520  df-fv 6574  df-ov 7454  df-tmd 24446  df-tgp 24447  df-trg 24534  df-tlm 24536

This theorem is used by:  cnmpt1vsca  24568  cnmpt2vsca  24569  tlmtgp  24570