Theorem tlmscatps 23413

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 23412	. 2 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopRing)
3		trgtps 23392	. 2 ⊢ (𝐹 ∈ TopRing → 𝐹 ∈ TopSp)
4	2, 3	syl 17	1 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  ‘cfv 6463  Scalarcsca 17032  TopSpctps 22152  TopRingctrg 23378  TopModctlm 23380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708  ax-nul 5243

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-rab 3405  df-v 3443  df-sbc 3726  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-br 5086  df-iota 6415  df-fv 6471  df-ov 7316  df-tmd 23294  df-tgp 23295  df-trg 23382  df-tlm 23384

This theorem is referenced by:  cnmpt1vsca  23416  cnmpt2vsca  23417  tlmtgp  23418