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Theorem tlmscatps 24215
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
StepHypRef Expression
1 tlmtrg.f . . 3 𝐹 = (Scalar‘𝑊)
21tlmtrg 24214 . 2 (𝑊 ∈ TopMod → 𝐹 ∈ TopRing)
3 trgtps 24194 . 2 (𝐹 ∈ TopRing → 𝐹 ∈ TopSp)
42, 3syl 17 1 (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cfv 6563  Scalarcsca 17301  TopSpctps 22954  TopRingctrg 24180  TopModctlm 24182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-tmd 24096  df-tgp 24097  df-trg 24184  df-tlm 24186
This theorem is referenced by:  cnmpt1vsca  24218  cnmpt2vsca  24219  tlmtgp  24220
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