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Theorem tlmscatps 24199
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 24198 . 2 (𝑊 ∈ TopMod → 𝐹 ∈ TopRing)
3 trgtps 24178 . 2 (𝐹 ∈ TopRing → 𝐹 ∈ TopSp)
42, 3syl 17 1 (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cfv 6561  Scalarcsca 17300  TopSpctps 22938  TopRingctrg 24164  TopModctlm 24166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-tmd 24080  df-tgp 24081  df-trg 24168  df-tlm 24170
This theorem is referenced by:  cnmpt1vsca  24202  cnmpt2vsca  24203  tlmtgp  24204
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