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Theorem tlmscatps 24123
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 = (Scalarβ€˜π‘Š)
Assertion
Ref Expression
tlmscatps (π‘Š ∈ TopMod β†’ 𝐹 ∈ TopSp)

Proof of Theorem tlmscatps
StepHypRef Expression
1 tlmtrg.f . . 3 𝐹 = (Scalarβ€˜π‘Š)
21tlmtrg 24122 . 2 (π‘Š ∈ TopMod β†’ 𝐹 ∈ TopRing)
3 trgtps 24102 . 2 (𝐹 ∈ TopRing β†’ 𝐹 ∈ TopSp)
42, 3syl 17 1 (π‘Š ∈ TopMod β†’ 𝐹 ∈ TopSp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  β€˜cfv 6553  Scalarcsca 17245  TopSpctps 22862  TopRingctrg 24088  TopModctlm 24090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-nul 5310
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429  df-tmd 24004  df-tgp 24005  df-trg 24092  df-tlm 24094
This theorem is referenced by:  cnmpt1vsca  24126  cnmpt2vsca  24127  tlmtgp  24128
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