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Theorem tlmscatps 22041
 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 22040 . 2 (𝑊 ∈ TopMod → 𝐹 ∈ TopRing)
3 trgtps 22020 . 2 (𝐹 ∈ TopRing → 𝐹 ∈ TopSp)
42, 3syl 17 1 (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  ‘cfv 5926  Scalarcsca 15991  TopSpctps 20784  TopRingctrg 22006  TopModctlm 22008 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-tmd 21923  df-tgp 21924  df-trg 22010  df-tlm 22012 This theorem is referenced by:  cnmpt1vsca  22044  cnmpt2vsca  22045  tlmtgp  22046
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