MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tlmscatps Structured version   Visualization version   GIF version

Theorem tlmscatps 22775
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 22774 . 2 (𝑊 ∈ TopMod → 𝐹 ∈ TopRing)
3 trgtps 22754 . 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 2114  cfv 6331  Scalarcsca 16547  TopSpctps 21516  TopRingctrg 22740  TopModctlm 22742
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-nul 5186
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-iota 6290  df-fv 6339  df-ov 7136  df-tmd 22656  df-tgp 22657  df-trg 22744  df-tlm 22746
This theorem is referenced by:  cnmpt1vsca  22778  cnmpt2vsca  22779  tlmtgp  22780
  Copyright terms: Public domain W3C validator