Theorem tlmscatps 22402

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

Step	Hyp	Ref	Expression
1		tlmtrg.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
2	1	tlmtrg 22401	. 2 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopRing)
3		trgtps 22381	. 2 ⊢ (𝐹 ∈ TopRing → 𝐹 ∈ TopSp)
4	2, 3	syl 17	1 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)

Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2107  ‘cfv 6135  Scalarcsca 16341  TopSpctps 21144  TopRingctrg 22367  TopModctlm 22369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-nul 5025

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-ov 6925  df-tmd 22284  df-tgp 22285  df-trg 22371  df-tlm 22373

This theorem is referenced by:  cnmpt1vsca  22405  cnmpt2vsca  22406  tlmtgp  22407