Theorem tlmtrg 24045

Description: The scalar ring of a topological module is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypothesis

Ref	Expression
tlmtrg.f	⊢ 𝐹 = (Scalar‘𝑊)

Assertion

Ref	Expression
tlmtrg	⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopRing)

Proof of Theorem tlmtrg

Step	Hyp	Ref	Expression
1		eqid 2726	. . . 4 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊)
2		eqid 2726	. . . 4 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊)
3		tlmtrg.f	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
4		eqid 2726	. . . 4 ⊢ (TopOpen‘𝐹) = (TopOpen‘𝐹)
5	1, 2, 3, 4	istlm 24040	. . 3 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ ( ·sf ‘𝑊) ∈ (((TopOpen‘𝐹) ×t (TopOpen‘𝑊)) Cn (TopOpen‘𝑊))))
6	5	simplbi 497	. 2 ⊢ (𝑊 ∈ TopMod → (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing))
7	6	simp3d 1141	1 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopRing)

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ‘cfv 6536  (class class class)co 7404  Scalarcsca 17207  TopOpenctopn 17374  LModclmod 20704   ·_sf cscaf 20705   Cn ccn 23079   ×_t ctx 23415  TopMndctmd 23925  TopRingctrg 24011  TopModctlm 24013

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 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407  df-tlm 24017

This theorem is referenced by:  tlmscatps  24046