Theorem tlmtrg 24221

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 2740	. . . 4 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊)
2		eqid 2740	. . . 4 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊)
3		tlmtrg.f	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
4		eqid 2740	. . . 4 ⊢ (TopOpen‘𝐹) = (TopOpen‘𝐹)
5	1, 2, 3, 4	istlm 24216	. . 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 1144	1 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopRing)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ‘cfv 6575  (class class class)co 7450  Scalarcsca 17316  TopOpenctopn 17483  LModclmod 20882   ·_sf cscaf 20883   Cn ccn 23255   ×_t ctx 23591  TopMndctmd 24101  TopRingctrg 24187  TopModctlm 24189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6527  df-fv 6583  df-ov 7453  df-tlm 24193

This theorem is referenced by:  tlmscatps  24222