Theorem tlmtrg 24173

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 2739	. . . 4 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊)
2		eqid 2739	. . . 4 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊)
3		tlmtrg.f	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
4		eqid 2739	. . . 4 ⊢ (TopOpen‘𝐹) = (TopOpen‘𝐹)
5	1, 2, 3, 4	istlm 24168	. . 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 1150	1 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopRing)

Syntax hints:   → wi 4   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ‘cfv 6485  (class class class)co 7356  Scalarcsca 17214  TopOpenctopn 17375  LModclmod 20850   ·_sf cscaf 20851   Cn ccn 23207   ×_t ctx 23543  TopMndctmd 24053  TopRingctrg 24139  TopModctlm 24141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-ov 7359  df-tlm 24145

This theorem is referenced by:  tlmscatps  24174