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Theorem tlmtrg 22504
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
StepHypRef Expression
1 eqid 2778 . . . 4 ( ·sf𝑊) = ( ·sf𝑊)
2 eqid 2778 . . . 4 (TopOpen‘𝑊) = (TopOpen‘𝑊)
3 tlmtrg.f . . . 4 𝐹 = (Scalar‘𝑊)
4 eqid 2778 . . . 4 (TopOpen‘𝐹) = (TopOpen‘𝐹)
51, 2, 3, 4istlm 22499 . . 3 (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ ( ·sf𝑊) ∈ (((TopOpen‘𝐹) ×t (TopOpen‘𝑊)) Cn (TopOpen‘𝑊))))
65simplbi 490 . 2 (𝑊 ∈ TopMod → (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing))
76simp3d 1124 1 (𝑊 ∈ TopMod → 𝐹 ∈ TopRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1068   = wceq 1507  wcel 2050  cfv 6190  (class class class)co 6978  Scalarcsca 16427  TopOpenctopn 16554  LModclmod 19359   ·sf cscaf 19360   Cn ccn 21539   ×t ctx 21875  TopMndctmd 22385  TopRingctrg 22470  TopModctlm 22472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-iota 6154  df-fv 6198  df-ov 6981  df-tlm 22476
This theorem is referenced by:  tlmscatps  22505
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