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Theorem tlmtrg 24107
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 = (Scalarβ€˜π‘Š)
Assertion
Ref Expression
tlmtrg (π‘Š ∈ TopMod β†’ 𝐹 ∈ TopRing)

Proof of Theorem tlmtrg
StepHypRef Expression
1 eqid 2728 . . . 4 ( Β·sf β€˜π‘Š) = ( Β·sf β€˜π‘Š)
2 eqid 2728 . . . 4 (TopOpenβ€˜π‘Š) = (TopOpenβ€˜π‘Š)
3 tlmtrg.f . . . 4 𝐹 = (Scalarβ€˜π‘Š)
4 eqid 2728 . . . 4 (TopOpenβ€˜πΉ) = (TopOpenβ€˜πΉ)
51, 2, 3, 4istlm 24102 . . 3 (π‘Š ∈ TopMod ↔ ((π‘Š ∈ TopMnd ∧ π‘Š ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ ( Β·sf β€˜π‘Š) ∈ (((TopOpenβ€˜πΉ) Γ—t (TopOpenβ€˜π‘Š)) Cn (TopOpenβ€˜π‘Š))))
65simplbi 497 . 2 (π‘Š ∈ TopMod β†’ (π‘Š ∈ TopMnd ∧ π‘Š ∈ LMod ∧ 𝐹 ∈ TopRing))
76simp3d 1142 1 (π‘Š ∈ TopMod β†’ 𝐹 ∈ TopRing)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  β€˜cfv 6548  (class class class)co 7420  Scalarcsca 17236  TopOpenctopn 17403  LModclmod 20743   Β·sf cscaf 20744   Cn ccn 23141   Γ—t ctx 23477  TopMndctmd 23987  TopRingctrg 24073  TopModctlm 24075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423  df-tlm 24079
This theorem is referenced by:  tlmscatps  24108
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