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Theorem istlm 24133
Description: The predicate "𝑊 is a topological left module". (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
istlm.s · = ( ·sf𝑊)
istlm.j 𝐽 = (TopOpen‘𝑊)
istlm.f 𝐹 = (Scalar‘𝑊)
istlm.k 𝐾 = (TopOpen‘𝐹)
Assertion
Ref Expression
istlm (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))

Proof of Theorem istlm
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 anass 467 . 2 (((𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ (𝐹 ∈ TopRing ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))))
2 df-3an 1086 . . . 4 ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ TopRing))
3 elin 3960 . . . . 5 (𝑊 ∈ (TopMnd ∩ LMod) ↔ (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod))
43anbi1i 622 . . . 4 ((𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ TopRing))
52, 4bitr4i 277 . . 3 ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing))
65anbi1i 622 . 2 (((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) ↔ ((𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
7 fveq2 6896 . . . . . 6 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
8 istlm.f . . . . . 6 𝐹 = (Scalar‘𝑊)
97, 8eqtr4di 2783 . . . . 5 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
109eleq1d 2810 . . . 4 (𝑤 = 𝑊 → ((Scalar‘𝑤) ∈ TopRing ↔ 𝐹 ∈ TopRing))
11 fveq2 6896 . . . . . 6 (𝑤 = 𝑊 → ( ·sf𝑤) = ( ·sf𝑊))
12 istlm.s . . . . . 6 · = ( ·sf𝑊)
1311, 12eqtr4di 2783 . . . . 5 (𝑤 = 𝑊 → ( ·sf𝑤) = · )
149fveq2d 6900 . . . . . . . 8 (𝑤 = 𝑊 → (TopOpen‘(Scalar‘𝑤)) = (TopOpen‘𝐹))
15 istlm.k . . . . . . . 8 𝐾 = (TopOpen‘𝐹)
1614, 15eqtr4di 2783 . . . . . . 7 (𝑤 = 𝑊 → (TopOpen‘(Scalar‘𝑤)) = 𝐾)
17 fveq2 6896 . . . . . . . 8 (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
18 istlm.j . . . . . . . 8 𝐽 = (TopOpen‘𝑊)
1917, 18eqtr4di 2783 . . . . . . 7 (𝑤 = 𝑊 → (TopOpen‘𝑤) = 𝐽)
2016, 19oveq12d 7437 . . . . . 6 (𝑤 = 𝑊 → ((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) = (𝐾 ×t 𝐽))
2120, 19oveq12d 7437 . . . . 5 (𝑤 = 𝑊 → (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)) = ((𝐾 ×t 𝐽) Cn 𝐽))
2213, 21eleq12d 2819 . . . 4 (𝑤 = 𝑊 → (( ·sf𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)) ↔ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
2310, 22anbi12d 630 . . 3 (𝑤 = 𝑊 → (((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤))) ↔ (𝐹 ∈ TopRing ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))))
24 df-tlm 24110 . . 3 TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}
2523, 24elrab2 3682 . 2 (𝑊 ∈ TopMod ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ (𝐹 ∈ TopRing ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))))
261, 6, 253bitr4ri 303 1 (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  cin 3943  cfv 6549  (class class class)co 7419  Scalarcsca 17239  TopOpenctopn 17406  LModclmod 20755   ·sf cscaf 20756   Cn ccn 23172   ×t ctx 23508  TopMndctmd 24018  TopRingctrg 24104  TopModctlm 24106
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 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-ov 7422  df-tlm 24110
This theorem is referenced by:  vscacn  24134  tlmtmd  24135  tlmlmod  24137  tlmtrg  24138  nlmtlm  24655
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