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Theorem istlm 21798
 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 679 . 2 (((𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ (𝐹 ∈ TopRing ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))))
2 df-3an 1033 . . . 4 ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ TopRing))
3 elin 3758 . . . . 5 (𝑊 ∈ (TopMnd ∩ LMod) ↔ (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod))
43anbi1i 727 . . . 4 ((𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ TopRing))
52, 4bitr4i 266 . . 3 ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing))
65anbi1i 727 . 2 (((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) ↔ ((𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
7 fveq2 6103 . . . . . 6 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
8 istlm.f . . . . . 6 𝐹 = (Scalar‘𝑊)
97, 8syl6eqr 2662 . . . . 5 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
109eleq1d 2672 . . . 4 (𝑤 = 𝑊 → ((Scalar‘𝑤) ∈ TopRing ↔ 𝐹 ∈ TopRing))
11 fveq2 6103 . . . . . 6 (𝑤 = 𝑊 → ( ·sf𝑤) = ( ·sf𝑊))
12 istlm.s . . . . . 6 · = ( ·sf𝑊)
1311, 12syl6eqr 2662 . . . . 5 (𝑤 = 𝑊 → ( ·sf𝑤) = · )
149fveq2d 6107 . . . . . . . 8 (𝑤 = 𝑊 → (TopOpen‘(Scalar‘𝑤)) = (TopOpen‘𝐹))
15 istlm.k . . . . . . . 8 𝐾 = (TopOpen‘𝐹)
1614, 15syl6eqr 2662 . . . . . . 7 (𝑤 = 𝑊 → (TopOpen‘(Scalar‘𝑤)) = 𝐾)
17 fveq2 6103 . . . . . . . 8 (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
18 istlm.j . . . . . . . 8 𝐽 = (TopOpen‘𝑊)
1917, 18syl6eqr 2662 . . . . . . 7 (𝑤 = 𝑊 → (TopOpen‘𝑤) = 𝐽)
2016, 19oveq12d 6567 . . . . . 6 (𝑤 = 𝑊 → ((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) = (𝐾 ×t 𝐽))
2120, 19oveq12d 6567 . . . . 5 (𝑤 = 𝑊 → (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)) = ((𝐾 ×t 𝐽) Cn 𝐽))
2213, 21eleq12d 2682 . . . 4 (𝑤 = 𝑊 → (( ·sf𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)) ↔ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
2310, 22anbi12d 743 . . 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 21775 . . 3 TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}
2523, 24elrab2 3333 . 2 (𝑊 ∈ TopMod ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ (𝐹 ∈ TopRing ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))))
261, 6, 253bitr4ri 292 1 (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∩ cin 3539  ‘cfv 5804  (class class class)co 6549  Scalarcsca 15771  TopOpenctopn 15905  LModclmod 18686   ·sf cscaf 18687   Cn ccn 20838   ×t ctx 21173  TopMndctmd 21684  TopRingctrg 21769  TopModctlm 21771 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-tlm 21775 This theorem is referenced by:  vscacn  21799  tlmtmd  21800  tlmlmod  21802  tlmtrg  21803  nlmtlm  22308
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