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Theorem istlm 21911
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 680 . 2 (((𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ (𝐹 ∈ TopRing ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))))
2 df-3an 1038 . . . 4 ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ TopRing))
3 elin 3779 . . . . 5 (𝑊 ∈ (TopMnd ∩ LMod) ↔ (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod))
43anbi1i 730 . . . 4 ((𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ TopRing))
52, 4bitr4i 267 . . 3 ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing))
65anbi1i 730 . 2 (((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) ↔ ((𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
7 fveq2 6153 . . . . . 6 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
8 istlm.f . . . . . 6 𝐹 = (Scalar‘𝑊)
97, 8syl6eqr 2673 . . . . 5 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
109eleq1d 2683 . . . 4 (𝑤 = 𝑊 → ((Scalar‘𝑤) ∈ TopRing ↔ 𝐹 ∈ TopRing))
11 fveq2 6153 . . . . . 6 (𝑤 = 𝑊 → ( ·sf𝑤) = ( ·sf𝑊))
12 istlm.s . . . . . 6 · = ( ·sf𝑊)
1311, 12syl6eqr 2673 . . . . 5 (𝑤 = 𝑊 → ( ·sf𝑤) = · )
149fveq2d 6157 . . . . . . . 8 (𝑤 = 𝑊 → (TopOpen‘(Scalar‘𝑤)) = (TopOpen‘𝐹))
15 istlm.k . . . . . . . 8 𝐾 = (TopOpen‘𝐹)
1614, 15syl6eqr 2673 . . . . . . 7 (𝑤 = 𝑊 → (TopOpen‘(Scalar‘𝑤)) = 𝐾)
17 fveq2 6153 . . . . . . . 8 (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
18 istlm.j . . . . . . . 8 𝐽 = (TopOpen‘𝑊)
1917, 18syl6eqr 2673 . . . . . . 7 (𝑤 = 𝑊 → (TopOpen‘𝑤) = 𝐽)
2016, 19oveq12d 6628 . . . . . 6 (𝑤 = 𝑊 → ((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) = (𝐾 ×t 𝐽))
2120, 19oveq12d 6628 . . . . 5 (𝑤 = 𝑊 → (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)) = ((𝐾 ×t 𝐽) Cn 𝐽))
2213, 21eleq12d 2692 . . . 4 (𝑤 = 𝑊 → (( ·sf𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)) ↔ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
2310, 22anbi12d 746 . . 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 21888 . . 3 TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}
2523, 24elrab2 3352 . 2 (𝑊 ∈ TopMod ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ (𝐹 ∈ TopRing ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))))
261, 6, 253bitr4ri 293 1 (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  cin 3558  cfv 5852  (class class class)co 6610  Scalarcsca 15876  TopOpenctopn 16014  LModclmod 18795   ·sf cscaf 18796   Cn ccn 20951   ×t ctx 21286  TopMndctmd 21797  TopRingctrg 21882  TopModctlm 21884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5815  df-fv 5860  df-ov 6613  df-tlm 21888
This theorem is referenced by:  vscacn  21912  tlmtmd  21913  tlmlmod  21915  tlmtrg  21916  nlmtlm  22421
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