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Theorem istlm 22400
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 462 . 2 (((𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ (𝐹 ∈ TopRing ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))))
2 df-3an 1073 . . . 4 ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ TopRing))
3 elin 4019 . . . . 5 (𝑊 ∈ (TopMnd ∩ LMod) ↔ (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod))
43anbi1i 617 . . . 4 ((𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ TopRing))
52, 4bitr4i 270 . . 3 ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing))
65anbi1i 617 . 2 (((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) ↔ ((𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
7 fveq2 6448 . . . . . 6 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
8 istlm.f . . . . . 6 𝐹 = (Scalar‘𝑊)
97, 8syl6eqr 2832 . . . . 5 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
109eleq1d 2844 . . . 4 (𝑤 = 𝑊 → ((Scalar‘𝑤) ∈ TopRing ↔ 𝐹 ∈ TopRing))
11 fveq2 6448 . . . . . 6 (𝑤 = 𝑊 → ( ·sf𝑤) = ( ·sf𝑊))
12 istlm.s . . . . . 6 · = ( ·sf𝑊)
1311, 12syl6eqr 2832 . . . . 5 (𝑤 = 𝑊 → ( ·sf𝑤) = · )
149fveq2d 6452 . . . . . . . 8 (𝑤 = 𝑊 → (TopOpen‘(Scalar‘𝑤)) = (TopOpen‘𝐹))
15 istlm.k . . . . . . . 8 𝐾 = (TopOpen‘𝐹)
1614, 15syl6eqr 2832 . . . . . . 7 (𝑤 = 𝑊 → (TopOpen‘(Scalar‘𝑤)) = 𝐾)
17 fveq2 6448 . . . . . . . 8 (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
18 istlm.j . . . . . . . 8 𝐽 = (TopOpen‘𝑊)
1917, 18syl6eqr 2832 . . . . . . 7 (𝑤 = 𝑊 → (TopOpen‘𝑤) = 𝐽)
2016, 19oveq12d 6942 . . . . . 6 (𝑤 = 𝑊 → ((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) = (𝐾 ×t 𝐽))
2120, 19oveq12d 6942 . . . . 5 (𝑤 = 𝑊 → (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)) = ((𝐾 ×t 𝐽) Cn 𝐽))
2213, 21eleq12d 2853 . . . 4 (𝑤 = 𝑊 → (( ·sf𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)) ↔ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
2310, 22anbi12d 624 . . 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 22377 . . 3 TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}
2523, 24elrab2 3576 . 2 (𝑊 ∈ TopMod ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ (𝐹 ∈ TopRing ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))))
261, 6, 253bitr4ri 296 1 (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  cin 3791  cfv 6137  (class class class)co 6924  Scalarcsca 16345  TopOpenctopn 16472  LModclmod 19259   ·sf cscaf 19260   Cn ccn 21440   ×t ctx 21776  TopMndctmd 22286  TopRingctrg 22371  TopModctlm 22373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-iota 6101  df-fv 6145  df-ov 6927  df-tlm 22377
This theorem is referenced by:  vscacn  22401  tlmtmd  22402  tlmlmod  22404  tlmtrg  22405  nlmtlm  22910
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