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Theorem istlm 24088
Description: The predicate "π‘Š is a topological left module". (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
istlm.s Β· = ( Β·sf β€˜π‘Š)
istlm.j 𝐽 = (TopOpenβ€˜π‘Š)
istlm.f 𝐹 = (Scalarβ€˜π‘Š)
istlm.k 𝐾 = (TopOpenβ€˜πΉ)
Assertion
Ref Expression
istlm (π‘Š ∈ TopMod ↔ ((π‘Š ∈ TopMnd ∧ π‘Š ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ Β· ∈ ((𝐾 Γ—t 𝐽) Cn 𝐽)))

Proof of Theorem istlm
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 anass 468 . 2 (((π‘Š ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ∧ Β· ∈ ((𝐾 Γ—t 𝐽) Cn 𝐽)) ↔ (π‘Š ∈ (TopMnd ∩ LMod) ∧ (𝐹 ∈ TopRing ∧ Β· ∈ ((𝐾 Γ—t 𝐽) Cn 𝐽))))
2 df-3an 1087 . . . 4 ((π‘Š ∈ TopMnd ∧ π‘Š ∈ LMod ∧ 𝐹 ∈ TopRing) ↔ ((π‘Š ∈ TopMnd ∧ π‘Š ∈ LMod) ∧ 𝐹 ∈ TopRing))
3 elin 3963 . . . . 5 (π‘Š ∈ (TopMnd ∩ LMod) ↔ (π‘Š ∈ TopMnd ∧ π‘Š ∈ LMod))
43anbi1i 623 . . . 4 ((π‘Š ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ↔ ((π‘Š ∈ TopMnd ∧ π‘Š ∈ LMod) ∧ 𝐹 ∈ TopRing))
52, 4bitr4i 278 . . 3 ((π‘Š ∈ TopMnd ∧ π‘Š ∈ LMod ∧ 𝐹 ∈ TopRing) ↔ (π‘Š ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing))
65anbi1i 623 . 2 (((π‘Š ∈ TopMnd ∧ π‘Š ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ Β· ∈ ((𝐾 Γ—t 𝐽) Cn 𝐽)) ↔ ((π‘Š ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ∧ Β· ∈ ((𝐾 Γ—t 𝐽) Cn 𝐽)))
7 fveq2 6897 . . . . . 6 (𝑀 = π‘Š β†’ (Scalarβ€˜π‘€) = (Scalarβ€˜π‘Š))
8 istlm.f . . . . . 6 𝐹 = (Scalarβ€˜π‘Š)
97, 8eqtr4di 2786 . . . . 5 (𝑀 = π‘Š β†’ (Scalarβ€˜π‘€) = 𝐹)
109eleq1d 2814 . . . 4 (𝑀 = π‘Š β†’ ((Scalarβ€˜π‘€) ∈ TopRing ↔ 𝐹 ∈ TopRing))
11 fveq2 6897 . . . . . 6 (𝑀 = π‘Š β†’ ( Β·sf β€˜π‘€) = ( Β·sf β€˜π‘Š))
12 istlm.s . . . . . 6 Β· = ( Β·sf β€˜π‘Š)
1311, 12eqtr4di 2786 . . . . 5 (𝑀 = π‘Š β†’ ( Β·sf β€˜π‘€) = Β· )
149fveq2d 6901 . . . . . . . 8 (𝑀 = π‘Š β†’ (TopOpenβ€˜(Scalarβ€˜π‘€)) = (TopOpenβ€˜πΉ))
15 istlm.k . . . . . . . 8 𝐾 = (TopOpenβ€˜πΉ)
1614, 15eqtr4di 2786 . . . . . . 7 (𝑀 = π‘Š β†’ (TopOpenβ€˜(Scalarβ€˜π‘€)) = 𝐾)
17 fveq2 6897 . . . . . . . 8 (𝑀 = π‘Š β†’ (TopOpenβ€˜π‘€) = (TopOpenβ€˜π‘Š))
18 istlm.j . . . . . . . 8 𝐽 = (TopOpenβ€˜π‘Š)
1917, 18eqtr4di 2786 . . . . . . 7 (𝑀 = π‘Š β†’ (TopOpenβ€˜π‘€) = 𝐽)
2016, 19oveq12d 7438 . . . . . 6 (𝑀 = π‘Š β†’ ((TopOpenβ€˜(Scalarβ€˜π‘€)) Γ—t (TopOpenβ€˜π‘€)) = (𝐾 Γ—t 𝐽))
2120, 19oveq12d 7438 . . . . 5 (𝑀 = π‘Š β†’ (((TopOpenβ€˜(Scalarβ€˜π‘€)) Γ—t (TopOpenβ€˜π‘€)) Cn (TopOpenβ€˜π‘€)) = ((𝐾 Γ—t 𝐽) Cn 𝐽))
2213, 21eleq12d 2823 . . . 4 (𝑀 = π‘Š β†’ (( Β·sf β€˜π‘€) ∈ (((TopOpenβ€˜(Scalarβ€˜π‘€)) Γ—t (TopOpenβ€˜π‘€)) Cn (TopOpenβ€˜π‘€)) ↔ Β· ∈ ((𝐾 Γ—t 𝐽) Cn 𝐽)))
2310, 22anbi12d 631 . . 3 (𝑀 = π‘Š β†’ (((Scalarβ€˜π‘€) ∈ TopRing ∧ ( Β·sf β€˜π‘€) ∈ (((TopOpenβ€˜(Scalarβ€˜π‘€)) Γ—t (TopOpenβ€˜π‘€)) Cn (TopOpenβ€˜π‘€))) ↔ (𝐹 ∈ TopRing ∧ Β· ∈ ((𝐾 Γ—t 𝐽) Cn 𝐽))))
24 df-tlm 24065 . . 3 TopMod = {𝑀 ∈ (TopMnd ∩ LMod) ∣ ((Scalarβ€˜π‘€) ∈ TopRing ∧ ( Β·sf β€˜π‘€) ∈ (((TopOpenβ€˜(Scalarβ€˜π‘€)) Γ—t (TopOpenβ€˜π‘€)) Cn (TopOpenβ€˜π‘€)))}
2523, 24elrab2 3685 . 2 (π‘Š ∈ TopMod ↔ (π‘Š ∈ (TopMnd ∩ LMod) ∧ (𝐹 ∈ TopRing ∧ Β· ∈ ((𝐾 Γ—t 𝐽) Cn 𝐽))))
261, 6, 253bitr4ri 304 1 (π‘Š ∈ TopMod ↔ ((π‘Š ∈ TopMnd ∧ π‘Š ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ Β· ∈ ((𝐾 Γ—t 𝐽) Cn 𝐽)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ∩ cin 3946  β€˜cfv 6548  (class class class)co 7420  Scalarcsca 17235  TopOpenctopn 17402  LModclmod 20742   Β·sf cscaf 20743   Cn ccn 23127   Γ—t ctx 23463  TopMndctmd 23973  TopRingctrg 24059  TopModctlm 24061
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 24065
This theorem is referenced by:  vscacn  24089  tlmtmd  24090  tlmlmod  24092  tlmtrg  24093  nlmtlm  24610
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