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Theorem istlm 23552
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 470 . 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 1090 . . . 4 ((π‘Š ∈ TopMnd ∧ π‘Š ∈ LMod ∧ 𝐹 ∈ TopRing) ↔ ((π‘Š ∈ TopMnd ∧ π‘Š ∈ LMod) ∧ 𝐹 ∈ TopRing))
3 elin 3927 . . . . 5 (π‘Š ∈ (TopMnd ∩ LMod) ↔ (π‘Š ∈ TopMnd ∧ π‘Š ∈ LMod))
43anbi1i 625 . . . 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 625 . 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 6843 . . . . . 6 (𝑀 = π‘Š β†’ (Scalarβ€˜π‘€) = (Scalarβ€˜π‘Š))
8 istlm.f . . . . . 6 𝐹 = (Scalarβ€˜π‘Š)
97, 8eqtr4di 2791 . . . . 5 (𝑀 = π‘Š β†’ (Scalarβ€˜π‘€) = 𝐹)
109eleq1d 2819 . . . 4 (𝑀 = π‘Š β†’ ((Scalarβ€˜π‘€) ∈ TopRing ↔ 𝐹 ∈ TopRing))
11 fveq2 6843 . . . . . 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 2791 . . . . 5 (𝑀 = π‘Š β†’ ( Β·sf β€˜π‘€) = Β· )
149fveq2d 6847 . . . . . . . 8 (𝑀 = π‘Š β†’ (TopOpenβ€˜(Scalarβ€˜π‘€)) = (TopOpenβ€˜πΉ))
15 istlm.k . . . . . . . 8 𝐾 = (TopOpenβ€˜πΉ)
1614, 15eqtr4di 2791 . . . . . . 7 (𝑀 = π‘Š β†’ (TopOpenβ€˜(Scalarβ€˜π‘€)) = 𝐾)
17 fveq2 6843 . . . . . . . 8 (𝑀 = π‘Š β†’ (TopOpenβ€˜π‘€) = (TopOpenβ€˜π‘Š))
18 istlm.j . . . . . . . 8 𝐽 = (TopOpenβ€˜π‘Š)
1917, 18eqtr4di 2791 . . . . . . 7 (𝑀 = π‘Š β†’ (TopOpenβ€˜π‘€) = 𝐽)
2016, 19oveq12d 7376 . . . . . 6 (𝑀 = π‘Š β†’ ((TopOpenβ€˜(Scalarβ€˜π‘€)) Γ—t (TopOpenβ€˜π‘€)) = (𝐾 Γ—t 𝐽))
2120, 19oveq12d 7376 . . . . 5 (𝑀 = π‘Š β†’ (((TopOpenβ€˜(Scalarβ€˜π‘€)) Γ—t (TopOpenβ€˜π‘€)) Cn (TopOpenβ€˜π‘€)) = ((𝐾 Γ—t 𝐽) Cn 𝐽))
2213, 21eleq12d 2828 . . . 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 632 . . 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 23529 . . 3 TopMod = {𝑀 ∈ (TopMnd ∩ LMod) ∣ ((Scalarβ€˜π‘€) ∈ TopRing ∧ ( Β·sf β€˜π‘€) ∈ (((TopOpenβ€˜(Scalarβ€˜π‘€)) Γ—t (TopOpenβ€˜π‘€)) Cn (TopOpenβ€˜π‘€)))}
2523, 24elrab2 3649 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ∩ cin 3910  β€˜cfv 6497  (class class class)co 7358  Scalarcsca 17141  TopOpenctopn 17308  LModclmod 20336   Β·sf cscaf 20337   Cn ccn 22591   Γ—t ctx 22927  TopMndctmd 23437  TopRingctrg 23523  TopModctlm 23525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-tlm 23529
This theorem is referenced by:  vscacn  23553  tlmtmd  23554  tlmlmod  23556  tlmtrg  23557  nlmtlm  24074
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