File: Mathlib/RingTheory/TwoSidedIdeal/Lattice.lean

full_name: TwoSidedIdeal.instSup
code: instance : Sup (TwoSidedIdeal R) 
goal: Sup (TwoSidedIdeal R)

full_name: TwoSidedIdeal.sup_ringCon
code: lemma sup_ringCon (I J : TwoSidedIdeal R) : (I ⊔ J).ringCon = I.ringCon ⊔ J.ringCon 
goal: (I ⊔ J).ringCon = I.ringCon ⊔ J.ringCon

full_name: TwoSidedIdeal.instSemilatticeSup
code: instance : SemilatticeSup (TwoSidedIdeal R) 
goal: SemilatticeSup (TwoSidedIdeal R)

full_name: TwoSidedIdeal.mem_sup_left
code: lemma mem_sup_left {I J : TwoSidedIdeal R} {x : R} (h : x ∈ I) :
    x ∈ I ⊔ J 
goal: x ∈ I ⊔ J

full_name: TwoSidedIdeal.mem_sup_right
code: lemma mem_sup_right {I J : TwoSidedIdeal R} {x : R} (h : x ∈ J) :
    x ∈ I ⊔ J 
goal: x ∈ I ⊔ J

full_name: TwoSidedIdeal.mem_sup
code: lemma mem_sup {I J : TwoSidedIdeal R} {x : R} :
    x ∈ I ⊔ J ↔ ∃ y ∈ I, ∃ z ∈ J, y + z = x 
goal: x ∈ I ⊔ J ↔ ∃ y ∈ I, ∃ z ∈ J, y + z = x

full_name: TwoSidedIdeal.instInf
code: instance : Inf (TwoSidedIdeal R) 
goal: Inf (TwoSidedIdeal R)

full_name: TwoSidedIdeal.inf_ringCon
code: lemma inf_ringCon (I J : TwoSidedIdeal R) : (I ⊓ J).ringCon = I.ringCon ⊓ J.ringCon 
goal: (I ⊓ J).ringCon = I.ringCon ⊓ J.ringCon

full_name: TwoSidedIdeal.instSemilatticeInf
code: instance : SemilatticeInf (TwoSidedIdeal R) 
goal: SemilatticeInf (TwoSidedIdeal R)

full_name: TwoSidedIdeal.mem_inf
code: lemma mem_inf {I J : TwoSidedIdeal R} {x : R} :
    x ∈ I ⊓ J ↔ x ∈ I ∧ x ∈ J 
goal: x ∈ I ⊓ J ↔ x ∈ I ∧ x ∈ J

full_name: TwoSidedIdeal.instSupSet
code: instance : SupSet (TwoSidedIdeal R) 
goal: SupSet (TwoSidedIdeal R)

full_name: TwoSidedIdeal.sSup_ringCon
code: lemma sSup_ringCon (S : Set (TwoSidedIdeal R)) :
    (sSup S).ringCon = sSup (TwoSidedIdeal.ringCon '' S) 
goal: (sSup S).ringCon = sSup (TwoSidedIdeal.ringCon '' S)

full_name: TwoSidedIdeal.iSup_ringCon
code: lemma iSup_ringCon {ι : Type*} (I : ι → TwoSidedIdeal R) :
    (⨆ i, I i).ringCon = ⨆ i, (I i).ringCon 
goal: (⨆ i, I i).ringCon = ⨆ i, (I i).ringCon

full_name: TwoSidedIdeal.instCompleteSemilatticeSup
code: instance : CompleteSemilatticeSup (TwoSidedIdeal R) 
goal: CompleteSemilatticeSup (TwoSidedIdeal R)

full_name: TwoSidedIdeal.instInfSet
code: instance : InfSet (TwoSidedIdeal R) 
goal: InfSet (TwoSidedIdeal R)

full_name: TwoSidedIdeal.sInf_ringCon
code: lemma sInf_ringCon (S : Set (TwoSidedIdeal R)) :
    (sInf S).ringCon = sInf (TwoSidedIdeal.ringCon '' S) 
goal: (sInf S).ringCon = sInf (TwoSidedIdeal.ringCon '' S)

full_name: TwoSidedIdeal.iInf_ringCon
code: lemma iInf_ringCon {ι : Type*} (I : ι → TwoSidedIdeal R) :
    (⨅ i, I i).ringCon = ⨅ i, (I i).ringCon 
goal: (⨅ i, I i).ringCon = ⨅ i, (I i).ringCon

full_name: TwoSidedIdeal.instCompleteSemilatticeInf
code: instance : CompleteSemilatticeInf (TwoSidedIdeal R) 
goal: CompleteSemilatticeInf (TwoSidedIdeal R)

full_name: TwoSidedIdeal.mem_iInf
code: lemma mem_iInf {ι : Type*} {I : ι → TwoSidedIdeal R} {x : R} :
    x ∈ iInf I ↔ ∀ i, x ∈ I i 
goal: x ∈ iInf I ↔ ∀ (i : ι), x ∈ I i

full_name: TwoSidedIdeal.mem_sInf
code: lemma mem_sInf {S : Set (TwoSidedIdeal R)} {x : R} :
    x ∈ sInf S ↔ ∀ I ∈ S, x ∈ I 
goal: x ∈ sInf S ↔ ∀ I ∈ S, x ∈ I

full_name: TwoSidedIdeal.instTop
code: instance : Top (TwoSidedIdeal R) 
goal: Top (TwoSidedIdeal R)

full_name: TwoSidedIdeal.top_ringCon
code: lemma top_ringCon : (⊤ : TwoSidedIdeal R).ringCon = ⊤ 
goal: ⊤.ringCon = ⊤

full_name: TwoSidedIdeal.instBot
code: instance : Bot (TwoSidedIdeal R) 
goal: Bot (TwoSidedIdeal R)

full_name: TwoSidedIdeal.bot_ringCon
code: lemma bot_ringCon : (⊥ : TwoSidedIdeal R).ringCon = ⊥ 
goal: ⊥.ringCon = ⊥

full_name: TwoSidedIdeal.mem_bot
code: lemma mem_bot {x : R} : x ∈ (⊥ : TwoSidedIdeal R) ↔ x = 0 
goal: x ∈ ⊥ ↔ x = 0

full_name: TwoSidedIdeal.instCompleteLattice
code: instance : CompleteLattice (TwoSidedIdeal R) 
goal: CompleteLattice (TwoSidedIdeal R)