Subsemiring of opposite semirings #

For every semiring R, we construct an equivalence between subsemirings of R and that of Rᵐᵒᵖ.

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def Subsemiring.op {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) :

Subsemiring R ᵐᵒᵖ

Pull a subsemiring back to an opposite subsemiring along MulOpposite.unop

Equations

S.op = { toSubmonoid := S.op, add_mem' := ⋯, zero_mem' := ⋯ }

Instances For

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@[simp]

theorem Subsemiring.coe_op {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) :

↑S.op = MulOpposite.unop ⁻¹' ↑S

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@[simp]

theorem Subsemiring.op_toSubmonoid {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) :

S.op.toSubmonoid = S.op

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@[simp]

theorem Subsemiring.mem_op {R : Type u_2} [NonAssocSemiring R] {x : R ᵐᵒᵖ} {S : Subsemiring R} :

x ∈ S.op ↔ MulOpposite.unop x ∈ S

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def Subsemiring.unop {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R ᵐᵒᵖ) :

Subsemiring R

Pull an opposite subsemiring back to a subsemiring along MulOpposite.op

Equations

S.unop = { toSubmonoid := S.unop, add_mem' := ⋯, zero_mem' := ⋯ }

Instances For

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@[simp]

theorem Subsemiring.coe_unop {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R ᵐᵒᵖ) :

↑S.unop = MulOpposite.op ⁻¹' ↑S

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@[simp]

theorem Subsemiring.unop_toSubmonoid {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R ᵐᵒᵖ) :

S.unop.toSubmonoid = S.unop

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@[simp]

theorem Subsemiring.mem_unop {R : Type u_2} [NonAssocSemiring R] {x : R} {S : Subsemiring R ᵐᵒᵖ} :

x ∈ S.unop ↔ MulOpposite.op x ∈ S

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@[simp]

theorem Subsemiring.unop_op {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) :

S.op.unop = S

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@[simp]

theorem Subsemiring.op_unop {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R ᵐᵒᵖ) :

S.unop.op = S

Lattice results #

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theorem Subsemiring.op_le_iff {R : Type u_2} [NonAssocSemiring R] {S₁ : Subsemiring R} {S₂ : Subsemiring R ᵐᵒᵖ} :

S₁.op ≤ S₂ ↔ S₁ ≤ S₂.unop

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theorem Subsemiring.le_op_iff {R : Type u_2} [NonAssocSemiring R] {S₁ : Subsemiring R ᵐᵒᵖ} {S₂ : Subsemiring R} :

S₁ ≤ S₂.op ↔ S₁.unop ≤ S₂

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@[simp]

theorem Subsemiring.op_le_op_iff {R : Type u_2} [NonAssocSemiring R] {S₁ S₂ : Subsemiring R} :

S₁.op ≤ S₂.op ↔ S₁ ≤ S₂

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@[simp]

theorem Subsemiring.unop_le_unop_iff {R : Type u_2} [NonAssocSemiring R] {S₁ S₂ : Subsemiring R ᵐᵒᵖ} :

S₁.unop ≤ S₂.unop ↔ S₁ ≤ S₂

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def Subsemiring.opEquiv {R : Type u_2} [NonAssocSemiring R] :

Subsemiring R ≃o Subsemiring R ᵐᵒᵖ

A subsemiring S of R determines a subsemiring S.op of the opposite ring Rᵐᵒᵖ.

Equations

Subsemiring.opEquiv = { toFun := Subsemiring.op, invFun := Subsemiring.unop, left_inv := ⋯, right_inv := ⋯, map_rel_iff' := ⋯ }

Instances For

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@[simp]

theorem Subsemiring.opEquiv_symm_apply {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R ᵐᵒᵖ) :

(RelIso.symm opEquiv) S = S.unop

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@[simp]

theorem Subsemiring.opEquiv_apply {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) :

opEquiv S = S.op

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theorem Subsemiring.op_injective {R : Type u_2} [NonAssocSemiring R] :

Function.Injective Subsemiring.op

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theorem Subsemiring.unop_injective {R : Type u_2} [NonAssocSemiring R] :

Function.Injective Subsemiring.unop

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@[simp]

theorem Subsemiring.op_inj {R : Type u_2} [NonAssocSemiring R] {S T : Subsemiring R} :

S.op = T.op ↔ S = T

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@[simp]

theorem Subsemiring.unop_inj {R : Type u_2} [NonAssocSemiring R] {S T : Subsemiring R ᵐᵒᵖ} :

S.unop = T.unop ↔ S = T

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@[simp]

theorem Subsemiring.op_bot {R : Type u_2} [NonAssocSemiring R] :

⊥.op = ⊥

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@[simp]

theorem Subsemiring.op_eq_bot {R : Type u_2} [NonAssocSemiring R] {S : Subsemiring R} :

S.op = ⊥ ↔ S = ⊥

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@[simp]

theorem Subsemiring.unop_bot {R : Type u_2} [NonAssocSemiring R] :

⊥.unop = ⊥

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@[simp]

theorem Subsemiring.unop_eq_bot {R : Type u_2} [NonAssocSemiring R] {S : Subsemiring R ᵐᵒᵖ} :

S.unop = ⊥ ↔ S = ⊥

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@[simp]

theorem Subsemiring.op_top {R : Type u_2} [NonAssocSemiring R] :

⊤.op = ⊤

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@[simp]

theorem Subsemiring.op_eq_top {R : Type u_2} [NonAssocSemiring R] {S : Subsemiring R} :

S.op = ⊤ ↔ S = ⊤

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@[simp]

theorem Subsemiring.unop_top {R : Type u_2} [NonAssocSemiring R] :

⊤.unop = ⊤

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@[simp]

theorem Subsemiring.unop_eq_top {R : Type u_2} [NonAssocSemiring R] {S : Subsemiring R ᵐᵒᵖ} :

S.unop = ⊤ ↔ S = ⊤

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theorem Subsemiring.op_sup {R : Type u_2} [NonAssocSemiring R] (S₁ S₂ : Subsemiring R) :

(S₁ ⊔ S₂).op = S₁.op ⊔ S₂.op

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theorem Subsemiring.unop_sup {R : Type u_2} [NonAssocSemiring R] (S₁ S₂ : Subsemiring R ᵐᵒᵖ) :

(S₁ ⊔ S₂).unop = S₁.unop ⊔ S₂.unop

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theorem Subsemiring.op_inf {R : Type u_2} [NonAssocSemiring R] (S₁ S₂ : Subsemiring R) :

(S₁ ⊓ S₂).op = S₁.op ⊓ S₂.op

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theorem Subsemiring.unop_inf {R : Type u_2} [NonAssocSemiring R] (S₁ S₂ : Subsemiring R ᵐᵒᵖ) :

(S₁ ⊓ S₂).unop = S₁.unop ⊓ S₂.unop

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theorem Subsemiring.op_sSup {R : Type u_2} [NonAssocSemiring R] (S : Set (Subsemiring R)) :

(sSup S).op = sSup (Subsemiring.unop ⁻¹' S)

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theorem Subsemiring.unop_sSup {R : Type u_2} [NonAssocSemiring R] (S : Set (Subsemiring R ᵐᵒᵖ)) :

(sSup S).unop = sSup (Subsemiring.op ⁻¹' S)

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theorem Subsemiring.op_sInf {R : Type u_2} [NonAssocSemiring R] (S : Set (Subsemiring R)) :

(sInf S).op = sInf (Subsemiring.unop ⁻¹' S)

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theorem Subsemiring.unop_sInf {R : Type u_2} [NonAssocSemiring R] (S : Set (Subsemiring R ᵐᵒᵖ)) :

(sInf S).unop = sInf (Subsemiring.op ⁻¹' S)

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theorem Subsemiring.op_iSup {ι : Sort u_1} {R : Type u_2} [NonAssocSemiring R] (S : ι → Subsemiring R) :

(iSup S).op = ⨆ (i : ι), (S i).op

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theorem Subsemiring.unop_iSup {ι : Sort u_1} {R : Type u_2} [NonAssocSemiring R] (S : ι → Subsemiring R ᵐᵒᵖ) :

(iSup S).unop = ⨆ (i : ι), (S i).unop

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theorem Subsemiring.op_iInf {ι : Sort u_1} {R : Type u_2} [NonAssocSemiring R] (S : ι → Subsemiring R) :

(iInf S).op = ⨅ (i : ι), (S i).op

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theorem Subsemiring.unop_iInf {ι : Sort u_1} {R : Type u_2} [NonAssocSemiring R] (S : ι → Subsemiring R ᵐᵒᵖ) :

(iInf S).unop = ⨅ (i : ι), (S i).unop

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theorem Subsemiring.op_closure {R : Type u_2} [NonAssocSemiring R] (s : Set R) :

(closure s).op = closure (MulOpposite.unop ⁻¹' s)

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theorem Subsemiring.unop_closure {R : Type u_2} [NonAssocSemiring R] (s : Set R ᵐᵒᵖ) :

(closure s).unop = closure (MulOpposite.op ⁻¹' s)

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def Subsemiring.addEquivOp {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) :

↥S ≃+ ↥S.op

Bijection between a subsemiring S and its opposite.

Equations

S.addEquivOp = { toEquiv := S.equivOp, map_add' := ⋯ }

Instances For

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@[simp]

theorem Subsemiring.addEquivOp_apply_coe {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) (a : ↥S.toSubmonoid) :

↑(S.addEquivOp a) = MulOpposite.op ↑a

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@[simp]

theorem Subsemiring.addEquivOp_symm_apply_coe {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) (b : ↥S.op) :

↑(S.addEquivOp.symm b) = MulOpposite.unop ↑b

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def Subsemiring.ringEquivOpMop {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) :

↥S ≃+* (↥S.op)ᵐᵒᵖ

Bijection between a subsemiring S and MulOpposite of its opposite.

Equations

S.ringEquivOpMop = { toEquiv := (S.addEquivOp.trans MulOpposite.opAddEquiv).toEquiv, map_mul' := ⋯, map_add' := ⋯ }

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@[simp]

theorem Subsemiring.ringEquivOpMop_apply {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) (a✝ : ↥S) :

S.ringEquivOpMop a✝ = MulOpposite.op (S.addEquivOp a✝)

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@[simp]

theorem Subsemiring.ringEquivOpMop_symm_apply_coe {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) (a✝ : (↥S.op)ᵐᵒᵖ) :

↑(S.ringEquivOpMop.symm a✝) = MulOpposite.unop ↑(MulOpposite.unop a✝)

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def Subsemiring.mopRingEquivOp {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) :

(↥S)ᵐᵒᵖ ≃+* ↥S.op

Bijection between MulOpposite of a subsemiring S and its opposite.

Equations

S.mopRingEquivOp = { toEquiv := (MulOpposite.opAddEquiv.symm.trans S.addEquivOp).toEquiv, map_mul' := ⋯, map_add' := ⋯ }

Instances For

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@[simp]

theorem Subsemiring.mopRingEquivOp_apply_coe {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) (a✝ : (↥S)ᵐᵒᵖ) :

↑(S.mopRingEquivOp a✝) = MulOpposite.op ↑(MulOpposite.unop a✝)

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@[simp]

theorem Subsemiring.mopRingEquivOp_symm_apply {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) (a✝ : ↥S.op) :

S.mopRingEquivOp.symm a✝ = MulOpposite.op (S.addEquivOp.symm a✝)

Documentation

Mathlib.Algebra.Ring.Subsemiring.MulOpposite

Subsemiring of opposite semirings #

Lattice results #