File: Mathlib/RingTheory/Valuation/Quotient.lean

full_name: Valuation.onQuotVal
code: /-- If `hJ : J ⊆ supp v` then `onQuotVal hJ` is the induced function on `R / J` as a function.
Note: it's just the function; the valuation is `onQuot hJ`. -/
def onQuotVal {J : Ideal R} (hJ : J ≤ supp v) : R ⧸ J → Γ₀ 
goal: R ⧸ J → Γ₀

full_name: Valuation.onQuot
code: /-- The extension of valuation `v` on `R` to valuation on `R / J` if `J ⊆ supp v`. -/
def onQuot {J : Ideal R} (hJ : J ≤ supp v) : Valuation (R ⧸ J) Γ₀ 
goal: Valuation (R ⧸ J) Γ₀

full_name: Valuation.onQuot_comap_eq
code: @[simp]
theorem onQuot_comap_eq {J : Ideal R} (hJ : J ≤ supp v) :
    (v.onQuot hJ).comap (Ideal.Quotient.mk J) = v 
goal: Valuation.comap (Ideal.Quotient.mk J) (v.onQuot hJ) = v

full_name: Valuation.self_le_supp_comap
code: theorem self_le_supp_comap (J : Ideal R) (v : Valuation (R ⧸ J) Γ₀) :
    J ≤ (v.comap (Ideal.Quotient.mk J)).supp 
goal: J ≤ (Valuation.comap (Ideal.Quotient.mk J) v).supp

full_name: Valuation.comap_onQuot_eq
code: @[simp]
theorem comap_onQuot_eq (J : Ideal R) (v : Valuation (R ⧸ J) Γ₀) :
    (v.comap (Ideal.Quotient.mk J)).onQuot (v.self_le_supp_comap J) = v 
goal: (Valuation.comap (Ideal.Quotient.mk J) v).onQuot ⋯ = v

full_name: Valuation.supp_quot
code: /-- The quotient valuation on `R / J` has support `(supp v) / J` if `J ⊆ supp v`. -/
theorem supp_quot {J : Ideal R} (hJ : J ≤ supp v) :
    supp (v.onQuot hJ) = (supp v).map (Ideal.Quotient.mk J) 
goal: (v.onQuot hJ).supp = Ideal.map (Ideal.Quotient.mk J) v.supp

full_name: Valuation.supp_quot_supp
code: theorem supp_quot_supp : supp (v.onQuot le_rfl) = 0 
goal: (v.onQuot ⋯).supp = 0

full_name: AddValuation.onQuotVal
code: /-- If `hJ : J ⊆ supp v` then `onQuotVal hJ` is the induced function on `R / J` as a function.
Note: it's just the function; the valuation is `onQuot hJ`. -/
def onQuotVal {J : Ideal R} (hJ : J ≤ supp v) : R ⧸ J → Γ₀ 
goal: R ⧸ J → Γ₀

full_name: AddValuation.onQuot
code: /-- The extension of valuation `v` on `R` to valuation on `R / J` if `J ⊆ supp v`. -/
def onQuot {J : Ideal R} (hJ : J ≤ supp v) : AddValuation (R ⧸ J) Γ₀ 
goal: AddValuation (R ⧸ J) Γ₀

full_name: AddValuation.onQuot_comap_eq
code: @[simp]
theorem onQuot_comap_eq {J : Ideal R} (hJ : J ≤ supp v) :
    (v.onQuot hJ).comap (Ideal.Quotient.mk J) = v 
goal: AddValuation.comap (Ideal.Quotient.mk J) (v.onQuot hJ) = v

full_name: AddValuation.comap_supp
code: theorem comap_supp {S : Type*} [CommRing S] (f : S →+* R) :
    supp (v.comap f) = Ideal.comap f v.supp 
goal: (AddValuation.comap f v).supp = Ideal.comap f v.supp

full_name: AddValuation.self_le_supp_comap
code: theorem self_le_supp_comap (J : Ideal R) (v : AddValuation (R ⧸ J) Γ₀) :
    J ≤ (v.comap (Ideal.Quotient.mk J)).supp 
goal: J ≤ (AddValuation.comap (Ideal.Quotient.mk J) v).supp

full_name: AddValuation.comap_onQuot_eq
code: @[simp]
theorem comap_onQuot_eq (J : Ideal R) (v : AddValuation (R ⧸ J) Γ₀) :
    (v.comap (Ideal.Quotient.mk J)).onQuot (v.self_le_supp_comap J) = v 
goal: (AddValuation.comap (Ideal.Quotient.mk J) v).onQuot ⋯ = v

full_name: AddValuation.supp_quot
code: /-- The quotient valuation on `R / J` has support `(supp v) / J` if `J ⊆ supp v`. -/
theorem supp_quot {J : Ideal R} (hJ : J ≤ supp v) :
    supp (v.onQuot hJ) = (supp v).map (Ideal.Quotient.mk J) 
goal: (v.onQuot hJ).supp = Ideal.map (Ideal.Quotient.mk J) v.supp

full_name: AddValuation.supp_quot_supp
code: theorem supp_quot_supp : supp ((Valuation.onQuot v) le_rfl) = 0 
goal: AddValuation.supp (Valuation.onQuot v ⋯) = 0