Theorems · Theorem · commutative algebra

IsLocalization.AtPrime.equivQuotMaximalIdeal_apply_mk

∀ {R : Type u_7} [inst : CommRing R] (p : Ideal R) [inst_1 : p.IsMaximal] (Rₚ : Type u_8) [inst_2 : CommRing Rₚ]
  [inst_3 : Algebra R Rₚ] [inst_4 : IsLocalization.AtPrime Rₚ p] [inst_5 : IsLocalRing Rₚ] (x : R),
  (IsLocalization.AtPrime.equivQuotMaximalIdeal p Rₚ) ((Ideal.Quotient.mk p) x) =
    (Ideal.Quotient.mk (IsLocalRing.maximalIdeal Rₚ)) ((algebraMap R Rₚ) x)

Defined in: Mathlib.RingTheory.Localization.AtPrime.Basic
Cited by: 1 results in Mathlib
Foundations: Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing Ideal.IsMaximal CommRing Algebra IsLocalization.AtPrime IsLocalRing

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Cites14

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement · cited by 62,936
CommRingstatement and proof · cited by 17,173
Algebrastatement and proof · cited by 11,388
RingHomstatement · cited by 10,189
Idealstatement and proof · cited by 4,748
Algebra.algebraMapstatement · cited by 4,706
HasQuotient.Quotientstatement · cited by 2,301
RingEquivstatement · cited by 1,147
Ideal.Quotient.mkstatement · cited by 610
Ideal.IsMaximalstatement and proof · cited by 452
IsLocalRingstatement and proof · cited by 339
IsLocalRing.maximalIdealstatement · cited by 297

Cited by1

Results whose statement or proof uses this declaration.

IsLocalization.AtPrime.equivQuotMaximalIdeal_symm_apply_mkproof · cited by 1