Theorems · Theorem · commutative algebra
FractionalIdeal.mem_coe
∀ {R : Type u_1} [inst : CommRing R] {S : Submonoid R} {P : Type u_2} [inst_1 : CommRing P] [inst_2 : Algebra R P]
{I : FractionalIdeal S P} {x : P}, x ∈ ↑I ↔ x ∈ I- Defined in
- Mathlib.RingTheory.FractionalIdeal.Basic
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 27 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Submodulestatement · cited by 7,192
- Submonoidstatement and proof · cited by 3,086
- FractionalIdealstatement and proof · cited by 423
- FractionalIdeal.coeToSubmodulestatement · cited by 130
Cited by9
Results whose statement or proof uses this declaration.
- FractionalIdeal.div_spanSingletonproof · cited by 4
- FractionalIdeal.spanSingleton_le_iff_memproof · cited by 3
- FractionalIdeal.one_leproof · cited by 2
- FractionalIdeal.den_mem_invproof · cited by 1
- dvd_differentIdeal_of_not_isSeparableproof · cited by 1
- pow_sub_one_dvd_differentIdeal_auxproof · cited by 1
- FractionalIdeal.mem_addproof · cited by 1
- conductor_mul_differentIdealproof · cited by 1
- FractionalIdeal.mem_span_mul_finite_of_mem_mulproof · cited by 0