Theorems · Theorem · commutative algebra
FractionalIdeal.mul_mem_mul
∀ {R : Type u_1} [inst : CommRing R] {S : Submonoid R} {P : Type u_2} [inst_1 : CommRing P] [inst_2 : Algebra R P]
{I J : FractionalIdeal S P} {i j : P}, i ∈ I → j ∈ J → i * j ∈ I * J- Defined in
- Mathlib.RingTheory.FractionalIdeal.Basic
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 79 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
- Submonoidstatement and proof · cited by 3,086
- FractionalIdealstatement and proof · cited by 423
- Submodule.mul_mem_mulproof · cited by 27
- FractionalIdeal.mul_defproof · cited by 6
Cited by6
Results whose statement or proof uses this declaration.
- FractionalIdeal.eq_one_div_of_mul_eq_one_rightproof · cited by 3
- FractionalIdeal.le_dual_inv_auxproof · cited by 2
- FractionalIdeal.coe_ideal_mul_invproof · cited by 1
- FractionalIdeal.isPrincipal_of_unit_of_comap_mul_span_singleton_eq_topproof · cited by 1
- FractionalIdeal.mem_singleton_mulproof · cited by 1
- FractionalIdeal.extended_mulproof · cited by 0