Theorems · Theorem · commutative algebra
Submodule.mem_one
∀ {R : Type u} [inst : CommSemiring R] {A : Type v} [inst_1 : Semiring A] [inst_2 : Algebra R A] {x : A},
x ∈ 1 ↔ ∃ y, (algebraMap R A) y = x- Defined in
- Mathlib.Algebra.Algebra.Operations
- Cited by
- 17 results in Mathlib
- Foundations
- Depth 32 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringSemiringAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Semiringstatement and proof · cited by 13,802
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- RingHomstatement · cited by 10,189
- Submodulestatement · cited by 7,192
- Algebra.algebraMapstatement and proof · cited by 4,706
- Submodule.one_eq_rangeproof · cited by 11
Cited by17
Results whose statement or proof uses this declaration.
- LinearMap.BilinForm.dualSubmodule_span_of_basisproof · cited by 7
- Submodule.pow_induction_on_left'proof · cited by 5
- FractionalIdeal.mem_dualproof · cited by 3
- CliffordAlgebra.even_inductionproof · cited by 3
- Submodule.span_singleton_eq_one_iffproof · cited by 3
- Submodule.pow_induction_on_right'proof · cited by 2
- IsIntegralClosure.range_le_span_dualBasisproof · cited by 2
- Submodule.traceDual_le_span_map_traceDualproof · cited by 2
- FractionalIdeal.le_dual_inv_auxproof · cited by 2
- MvPolynomial.homogeneousSubmodule_zeroproof · cited by 1
- FractionalIdeal.isPrincipal_of_unit_of_comap_mul_span_singleton_eq_topproof · cited by 1
- Submodule.mem_traceDualproof · cited by 1