Theorems · Theorem · ring theory
Submodule.iSup_eq_toSubmodule_range
∀ {ι : Type u_1} {S : Type u_3} {R : Type u_4} [inst : DecidableEq ι] [inst_1 : AddMonoid ι] [inst_2 : CommSemiring S]
[inst_3 : Semiring R] [inst_4 : Algebra S R] (A : ι → Submodule S R) [inst_5 : SetLike.GradedMonoid A],
⨆ i, A i = Subalgebra.toSubmodule (DirectSum.coeAlgHom A).rangeThe supremum of submodules that form a graded monoid is a subalgebra, and equal to the range of
DirectSum.coeAlgHom.
- Defined in
- Mathlib.Algebra.DirectSum.Internal
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 99 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites19
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
- Submodulestatement and proof · cited by 7,192
- AddMonoidstatement and proof · cited by 2,864
- iSupstatement · cited by 2,415
- Subalgebrastatement · cited by 1,353
- LinearMap.rangeproof · cited by 893
- OrderEmbeddingstatement · cited by 619
- Submodule.subtypeproof · cited by 480
- DirectSumstatement · cited by 446
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