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Theorems · Theorem · ring theory

Algebra.toSubmodule_bot

∀ {R : Type u} {A : Type v} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A],
  Subalgebra.toSubmodule ⊥ = 1

TODO: change proof to rfl when fixing https://github.com/leanprover-community/mathlib4/issues/18110.

Defined in
Mathlib.Algebra.Algebra.Subalgebra.Lattice
Cited by
10 results in Mathlib
Foundations
Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
CommSemiringSemiringAlgebra
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

fg_adjoin_of_finite · cited by 9fg_adjoin_of_finiteSubalgebra.LinearDisjoint.bot_right · cited by 3LinearDisjoint.bot_rightSubalgebra.rank_eq_one_iff · cited by 3Subalgebra.rank_eq_one_iffSubmodule.LinearDisjoint.one_left · cited by 2LinearDisjoint.one_leftSubmodule.LinearDisjoint.one_right · cited by 2LinearDisjoint.one_rightSubalgebra.LinearDisjoint.bot_left · cited by 2LinearDisjoint.bot_leftIsCyclotomicExtension.finite · cited by 1IsCyclotomicExtension.fin…Subalgebra.rank_bot · cited by 1Subalgebra.rank_botAlgebra.map_bot · cited by 1Algebra.map_botSubalgebra.fg_bot_toSubmodule · cited by 0Subalgebra.fg_bot_toSubmo…DFunLike.coe · cited by 62936DFunLike.coeSemiring · cited by 13802SemiringAlgebra · cited by 11388AlgebraCommSemiring · cited by 10911CommSemiringSubmodule · cited by 7192SubmoduleBot.bot · cited by 4720Bot.botSubalgebra · cited by 1353SubalgebraOrderEmbedding · cited by 619OrderEmbeddingSubalgebra.toSubmodule · cited by 141Subalgebra.toSubmoduleSubmodule.one_eq_range · cited by 11Submodule.one_eq_rangeAlgebra.toSubmodule_botCITED BYCITES

Cites10

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

Cited by10

Results whose statement or proof uses this declaration.