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Theorems · Theorem · commutative algebra

AddSubmonoid.addSubmonoid_smul_bot

∀ {R : Type u_2} {A : Type u_3} [inst : AddMonoid R] [inst_1 : AddMonoid A] [inst_2 : DistribSMul R A]
  (S : AddSubmonoid R), S • ⊥ = ⊥
Defined in
Mathlib.Algebra.Ring.Submonoid.Pointwise
Cited by
2 results in Mathlib
Foundations
Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
AddMonoidAddMonoidDistribSMul
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Around this declaration

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

Submodule.smul_bot · cited by 2Submodule.smul_botAddSubmonoid.mul_bot · cited by 0AddSubmonoid.mul_botBot.bot · cited by 4720Bot.botAddMonoid · cited by 2864AddMonoidAddSubmonoid · cited by 1178AddSubmonoidsmul_zero · cited by 665smul_zeroeq_bot_iff · cited by 159eq_bot_iffDistribSMul · cited by 117DistribSMulAddSubmonoid.mem_bot · cited by 12AddSubmonoid.mem_botAddSubmonoid.smul · cited by 11AddSubmonoid.smulAddSubmonoid.smul_le · cited by 8AddSubmonoid.smul_leAddSubmonoid.addSubmonoid_smu…CITED BYCITES

Cites9

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Cited by2

Results whose statement or proof uses this declaration.