Theorems · Theorem · commutative algebra
inv_natCast_smul_comm
∀ {α : Type u_5} {E : Type u_6} (R : Type u_7) [inst : AddCommMonoid E] [inst_1 : DivisionSemiring R]
[inst_2 : Module R E] [inst_3 : DistribSMul α E] (n : ℕ) (s : α) (x : E), (↑n)⁻¹ • s • x = s • (↑n)⁻¹ • xIf E is a vector space over a division semiring R and has a monoid action by α, then that
action commutes by scalar multiplication of inverses of natural numbers in R.
- Defined in
- Mathlib.Algebra.Module.Basic
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 29 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- AddCommMonoidstatement and proof · cited by 12,281
- DivisionSemiringstatement and proof · cited by 216
- DistribSMulstatement and proof · cited by 117
- DistribSMul.toAddMonoidHomproof · cited by 20
- map_inv_natCast_smulproof · cited by 8
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