cardinalMk_algHom_le_rank — Mathlib

English

The cardinality of the algebra hom set is bounded by the rank of M over K, after lifting.

Русский

Кардинал множества алгомоморфизм ограничен рангом M над K после подъема.

LaTeX

$$$$ \#(\mathrm{AlgHom}_K(M, L)) \le \mathrm{lift}(\mathrm{rank}_K M). $$$$

Lean4

theorem cardinalMk_algHom_le_rank : #(M →ₐ[K] L) ≤ lift.{v} (Module.rank K M) :=
  by
  convert (linearIndependent_algHom_toLinearMap K M L).cardinal_lift_le_rank
  · rw [lift_id]
  · have := Module.nontrivial K L
    rw [lift_id, Module.rank_linearMap_self]

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