English
Let W be a K-vector space. If rank_K V = n • rank_K W for some n ≠ 0, then FiniteDimensional K V ↔ FiniteDimensional K W.
Русский
Пусть W — векторное пространство над K. Если ранк_K V = n • ранк_K W для некоторого n ≠ 0, то FiniteDimensional K V эквивалентно FiniteDimensional K W.
LaTeX
$$∃ n ≠ 0, Module.rank K V = n • Module.rank K W ⇒ FiniteDimensional K V ↔ FiniteDimensional K W$$
Lean4
/-- The dimension of a strict submodule is strictly bounded by the dimension of the ambient
space.
See also `Submodule.length_lt`. -/
theorem finrank_lt [FiniteDimensional K V] {s : Submodule K V} (h : s ≠ ⊤) : finrank K s < finrank K V :=
by
rw [← s.finrank_quotient_add_finrank, add_comm]
exact Nat.lt_add_of_pos_right (finrank_pos_iff.mpr (Quotient.nontrivial_of_lt_top _ h.lt_top))
