finrank_ker_add_one_of_ne_zero — Mathlib

English

If f ≠ 0 and V is finite-dimensional, and p is a subspace of V₁ that is disjoint from ker f and nontrivial, then V₁ splits as a direct sum of ker f and p.

Русский

Если f ≠ 0 и V конечномерно, и p — подпространство V с дополнением к ker f (потревоженное, но непустое), то V делится как прямая сумма ker f ⊕ p.

LaTeX

$$IsCompl(ker f, p)$$

Lean4

theorem finrank_ker_add_one_of_ne_zero : finrank K (LinearMap.ker f) + 1 = finrank K V₁ :=
  by
  suffices finrank K (LinearMap.range f) = 1 by
    rw [← (LinearMap.ker f).finrank_quotient_add_finrank, add_comm, add_left_inj, f.quotKerEquivRange.finrank_eq, this]
  rw [range_eq_top_of_ne_zero hf, finrank_top, finrank_self]

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