English
For the 1-matrix, cRank equals the lift of the cardinality of the index set, i.e., cRank(1) = lift #{m}.
Русский
Для единичной матрицы размерности m×m ранг по столбцам равен переносимойКардинАЛом мощности множества m: cRank(1) = lift #{m}.
LaTeX
$$$\operatorname{cRank}(1) = \operatorname{lift}\,\#m.$$$
Lean4
@[simp]
theorem cRank_one [Nontrivial R] [DecidableEq m] : (cRank (1 : Matrix m m R)) = lift.{uR} #m :=
by
have h : LinearIndependent R (1 : Matrix m m R)ᵀ :=
by
convert Pi.linearIndependent_single_one m R
simp [funext_iff, Matrix.one_eq_pi_single]
rw [cRank, rank_span h, ← lift_umax, ← Cardinal.mk_range_eq_of_injective h.injective, lift_id']
