Theorems · Theorem · linear algebra
Matrix.rank_mul_eq_left_of_det_mem_nonZeroDivisors
∀ {m : Type um} {n : Type un} [inst : Fintype n] {R : Type u_1} [inst_1 : CommRing R] [inst_2 : DecidableEq n]
(A : Matrix n n R) (B : Matrix m n R), A.det ∈ nonZeroDivisors R → (B * A).rank = B.rank- Defined in
- Mathlib.LinearAlgebra.Matrix.Rank
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 120 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FintypeCommRingDecidableEq
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Fintypestatement and proof · cited by 7,736
- Matrixstatement and proof · cited by 4,303
- Submonoidstatement · cited by 3,086
- Nontrivialproof · cited by 2,416
- le_antisymmproof · cited by 2,068
- nonZeroDivisorsstatement and proof · cited by 895
- Matrix.detstatement and proof · cited by 665
- Matrix.adjugateproof · cited by 64
- Matrix.mul_assocproof · cited by 56
- Matrix.rankstatement and proof · cited by 48
- Matrix.mul_oneproof · cited by 45
Cited by2
Results whose statement or proof uses this declaration.
- Matrix.rank_mul_eq_left_of_isUnit_detproof · cited by 2
- Matrix.rank_mul_eq_left_of_det_ne_zeroproof · cited by 0