Theorems · Theorem · linear algebra

LinearMap.IsProj.trace

∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M] {p : Submodule R M}
  {f : M →ₗ[R] M},
  LinearMap.IsProj p f →
    ∀ [Module.Free R ↥p] [Module.Finite R ↥p] [Module.Free R ↥f.ker] [Module.Finite R ↥f.ker],
      (LinearMap.trace R M) f = ↑(Module.finrank R ↥p)

Defined in: Mathlib.LinearAlgebra.Trace
Cited by: 3 results in Mathlib
Foundations: Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing AddCommGroup Module Module.Free Module.Finite Module.Free Module.Finite

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Cites23

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
Modulestatement and proof · cited by 20,661
RingHom.idstatement and proof · cited by 18,349
CommRingstatement and proof · cited by 17,173
AddCommGroupstatement and proof · cited by 12,871
LinearMapstatement and proof · cited by 10,215
Submodulestatement and proof · cited by 7,192
add_zeroproof · cited by 2,707
Module.finrankstatement and proof · cited by 1,770
map_zeroproof · cited by 1,614
Module.Finitestatement and proof · cited by 1,032
LinearMap.kerstatement and proof · cited by 848

Cited by3

Results whose statement or proof uses this declaration.

FDRep.average_char_eq_finrank_invariantsproof · cited by 1
Representation.card_inv_mul_sum_char_eq_finrankproof · cited by 1
LinearMap.IsIdempotentElem.trace_eq_zero_iffproof · cited by 1