trace_prodMap' — Mathlib

English

For endomorphisms f on M and g on N, the trace on End(M×N) of the product map equals the sum of the traces: tr_{M×N}(f.prodMap g) = tr_M(f) + tr_N(g).

Русский

Для концевых отображений f: M→M и g: N→N вернётся: tr_{M×N}(f.prodMap g) = tr_M(f) + tr_N(g).

LaTeX

$$$ \operatorname{trace}_R(M \times N)(f.prodMap g) = \operatorname{trace}_R M f + \operatorname{trace}_R N g. $$$

Lean4

theorem trace_prodMap' (f : M →ₗ[R] M) (g : N →ₗ[R] N) : trace R (M × N) (prodMap f g) = trace R M f + trace R N g :=
  by
  have h := LinearMap.ext_iff.1 (trace_prodMap R M N) (f, g)
  simp only [coe_comp, Function.comp_apply, prodMap_apply, coprod_apply, id, prodMapLinear_apply] at h
  exact h

Похожие утверждения