trace_comp_comm' — Mathlib

English

For 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 f на M и g на N трace на End(M×N) от prodMap f g равен сумме следов: tr_M(f) + tr_N(g).

LaTeX

$$$ \operatorname{trace}_R (\mathrm{Prod}\;M\;N)(f,g) = \operatorname{trace}_R M f + \operatorname{trace}_R N g. $$$

Lean4

theorem trace_comp_comm' (f : M →ₗ[R] N) (g : N →ₗ[R] M) : trace R M (g ∘ₗ f) = trace R N (f ∘ₗ g) :=
  by
  have h := LinearMap.ext_iff.1 (LinearMap.ext_iff.1 (trace_comp_comm R M N) g) f
  simp only [llcomp_apply', compr₂_apply, flip_apply] at h
  exact h

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