tmul_comp_tensorComm — Mathlib

English

CategoryTheory.Iso.toIsometryEquiv is compatible with composition of isomorphisms, i.e., for e: X ≅ Y and f: Y ≅ Z, toIsometryEquiv (e ≪≫ f) equals e.toIsometryEquiv.trans f.toIsometryEquiv.

Русский

CategoryTheory.Iso.toIsometryEquiv совместим с композициями изоморфизмов: toIsometryEquiv (e ≪≫ f) = e.toIsometryEquiv.trans f.toIsometryEquiv.

LaTeX

$$$\\mathrm{toIsometryEquiv}(e \\overset{\\mathrm{trans}}{\\circ} f) = (\\mathrm{toIsometryEquiv}(e)) \\mathrm{.trans} (\\mathrm{toIsometryEquiv}(f))$$$

Lean4

@[simp]
theorem tmul_comp_tensorComm (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) :
    (Q₂.tmul Q₁).comp (TensorProduct.comm R M₁ M₂) = Q₁.tmul Q₂ :=
  by
  refine (QuadraticMap.associated_rightInverse R).injective ?_
  ext m₁ m₂ m₁' m₂'
  dsimp [-associated_apply]
  simp only [associated_tmul, QuadraticMap.associated_comp]
  exact mul_comm _ _

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