closure_range_union_range_eq_top — Mathlib

English

The symm of tensorTensorTensorComm is equal to the swapped version of tensorTensorTensorComm.

Русский

Обратная конгруэнция тензор-тензор равна swapped версии tensorTensorTensorComm.

LaTeX

$$tensorTensorTensorComm_symm = tensorTensorTensorComm R S R' T A C B D$$

Lean4

theorem closure_range_union_range_eq_top [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] :
    Subring.closure
        (Set.range (Algebra.TensorProduct.includeLeft : A →ₐ[R] A ⊗[R] B) ∪
          Set.range Algebra.TensorProduct.includeRight) =
      ⊤ :=
  by
  rw [← top_le_iff]
  rintro x -
  induction x with
  | zero => exact zero_mem _
  | tmul x
    y =>
    convert_to (Algebra.TensorProduct.includeLeftRingHom (R := R) x) * (Algebra.TensorProduct.includeRight y) ∈ _
    · simp
    · exact mul_mem (Subring.subset_closure (.inl ⟨x, rfl⟩)) (Subring.subset_closure (.inr ⟨_, rfl⟩))
  | add x y _ _ => exact add_mem ‹_› ‹_›

Похожие утверждения