English
There exists a linear map M ⊗R N →A P corresponding to a bilinear map M →A N →R P, which is universal with respect to composing with the canonical bilinear map M →A N →R M ⊗R N.
Русский
Существует линейное отображение M ⊗R N →A P, соответствующее дву-кUSTOM BilinearMap M →A N →R P, которое является универсальным по отношению к композиции с каноническим билинейным отображением M →A N →R M ⊗R N.
LaTeX
$$$\exists \ell : M ⊗_R N \to_A P \,\text{ such that }\ \ell \circ {\text{mk}}_{M,N} = f$$$
Lean4
/-- Heterobasic version of `TensorProduct.lift`:
Constructing a linear map `M ⊗[R] N →[A] P` given a bilinear map `M →[A] N →[R] P` with the
property that its composition with the canonical bilinear map `M →[A] N →[R] M ⊗[R] N` is
the given bilinear map `M →[A] N →[R] P`. -/
nonrec def lift (f : M →ₗ[A] N →ₗ[R] P) : M ⊗[R] N →ₗ[A] P :=
{ lift (f.restrictScalars R) with
map_smul' := fun c =>
show
∀ x : M ⊗[R] N,
(lift (f.restrictScalars R)).comp (lsmul R R _ c) x = (lsmul R R _ c).comp (lift (f.restrictScalars R)) x
from
LinearMap.ext_iff.1 <|
TensorProduct.ext' fun x y => by
simp only [comp_apply, Algebra.lsmul_coe, smul_tmul', lift.tmul, coe_restrictScalars, f.map_smul,
smul_apply] }
