feat: add Subalgebra.finite_(bot|sup) (#12025)

... and deprecated `Subalgebra.finiteDimensional_(bot|sup)`

Author: acmepjz

Mathlib/FieldTheory/Adjoin.lean

@@ -1194,7 +1194,7 @@ theorem exists_lt_finrank_of_infinite_dimensional
     (halg : Algebra.IsAlgebraic F E) (hnfd : ¬ FiniteDimensional F E) (n : ℕ) :
     ∃ L : IntermediateField F E, FiniteDimensional F L ∧ n < finrank F L := by
   induction' n with n ih
-  · exact ⟨⊥, Subalgebra.finiteDimensional_bot, finrank_pos⟩
+  · exact ⟨⊥, Subalgebra.finite_bot, finrank_pos⟩
   obtain ⟨L, fin, hn⟩ := ih
   obtain ⟨x, hx⟩ : ∃ x : E, x ∉ L := by
     contrapose! hnfd

Mathlib/LinearAlgebra/FiniteDimensional.lean

@@ -1143,9 +1143,9 @@ instance FiniteDimensional.finiteDimensional_subalgebra [FiniteDimensional F E]
   FiniteDimensional.of_subalgebra_toSubmodule inferInstance
 #align finite_dimensional.finite_dimensional_subalgebra FiniteDimensional.finiteDimensional_subalgebra
 
-instance Subalgebra.finiteDimensional_bot : FiniteDimensional F (⊥ : Subalgebra F E) := by
-  nontriviality E
-  exact .of_rank_eq_one Subalgebra.rank_bot
+@[deprecated Subalgebra.finite_bot] -- 2024-04-11
+theorem Subalgebra.finiteDimensional_bot : FiniteDimensional F (⊥ : Subalgebra F E) :=
+  Subalgebra.finite_bot
 #align subalgebra.finite_dimensional_bot Subalgebra.finiteDimensional_bot
 
 theorem Subalgebra.eq_bot_of_rank_le_one {S : Subalgebra F E} (h : Module.rank F S ≤ 1) :

Mathlib/RingTheory/Finiteness.lean

@@ -890,3 +890,6 @@ theorem of_comp_finite {f : A →ₐ[R] B} {g : B →ₐ[R] C} (h : (g.comp f).F
 end Finite
 
 end AlgHom
+
+instance Subalgebra.finite_bot {F E : Type*} [CommSemiring F] [Semiring E] [Algebra F E] :
+    Module.Finite F (⊥ : Subalgebra F E) := Module.Finite.range (Algebra.linearMap F E)

Mathlib/RingTheory/TensorProduct/Basic.lean

@@ -1154,11 +1154,16 @@ theorem endTensorEndAlgHom_apply (f : End A M) (g : End R N) :
 
 end Module
 
+theorem Subalgebra.finite_sup {K L : Type*} [CommRing K] [CommRing L] [Algebra K L]
+    (E1 E2 : Subalgebra K L) [Module.Finite K E1] [Module.Finite K E2] :
+    Module.Finite K ↥(E1 ⊔ E2) := by
+  rw [← E1.range_val, ← E2.range_val, ← Algebra.TensorProduct.productMap_range]
+  exact Module.Finite.range (Algebra.TensorProduct.productMap E1.val E2.val).toLinearMap
+
+@[deprecated Subalgebra.finite_sup] -- 2024-04-11
 theorem Subalgebra.finiteDimensional_sup {K L : Type*} [Field K] [CommRing L] [Algebra K L]
     (E1 E2 : Subalgebra K L) [FiniteDimensional K E1] [FiniteDimensional K E2] :
-    FiniteDimensional K (E1 ⊔ E2 : Subalgebra K L) := by
-  rw [← E1.range_val, ← E2.range_val, ← Algebra.TensorProduct.productMap_range]
-  exact (Algebra.TensorProduct.productMap E1.val E2.val).toLinearMap.finiteDimensional_range
+    FiniteDimensional K (E1 ⊔ E2 : Subalgebra K L) := Subalgebra.finite_sup E1 E2
 #align subalgebra.finite_dimensional_sup Subalgebra.finiteDimensional_sup
 
 namespace TensorProduct.Algebra