feat(linear_algebra): elements of a dim zero space (#3054)

Co-authored-by: Rob Lewis <rob.y.lewis@gmail.com>

src/linear_algebra/dimension.lean

@@ -412,6 +412,26 @@ by rw [rank, rank, linear_map.range_comp]; exact dim_map_le _ _
 
 end rank
 
+lemma dim_zero_iff_forall_zero : vector_space.dim K V = 0 ↔ ∀ x : V, x = 0 :=
+begin
+  split,
+  { intros h x,
+    cases exists_is_basis K V with w hw,
+    have card_mk_range := hw.mk_range_eq_dim,
+    rw [h, cardinal.mk_emptyc_iff, set.range_coe_subtype] at card_mk_range,
+    simpa [card_mk_range] using hw.mem_span x },
+  { intro h,
+    have : (⊤ : submodule K V) = ⊥,
+    { ext x, simp [h x] },
+    rw [←dim_top, this, dim_bot] }
+end
+
+lemma dim_pos_iff_exists_ne_zero : 0 < vector_space.dim K V  ↔ ∃ x : V, x ≠ 0 :=
+begin
+  rw ←not_iff_not,
+  simpa using dim_zero_iff_forall_zero
+end
+
 end vector_space
 
 section unconstrained_universes

src/linear_algebra/finite_dimensional.lean

@@ -366,3 +366,40 @@ theorem findim_range_add_findim_ker [finite_dimensional K V] (f : V →ₗ[K] V
 by { rw [← f.quot_ker_equiv_range.findim_eq], exact submodule.findim_quotient_add_findim _ }
 
 end linear_map
+
+section zero_dim
+
+open vector_space finite_dimensional
+
+lemma finite_dimensional_of_dim_eq_zero (h : vector_space.dim K V = 0) : finite_dimensional K V :=
+by rw [finite_dimensional_iff_dim_lt_omega, h]; exact cardinal.omega_pos
+
+lemma findim_eq_zero_of_dim_eq_zero [finite_dimensional K V] (h : vector_space.dim K V = 0) :
+  findim K V = 0 :=
+begin
+  convert findim_eq_dim K V,
+  rw h, norm_cast
+end
+
+variables (K V)
+
+lemma finite_dimensional_bot : finite_dimensional K (⊥ : submodule K V) :=
+finite_dimensional_of_dim_eq_zero $ by simp
+
+lemma findim_bot : findim K (⊥ : submodule K V) = 0 :=
+begin
+  haveI := finite_dimensional_bot K V,
+  convert findim_eq_dim K (⊥ : submodule K V),
+  rw dim_bot, norm_cast
+end
+
+variables {K V}
+
+lemma bot_eq_top_of_dim_eq_zero (h : vector_space.dim K V = 0) : (⊥ : submodule K V) = ⊤ :=
+begin
+  haveI := finite_dimensional_of_dim_eq_zero h,
+  apply eq_top_of_findim_eq,
+  rw [findim_bot, findim_eq_zero_of_dim_eq_zero h]
+end
+
+end zero_dim

src/set_theory/cardinal.lean

@@ -853,6 +853,17 @@ theorem mk_subtype_le_of_subset {α : Type u} {p q : α → Prop} (h : ∀ ⦃x
 @[simp] theorem mk_emptyc (α : Type u) : mk (∅ : set α) = 0 :=
 quotient.sound ⟨equiv.set.pempty α⟩
 
+lemma mk_emptyc_iff {α : Type u} {s : set α} : mk s = 0 ↔ s = ∅ :=
+begin
+  split,
+  { intro h,
+    have h2 : cardinal.mk s = cardinal.mk pempty, by simp [h],
+    refine set.eq_empty_iff_forall_not_mem.mpr (λ _ hx, _),
+    rcases cardinal.eq.mp h2 with ⟨f, _⟩,
+    cases f ⟨_, hx⟩ },
+  { intro, convert mk_emptyc _ }
+end
+
 theorem mk_univ {α : Type u} : mk (@univ α) = mk α :=
 quotient.sound ⟨equiv.set.univ α⟩