feat(linear_algebra/affine_space/affine_subspace): prove that a set whose affine span is top cannot be empty. (#9113)

The lemma `finset.card_sdiff_add_card` is unrelated but I've been meaning to add it
and now seemed like a good time since I'm touching `data/finset/basic.lean` anyway.

Author: ocfnash

src/data/finset/basic.lean

@@ -313,6 +313,8 @@ protected def nonempty (s : finset α) : Prop := ∃ x:α, x ∈ s
 
 @[simp, norm_cast] lemma coe_nonempty {s : finset α} : (s:set α).nonempty ↔ s.nonempty := iff.rfl
 
+@[simp] lemma nonempty_coe_sort (s : finset α) : nonempty ↥s ↔ s.nonempty := nonempty_subtype
+
 alias coe_nonempty ↔ _ finset.nonempty.to_set
 
 lemma nonempty.bex {s : finset α} (h : s.nonempty) : ∃ x:α, x ∈ s := h
@@ -2862,6 +2864,9 @@ theorem card_sdiff {s t : finset α} (h : s ⊆ t) : card (t \ s) = card t - car
 suffices card (t \ s) = card ((t \ s) ∪ s) - card s, by rwa sdiff_union_of_subset h at this,
 by rw [card_disjoint_union sdiff_disjoint, nat.add_sub_cancel]
 
+lemma card_sdiff_add_card {s t : finset α} : (s \ t).card + t.card = (s ∪ t).card :=
+by rw [← card_disjoint_union sdiff_disjoint, sdiff_union_self_eq_union]
+
 lemma disjoint_filter {s : finset α} {p q : α → Prop} [decidable_pred p] [decidable_pred q] :
     disjoint (s.filter p) (s.filter q) ↔ (∀ x ∈ s, p x → ¬ q x) :=
 by split; simp [disjoint_left] {contextual := tt}

src/data/set/basic.lean

@@ -291,6 +291,8 @@ in theorem assumptions instead of `∃ x, x ∈ s` or `s ≠ ∅` as it gives ac
 to the dot notation. -/
 protected def nonempty (s : set α) : Prop := ∃ x, x ∈ s
 
+@[simp] lemma nonempty_coe_sort (s : set α) : nonempty ↥s ↔ s.nonempty := nonempty_subtype
+
 lemma nonempty_def : s.nonempty ↔ ∃ x, x ∈ s := iff.rfl
 
 lemma nonempty_of_mem {x} (h : x ∈ s) : s.nonempty := ⟨x, h⟩

src/linear_algebra/affine_space/affine_subspace.lean

@@ -351,6 +351,10 @@ begin
   exact h
 end
 
+@[simp] lemma ext_iff (s₁ s₂ : affine_subspace k P) :
+  (s₁ : set P) = s₂ ↔ s₁ = s₂ :=
+⟨ext, by tidy⟩
+
 /-- Two affine subspaces with the same direction and nonempty
 intersection are equal. -/
 lemma ext_of_direction_eq {s1 s2 : affine_subspace k P} (hd : s1.direction = s2.direction)
@@ -667,6 +671,21 @@ end
 @[simp] lemma bot_coe : ((⊥ : affine_subspace k P) : set P) = ∅ :=
 rfl
 
+lemma bot_ne_top : (⊥ : affine_subspace k P) ≠ ⊤ :=
+begin
+  intros contra,
+  rw [← ext_iff, bot_coe, top_coe] at contra,
+  exact set.empty_ne_univ contra,
+end
+
+lemma nonempty_of_affine_span_eq_top {s : set P} (h : affine_span k s = ⊤) : s.nonempty :=
+begin
+  rw ← set.ne_empty_iff_nonempty,
+  rintros rfl,
+  rw affine_subspace.span_empty at h,
+  exact bot_ne_top k V P h,
+end
+
 variables {P}
 
 /-- No points are in `⊥`. -/