File: Mathlib/Data/Finset/Sum.lean

full_name: Finset.disjSum
code: /-- Disjoint sum of finsets. -/
def disjSum : Finset (α ⊕ β) 
goal: Finset (α ⊕ β)

full_name: Finset.val_disjSum
code: @[simp]
theorem val_disjSum : (s.disjSum t).1 = s.1.disjSum t.1 
goal: (s.disjSum t).val = s.val.disjSum t.val

full_name: Finset.empty_disjSum
code: @[simp]
theorem empty_disjSum : (∅ : Finset α).disjSum t = t.map Embedding.inr 
goal: ∅.disjSum t = Finset.map Function.Embedding.inr t

full_name: Finset.disjSum_empty
code: @[simp]
theorem disjSum_empty : s.disjSum (∅ : Finset β) = s.map Embedding.inl 
goal: s.disjSum ∅ = Finset.map Function.Embedding.inl s

full_name: Finset.card_disjSum
code: @[simp]
theorem card_disjSum : (s.disjSum t).card = s.card + t.card 
goal: (s.disjSum t).card = s.card + t.card

full_name: Finset.disjoint_map_inl_map_inr
code: theorem disjoint_map_inl_map_inr : Disjoint (s.map Embedding.inl) (t.map Embedding.inr) 
goal: Disjoint (Finset.map Function.Embedding.inl s) (Finset.map Function.Embedding.inr t)

full_name: Finset.map_inl_disjUnion_map_inr
code: @[simp]
theorem map_inl_disjUnion_map_inr :
    (s.map Embedding.inl).disjUnion (t.map Embedding.inr) (disjoint_map_inl_map_inr _ _) =
      s.disjSum t 
goal: (Finset.map Function.Embedding.inl s).disjUnion (Finset.map Function.Embedding.inr t) ⋯ = s.disjSum t

full_name: Finset.mem_disjSum
code: theorem mem_disjSum : x ∈ s.disjSum t ↔ (∃ a, a ∈ s ∧ inl a = x) ∨ ∃ b, b ∈ t ∧ inr b = x 
goal: x ∈ s.disjSum t ↔ (∃ a ∈ s, Sum.inl a = x) ∨ ∃ b ∈ t, Sum.inr b = x

full_name: Finset.inl_mem_disjSum
code: @[simp]
theorem inl_mem_disjSum : inl a ∈ s.disjSum t ↔ a ∈ s 
goal: Sum.inl a ∈ s.disjSum t ↔ a ∈ s

full_name: Finset.inr_mem_disjSum
code: @[simp]
theorem inr_mem_disjSum : inr b ∈ s.disjSum t ↔ b ∈ t 
goal: Sum.inr b ∈ s.disjSum t ↔ b ∈ t

full_name: Finset.disjSum_eq_empty
code: @[simp]
theorem disjSum_eq_empty : s.disjSum t = ∅ ↔ s = ∅ ∧ t = ∅ 
goal: s.disjSum t = ∅ ↔ s = ∅ ∧ t = ∅

full_name: Finset.disjSum_mono
code: theorem disjSum_mono (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t₂) : s₁.disjSum t₁ ⊆ s₂.disjSum t₂ 
goal: s₁.disjSum t₁ ⊆ s₂.disjSum t₂

full_name: Finset.disjSum_mono_left
code: theorem disjSum_mono_left (t : Finset β) : Monotone fun s : Finset α => s.disjSum t 
goal: Monotone fun s => s.disjSum t

full_name: Finset.disjSum_mono_right
code: theorem disjSum_mono_right (s : Finset α) : Monotone (s.disjSum : Finset β → Finset (α ⊕ β)) 
goal: Monotone s.disjSum

full_name: Finset.disjSum_ssubset_disjSum_of_ssubset_of_subset
code: theorem disjSum_ssubset_disjSum_of_ssubset_of_subset (hs : s₁ ⊂ s₂) (ht : t₁ ⊆ t₂) :
    s₁.disjSum t₁ ⊂ s₂.disjSum t₂ 
goal: s₁.disjSum t₁ ⊂ s₂.disjSum t₂

full_name: Finset.disjSum_ssubset_disjSum_of_subset_of_ssubset
code: theorem disjSum_ssubset_disjSum_of_subset_of_ssubset (hs : s₁ ⊆ s₂) (ht : t₁ ⊂ t₂) :
    s₁.disjSum t₁ ⊂ s₂.disjSum t₂ 
goal: s₁.disjSum t₁ ⊂ s₂.disjSum t₂

full_name: Finset.disjSum_strictMono_left
code: theorem disjSum_strictMono_left (t : Finset β) : StrictMono fun s : Finset α => s.disjSum t 
goal: StrictMono fun s => s.disjSum t

full_name: Finset.disj_sum_strictMono_right
code: theorem disj_sum_strictMono_right (s : Finset α) :
    StrictMono (s.disjSum : Finset β → Finset (α ⊕ β)) 
goal: StrictMono s.disjSum