File: Mathlib/Data/Set/SMulAntidiagonal.lean

full_name: Set.smulAntidiagonal
code: /-- `smulAntidiagonal s t a` is the set of all pairs of an element in `s` and an
      element in `t` that scalar multiply to `a`.-/
@[to_additive "`vaddAntidiagonal s t a` is the set of all pairs of an element in `s` and an
      element in `t` that vector-add to `a`."]
def smulAntidiagonal (s : Set G) (t : Set P) (a : P) : Set (G × P) 
goal: Set (G × P)

full_name: Set.vaddAntidiagonal
code: /-- `smulAntidiagonal s t a` is the set of all pairs of an element in `s` and an
      element in `t` that scalar multiply to `a`.-/
@[to_additive "`vaddAntidiagonal s t a` is the set of all pairs of an element in `s` and an
      element in `t` that vector-add to `a`."]
def smulAntidiagonal (s : Set G) (t : Set P) (a : P) : Set (G × P) 
goal: Set (G × P)

full_name: Set.mem_smulAntidiagonal
code: @[to_additive (attr := simp)]
theorem mem_smulAntidiagonal : x ∈ smulAntidiagonal s t a ↔ x.1 ∈ s ∧ x.2 ∈ t ∧ x.1 • x.2 = a 
goal: x ∈ s.smulAntidiagonal t a ↔ x.1 ∈ s ∧ x.2 ∈ t ∧ x.1 • x.2 = a

full_name: Set.mem_vaddAntidiagonal
code: @[to_additive (attr := simp)]
theorem mem_smulAntidiagonal : x ∈ smulAntidiagonal s t a ↔ x.1 ∈ s ∧ x.2 ∈ t ∧ x.1 • x.2 = a 
goal: x ∈ s.vaddAntidiagonal t a ↔ x.1 ∈ s ∧ x.2 ∈ t ∧ x.1 +ᵥ x.2 = a

full_name: Set.smulAntidiagonal_mono_left
code: @[to_additive]
theorem smulAntidiagonal_mono_left (h : s₁ ⊆ s₂) :
    smulAntidiagonal s₁ t a ⊆ smulAntidiagonal s₂ t a 
goal: s₁.smulAntidiagonal t a ⊆ s₂.smulAntidiagonal t a

full_name: Set.vaddAntidiagonal_mono_left
code: @[to_additive]
theorem smulAntidiagonal_mono_left (h : s₁ ⊆ s₂) :
    smulAntidiagonal s₁ t a ⊆ smulAntidiagonal s₂ t a 
goal: s₁.vaddAntidiagonal t a ⊆ s₂.vaddAntidiagonal t a

full_name: Set.vaddAntidiagonal_mono_right
code: @[to_additive]
theorem smulAntidiagonal_mono_right (h : t₁ ⊆ t₂) :
    smulAntidiagonal s t₁ a ⊆ smulAntidiagonal s t₂ a 
goal: s.vaddAntidiagonal t₁ a ⊆ s.vaddAntidiagonal t₂ a

full_name: Set.smulAntidiagonal_mono_right
code: @[to_additive]
theorem smulAntidiagonal_mono_right (h : t₁ ⊆ t₂) :
    smulAntidiagonal s t₁ a ⊆ smulAntidiagonal s t₂ a 
goal: s.smulAntidiagonal t₁ a ⊆ s.smulAntidiagonal t₂ a

full_name: Set.VAddAntidiagonal.fst_eq_fst_iff_snd_eq_snd
code: @[to_additive VAddAntidiagonal.fst_eq_fst_iff_snd_eq_snd]
theorem fst_eq_fst_iff_snd_eq_snd :
    (x : G × P).1 = (y : G × P).1 ↔ (x : G × P).2 = (y : G × P).2 
goal: (↑x).1 = (↑y).1 ↔ (↑x).2 = (↑y).2

full_name: Set.SMulAntidiagonal.fst_eq_fst_iff_snd_eq_snd
code: @[to_additive VAddAntidiagonal.fst_eq_fst_iff_snd_eq_snd]
theorem fst_eq_fst_iff_snd_eq_snd :
    (x : G × P).1 = (y : G × P).1 ↔ (x : G × P).2 = (y : G × P).2 
goal: (↑x).1 = (↑y).1 ↔ (↑x).2 = (↑y).2

full_name: Set.SMulAntidiagonal.eq_of_fst_eq_fst
code: @[to_additive VAddAntidiagonal.eq_of_fst_eq_fst]
theorem eq_of_fst_eq_fst (h : (x : G × P).fst = (y : G × P).fst) : x = y 
goal: x = y

full_name: Set.VAddAntidiagonal.eq_of_fst_eq_fst
code: @[to_additive VAddAntidiagonal.eq_of_fst_eq_fst]
theorem eq_of_fst_eq_fst (h : (x : G × P).fst = (y : G × P).fst) : x = y 
goal: x = y

full_name: Set.VAddAntidiagonal.eq_of_snd_eq_snd
code: @[to_additive VAddAntidiagonal.eq_of_snd_eq_snd]
theorem eq_of_snd_eq_snd (h : (x : G × P).snd = (y : G × P).snd) : x = y 
goal: x = y

full_name: Set.SMulAntidiagonal.eq_of_snd_eq_snd
code: @[to_additive VAddAntidiagonal.eq_of_snd_eq_snd]
theorem eq_of_snd_eq_snd (h : (x : G × P).snd = (y : G × P).snd) : x = y 
goal: x = y

full_name: Set.VAddAntidiagonal.eq_of_fst_le_fst_of_snd_le_snd
code: @[to_additive VAddAntidiagonal.eq_of_fst_le_fst_of_snd_le_snd]
theorem eq_of_fst_le_fst_of_snd_le_snd (h₁ : (x : G × P).1 ≤ (y : G × P).1)
    (h₂ : (x : G × P).2 ≤ (y : G × P).2) : x = y 
goal: x = y

full_name: Set.SMulAntidiagonal.eq_of_fst_le_fst_of_snd_le_snd
code: @[to_additive VAddAntidiagonal.eq_of_fst_le_fst_of_snd_le_snd]
theorem eq_of_fst_le_fst_of_snd_le_snd (h₁ : (x : G × P).1 ≤ (y : G × P).1)
    (h₂ : (x : G × P).2 ≤ (y : G × P).2) : x = y 
goal: x = y

full_name: Set.VAddAntidiagonal.finite_of_isPWO
code: @[to_additive VAddAntidiagonal.finite_of_isPWO]
theorem finite_of_isPWO (hs : s.IsPWO) (ht : t.IsPWO) (a) : (smulAntidiagonal s t a).Finite 
goal: (s.vaddAntidiagonal t a).Finite

full_name: Set.SMulAntidiagonal.finite_of_isPWO
code: @[to_additive VAddAntidiagonal.finite_of_isPWO]
theorem finite_of_isPWO (hs : s.IsPWO) (ht : t.IsPWO) (a) : (smulAntidiagonal s t a).Finite 
goal: (s.smulAntidiagonal t a).Finite