state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : ι → Set α
hs : Directed (fun x x_1 => x ⊆ x_1) s
f : α → β
hf : ∀ (i : ι), InjOn f (s i)
x : α
hx : x ∈ ⋃ i, s i
y : α
hy : y ∈ ⋃ i, s i
hxy : f x = f y
⊢ x = y | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rcases mem_iUnion.1 hx with ⟨i, hx⟩ | theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) := by
intro x hx y hy hxy
| Mathlib.Data.Set.Lattice.1708_0.5mONj49h3SYSDwc | theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) | Mathlib_Data_Set_Lattice |
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : ι → Set α
hs : Directed (fun x x_1 => x ⊆ x_1) s
f : α → β
hf : ∀ (i : ι), InjOn f (s i)
x : α
hx✝ : x ∈ ⋃ i, s i
y : α
hy : y ∈ ⋃ i, s i
hxy : f x = f ... | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rcases mem_iUnion.1 hy with ⟨j, hy⟩ | theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) := by
intro x hx y hy hxy
rcases mem_iUnion.1 hx with ⟨i, hx⟩
| Mathlib.Data.Set.Lattice.1708_0.5mONj49h3SYSDwc | theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) | Mathlib_Data_Set_Lattice |
case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : ι → Set α
hs : Directed (fun x x_1 => x ⊆ x_1) s
f : α → β
hf : ∀ (i : ι), InjOn f (s i)
x : α
hx✝ : x ∈ ⋃ i, s i
y : α
hy✝ : y ∈ ⋃ i, s i
hxy : f... | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rcases hs i j with ⟨k, hi, hj⟩ | theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) := by
intro x hx y hy hxy
rcases mem_iUnion.1 hx with ⟨i, hx⟩
rcases mem_iUnion.1 hy with ⟨j, hy⟩
| Mathlib.Data.Set.Lattice.1708_0.5mONj49h3SYSDwc | theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) | Mathlib_Data_Set_Lattice |
case intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : ι → Set α
hs : Directed (fun x x_1 => x ⊆ x_1) s
f : α → β
hf : ∀ (i : ι), InjOn f (s i)
x : α
hx✝ : x ∈ ⋃ i, s i
y : α
hy✝ : y ∈ ⋃ i,... | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | exact hf k (hi hx) (hj hy) hxy | theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) := by
intro x hx y hy hxy
rcases mem_iUnion.1 hx with ⟨i, hx⟩
rcases mem_iUnion.1 hy with ⟨j, hy⟩
rcases hs i j with ⟨k, hi, hj⟩
| Mathlib.Data.Set.Lattice.1708_0.5mONj49h3SYSDwc | theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : Nonempty ι
s : ι → Set α
t : Set β
f : α → β
H : ∀ (i : ι), SurjOn f (s i) t
Hinj : InjOn f (⋃ i, s i)
⊢ SurjOn f (⋂ i, s i) t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | intro y hy | theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t := by
| Mathlib.Data.Set.Lattice.1747_0.5mONj49h3SYSDwc | theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : Nonempty ι
s : ι → Set α
t : Set β
f : α → β
H : ∀ (i : ι), SurjOn f (s i) t
Hinj : InjOn f (⋃ i, s i)
y : β
hy : y ∈ t
⊢ y ∈ f '' ⋂ i, s i | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [Hinj.image_iInter_eq, mem_iInter] | theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t := by
intro y hy
| Mathlib.Data.Set.Lattice.1747_0.5mONj49h3SYSDwc | theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : Nonempty ι
s : ι → Set α
t : Set β
f : α → β
H : ∀ (i : ι), SurjOn f (s i) t
Hinj : InjOn f (⋃ i, s i)
y : β
hy : y ∈ t
⊢ ∀ (i : ι), y ∈ f '' s i | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | exact fun i => H i hy | theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t := by
intro y hy
rw [Hinj.image_iInter_eq, mem_iInter]
| Mathlib.Data.Set.Lattice.1747_0.5mONj49h3SYSDwc | theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β
s : ι → Set α
⊢ f '' ⋃ i, s i = ⋃ i, f '' s i | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | ext1 x | theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i := by
| Mathlib.Data.Set.Lattice.1791_0.5mONj49h3SYSDwc | theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i | Mathlib_Data_Set_Lattice |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β
s : ι → Set α
x : β
⊢ x ∈ f '' ⋃ i, s i ↔ x ∈ ⋃ i, f '' s i | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp only [mem_image, mem_iUnion, ← exists_and_right, ← exists_and_left] | theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i := by
ext1 x
| Mathlib.Data.Set.Lattice.1791_0.5mONj49h3SYSDwc | theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i | Mathlib_Data_Set_Lattice |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β
s : ι → Set α
x : β
⊢ (∃ x_1 x_2, x_1 ∈ s x_2 ∧ f x_1 = x) ↔ ∃ i, ∃ x_1 ∈ s i, f x_1 = x | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [exists_swap] | theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i := by
ext1 x
simp only [mem_image, mem_iUnion, ← exists_and_right, ← exists_and_left]
--Porting note: `exists_swap` causes a `simp` loop in Lean4 so we use `rw` instead.
| Mathlib.Data.Set.Lattice.1791_0.5mONj49h3SYSDwc | theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β
s : (i : ι) → κ i → Set α
⊢ f '' ⋃ i, ⋃ j, s i j = ⋃ i, ⋃ j, f '' s i j | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp_rw [image_iUnion] | theorem image_iUnion₂ (f : α → β) (s : ∀ i, κ i → Set α) :
(f '' ⋃ (i) (j), s i j) = ⋃ (i) (j), f '' s i j := by | Mathlib.Data.Set.Lattice.1800_0.5mONj49h3SYSDwc | theorem image_iUnion₂ (f : α → β) (s : ∀ i, κ i → Set α) :
(f '' ⋃ (i) (j), s i j) = ⋃ (i) (j), f '' s i j | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
p : α → Prop
x✝ : Subtype p
x : α
h : p x
⊢ { val := x, property := h } ∈ univ ↔ { val := x, property := h } ∈ ⋃ x, ⋃ (h : p x), {{ val := x, property := h }} | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp [h] | theorem univ_subtype {p : α → Prop} : (univ : Set (Subtype p)) = ⋃ (x) (h : p x), {⟨x, h⟩} :=
Set.ext fun ⟨x, h⟩ => by | Mathlib.Data.Set.Lattice.1804_0.5mONj49h3SYSDwc | theorem univ_subtype {p : α → Prop} : (univ : Set (Subtype p)) = ⋃ (x) (h : p x), {⟨x, h⟩} | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι✝ → Sort u_7
κ₁ : ι✝ → Sort u_8
κ₂ : ι✝ → Sort u_9
κ' : ι' → Sort u_10
ι : Sort u_11
f : ι → α
a : α
⊢ a ∈ range f ↔ a ∈ ⋃ i, {f i} | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp [@eq_comm α a] | theorem range_eq_iUnion {ι} (f : ι → α) : range f = ⋃ i, {f i} :=
Set.ext fun a => by | Mathlib.Data.Set.Lattice.1808_0.5mONj49h3SYSDwc | theorem range_eq_iUnion {ι} (f : ι → α) : range f = ⋃ i, {f i} | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β
s : Set α
b : β
⊢ b ∈ f '' s ↔ b ∈ ⋃ i ∈ s, {f i} | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp [@eq_comm β b] | theorem image_eq_iUnion (f : α → β) (s : Set α) : f '' s = ⋃ i ∈ s, {f i} :=
Set.ext fun b => by | Mathlib.Data.Set.Lattice.1812_0.5mONj49h3SYSDwc | theorem image_eq_iUnion (f : α → β) (s : Set α) : f '' s = ⋃ i ∈ s, {f i} | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : ι → α
g : α → Set β
⊢ ⋃ x, ⋃ y, ⋃ (_ : f y = x), g x = ⋃ y, g (f y) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simpa using biUnion_range | @[simp]
theorem iUnion_iUnion_eq' {f : ι → α} {g : α → Set β} :
⋃ (x) (y) (_ : f y = x), g x = ⋃ y, g (f y) := by | Mathlib.Data.Set.Lattice.1821_0.5mONj49h3SYSDwc | @[simp]
theorem iUnion_iUnion_eq' {f : ι → α} {g : α → Set β} :
⋃ (x) (y) (_ : f y = x), g x = ⋃ y, g (f y) | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : ι → α
g : α → Set β
⊢ ⋂ x, ⋂ y, ⋂ (_ : f y = x), g x = ⋂ y, g (f y) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simpa using biInter_range | @[simp]
theorem iInter_iInter_eq' {f : ι → α} {g : α → Set β} :
⋂ (x) (y) (_ : f y = x), g x = ⋂ y, g (f y) := by | Mathlib.Data.Set.Lattice.1831_0.5mONj49h3SYSDwc | @[simp]
theorem iInter_iInter_eq' {f : ι → α} {g : α → Set β} :
⋂ (x) (y) (_ : f y = x), g x = ⋂ y, g (f y) | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β
s : ι → Set β
⊢ ∀ (x : α), x ∈ f ⁻¹' ⋃ i, s i ↔ x ∈ ⋃ i, f ⁻¹' s i | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp [preimage] | @[simp]
theorem preimage_iUnion {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋃ i, s i) = ⋃ i, f ⁻¹' s i :=
Set.ext <| by | Mathlib.Data.Set.Lattice.1853_0.5mONj49h3SYSDwc | @[simp]
theorem preimage_iUnion {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋃ i, s i) = ⋃ i, f ⁻¹' s i | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β
s : (i : ι) → κ i → Set β
⊢ f ⁻¹' ⋃ i, ⋃ j, s i j = ⋃ i, ⋃ j, f ⁻¹' s i j | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp_rw [preimage_iUnion] | theorem preimage_iUnion₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋃ (i) (j), s i j) = ⋃ (i) (j), f ⁻¹' s i j := by | Mathlib.Data.Set.Lattice.1860_0.5mONj49h3SYSDwc | theorem preimage_iUnion₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋃ (i) (j), s i j) = ⋃ (i) (j), f ⁻¹' s i j | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β
s : Set (Set β)
⊢ f ⁻¹' ⋃₀ s = ⋃ t ∈ s, f ⁻¹' t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [sUnion_eq_biUnion, preimage_iUnion₂] | @[simp]
theorem preimage_sUnion {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋃₀s = ⋃ t ∈ s, f ⁻¹' t := by
| Mathlib.Data.Set.Lattice.1864_0.5mONj49h3SYSDwc | @[simp]
theorem preimage_sUnion {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋃₀s = ⋃ t ∈ s, f ⁻¹' t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β
s : ι → Set β
⊢ f ⁻¹' ⋂ i, s i = ⋂ i, f ⁻¹' s i | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | ext | theorem preimage_iInter {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋂ i, s i) = ⋂ i, f ⁻¹' s i := by
| Mathlib.Data.Set.Lattice.1869_0.5mONj49h3SYSDwc | theorem preimage_iInter {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋂ i, s i) = ⋂ i, f ⁻¹' s i | Mathlib_Data_Set_Lattice |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β
s : ι → Set β
x✝ : α
⊢ x✝ ∈ f ⁻¹' ⋂ i, s i ↔ x✝ ∈ ⋂ i, f ⁻¹' s i | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp | theorem preimage_iInter {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋂ i, s i) = ⋂ i, f ⁻¹' s i := by
ext; | Mathlib.Data.Set.Lattice.1869_0.5mONj49h3SYSDwc | theorem preimage_iInter {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋂ i, s i) = ⋂ i, f ⁻¹' s i | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β
s : (i : ι) → κ i → Set β
⊢ f ⁻¹' ⋂ i, ⋂ j, s i j = ⋂ i, ⋂ j, f ⁻¹' s i j | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp_rw [preimage_iInter] | theorem preimage_iInter₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋂ (i) (j), s i j) = ⋂ (i) (j), f ⁻¹' s i j := by | Mathlib.Data.Set.Lattice.1875_0.5mONj49h3SYSDwc | theorem preimage_iInter₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋂ (i) (j), s i j) = ⋂ (i) (j), f ⁻¹' s i j | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β
s : Set (Set β)
⊢ f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [sInter_eq_biInter, preimage_iInter₂] | @[simp]
theorem preimage_sInter {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t := by
| Mathlib.Data.Set.Lattice.1879_0.5mONj49h3SYSDwc | @[simp]
theorem preimage_sInter {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β
s : Set β
⊢ ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [← preimage_iUnion₂, biUnion_of_singleton] | @[simp]
theorem biUnion_preimage_singleton (f : α → β) (s : Set β) : ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s := by
| Mathlib.Data.Set.Lattice.1884_0.5mONj49h3SYSDwc | @[simp]
theorem biUnion_preimage_singleton (f : α → β) (s : Set β) : ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β
⊢ ⋃ y ∈ range f, f ⁻¹' {y} = univ | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [biUnion_preimage_singleton, preimage_range] | theorem biUnion_range_preimage_singleton (f : α → β) : ⋃ y ∈ range f, f ⁻¹' {y} = univ := by
| Mathlib.Data.Set.Lattice.1889_0.5mONj49h3SYSDwc | theorem biUnion_range_preimage_singleton (f : α → β) : ⋃ y ∈ range f, f ⁻¹' {y} = univ | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : Set α
t : ι → Set β
⊢ s ×ˢ ⋃ i, t i = ⋃ i, s ×ˢ t i | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | ext | theorem prod_iUnion {s : Set α} {t : ι → Set β} : (s ×ˢ ⋃ i, t i) = ⋃ i, s ×ˢ t i := by
| Mathlib.Data.Set.Lattice.1899_0.5mONj49h3SYSDwc | theorem prod_iUnion {s : Set α} {t : ι → Set β} : (s ×ˢ ⋃ i, t i) = ⋃ i, s ×ˢ t i | Mathlib_Data_Set_Lattice |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : Set α
t : ι → Set β
x✝ : α × β
⊢ x✝ ∈ s ×ˢ ⋃ i, t i ↔ x✝ ∈ ⋃ i, s ×ˢ t i | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp | theorem prod_iUnion {s : Set α} {t : ι → Set β} : (s ×ˢ ⋃ i, t i) = ⋃ i, s ×ˢ t i := by
ext
| Mathlib.Data.Set.Lattice.1899_0.5mONj49h3SYSDwc | theorem prod_iUnion {s : Set α} {t : ι → Set β} : (s ×ˢ ⋃ i, t i) = ⋃ i, s ×ˢ t i | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : Set α
t : (i : ι) → κ i → Set β
⊢ s ×ˢ ⋃ i, ⋃ j, t i j = ⋃ i, ⋃ j, s ×ˢ t i j | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp_rw [prod_iUnion] | theorem prod_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} :
(s ×ˢ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ×ˢ t i j := by | Mathlib.Data.Set.Lattice.1908_0.5mONj49h3SYSDwc | theorem prod_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} :
(s ×ˢ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ×ˢ t i j | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : Set α
C : Set (Set β)
⊢ s ×ˢ ⋃₀ C = ⋃₀ ((fun t => s ×ˢ t) '' C) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp_rw [sUnion_eq_biUnion, biUnion_image, prod_iUnion₂] | theorem prod_sUnion {s : Set α} {C : Set (Set β)} : s ×ˢ ⋃₀C = ⋃₀((fun t => s ×ˢ t) '' C) := by
| Mathlib.Data.Set.Lattice.1914_0.5mONj49h3SYSDwc | theorem prod_sUnion {s : Set α} {C : Set (Set β)} : s ×ˢ ⋃₀C = ⋃₀((fun t => s ×ˢ t) '' C) | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : ι → Set α
t : Set β
⊢ (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | ext | theorem iUnion_prod_const {s : ι → Set α} {t : Set β} : (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t := by
| Mathlib.Data.Set.Lattice.1920_0.5mONj49h3SYSDwc | theorem iUnion_prod_const {s : ι → Set α} {t : Set β} : (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t | Mathlib_Data_Set_Lattice |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : ι → Set α
t : Set β
x✝ : α × β
⊢ x✝ ∈ (⋃ i, s i) ×ˢ t ↔ x✝ ∈ ⋃ i, s i ×ˢ t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp | theorem iUnion_prod_const {s : ι → Set α} {t : Set β} : (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t := by
ext
| Mathlib.Data.Set.Lattice.1920_0.5mONj49h3SYSDwc | theorem iUnion_prod_const {s : ι → Set α} {t : Set β} : (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : (i : ι) → κ i → Set α
t : Set β
⊢ (⋃ i, ⋃ j, s i j) ×ˢ t = ⋃ i, ⋃ j, s i j ×ˢ t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp_rw [iUnion_prod_const] | theorem iUnion₂_prod_const {s : ∀ i, κ i → Set α} {t : Set β} :
(⋃ (i) (j), s i j) ×ˢ t = ⋃ (i) (j), s i j ×ˢ t := by | Mathlib.Data.Set.Lattice.1929_0.5mONj49h3SYSDwc | theorem iUnion₂_prod_const {s : ∀ i, κ i → Set α} {t : Set β} :
(⋃ (i) (j), s i j) ×ˢ t = ⋃ (i) (j), s i j ×ˢ t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
C : Set (Set α)
t : Set β
⊢ ⋃₀ C ×ˢ t = ⋃₀ ((fun s => s ×ˢ t) '' C) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp only [sUnion_eq_biUnion, iUnion₂_prod_const, biUnion_image] | theorem sUnion_prod_const {C : Set (Set α)} {t : Set β} :
⋃₀C ×ˢ t = ⋃₀((fun s : Set α => s ×ˢ t) '' C) := by
| Mathlib.Data.Set.Lattice.1935_0.5mONj49h3SYSDwc | theorem sUnion_prod_const {C : Set (Set α)} {t : Set β} :
⋃₀C ×ˢ t = ⋃₀((fun s : Set α => s ×ˢ t) '' C) | Mathlib_Data_Set_Lattice |
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
ι✝ : Sort u_4
ι'✝ : Sort u_5
ι₂ : Sort u_6
κ : ι✝ → Sort u_7
κ₁ : ι✝ → Sort u_8
κ₂ : ι✝ → Sort u_9
κ' : ι'✝ → Sort u_10
ι : Type u_11
ι' : Type u_12
α : Type u_13
β : Type u_14
s : ι → Set α
t : ι' → Set β
⊢ ⋃ x, s x.1 ×ˢ t x.2 = (⋃ i, s i) ×ˢ ⋃ i, t i | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | ext | theorem iUnion_prod {ι ι' α β} (s : ι → Set α) (t : ι' → Set β) :
⋃ x : ι × ι', s x.1 ×ˢ t x.2 = (⋃ i : ι, s i) ×ˢ ⋃ i : ι', t i := by
| Mathlib.Data.Set.Lattice.1942_0.5mONj49h3SYSDwc | theorem iUnion_prod {ι ι' α β} (s : ι → Set α) (t : ι' → Set β) :
⋃ x : ι × ι', s x.1 ×ˢ t x.2 = (⋃ i : ι, s i) ×ˢ ⋃ i : ι', t i | Mathlib_Data_Set_Lattice |
case h
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
ι✝ : Sort u_4
ι'✝ : Sort u_5
ι₂ : Sort u_6
κ : ι✝ → Sort u_7
κ₁ : ι✝ → Sort u_8
κ₂ : ι✝ → Sort u_9
κ' : ι'✝ → Sort u_10
ι : Type u_11
ι' : Type u_12
α : Type u_13
β : Type u_14
s : ι → Set α
t : ι' → Set β
x✝ : α × β
⊢ x✝ ∈ ⋃ x, s x.1 ×ˢ t x.2 ↔ x✝ ∈ (⋃ i, s i) ×ˢ ⋃ i, t ... | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp | theorem iUnion_prod {ι ι' α β} (s : ι → Set α) (t : ι' → Set β) :
⋃ x : ι × ι', s x.1 ×ˢ t x.2 = (⋃ i : ι, s i) ×ˢ ⋃ i : ι', t i := by
ext
| Mathlib.Data.Set.Lattice.1942_0.5mONj49h3SYSDwc | theorem iUnion_prod {ι ι' α β} (s : ι → Set α) (t : ι' → Set β) :
⋃ x : ι × ι', s x.1 ×ˢ t x.2 = (⋃ i : ι, s i) ×ˢ ⋃ i : ι', t i | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : SemilatticeSup α
s : α → Set β
t : α → Set γ
hs : Monotone s
ht : Monotone t
⊢ ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | ext ⟨z, w⟩ | theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
| Mathlib.Data.Set.Lattice.1954_0.5mONj49h3SYSDwc | theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x | Mathlib_Data_Set_Lattice |
case h.mk
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : SemilatticeSup α
s : α → Set β
t : α → Set γ
hs : Monotone s
ht : Monotone t
z : β
w : γ
⊢ (z, w) ∈ ⋃ x, s x ×ˢ t x ↔ (z, w) ∈ (⋃ x, s x) ×ˢ ⋃ x, t x | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def] | theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
ext ⟨z, w⟩; | Mathlib.Data.Set.Lattice.1954_0.5mONj49h3SYSDwc | theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x | Mathlib_Data_Set_Lattice |
case h.mk
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : SemilatticeSup α
s : α → Set β
t : α → Set γ
hs : Monotone s
ht : Monotone t
z : β
w : γ
⊢ (∀ (x : α), z ∈ s x → w ∈ t x → (∃ i, z ∈ s i) ∧ ∃ i, w ∈ ... | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | constructor | theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
ext ⟨z, w⟩; simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]; | Mathlib.Data.Set.Lattice.1954_0.5mONj49h3SYSDwc | theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x | Mathlib_Data_Set_Lattice |
case h.mk.left
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : SemilatticeSup α
s : α → Set β
t : α → Set γ
hs : Monotone s
ht : Monotone t
z : β
w : γ
⊢ ∀ (x : α), z ∈ s x → w ∈ t x → (∃ i, z ∈ s i) ∧ ∃ i, ... | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | intro x hz hw | theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
ext ⟨z, w⟩; simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]; constructor
· | Mathlib.Data.Set.Lattice.1954_0.5mONj49h3SYSDwc | theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x | Mathlib_Data_Set_Lattice |
case h.mk.left
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : SemilatticeSup α
s : α → Set β
t : α → Set γ
hs : Monotone s
ht : Monotone t
z : β
w : γ
x : α
hz : z ∈ s x
hw : w ∈ t x
⊢ (∃ i, z ∈ s i) ∧ ∃ i,... | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | exact ⟨⟨x, hz⟩, x, hw⟩ | theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
ext ⟨z, w⟩; simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]; constructor
· intro x hz hw
| Mathlib.Data.Set.Lattice.1954_0.5mONj49h3SYSDwc | theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x | Mathlib_Data_Set_Lattice |
case h.mk.right
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : SemilatticeSup α
s : α → Set β
t : α → Set γ
hs : Monotone s
ht : Monotone t
z : β
w : γ
⊢ ∀ (x : α), z ∈ s x → ∀ (x : α), w ∈ t x → ∃ i, z ∈ s... | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | intro x hz x' hw | theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
ext ⟨z, w⟩; simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]; constructor
· intro x hz hw
exact ⟨⟨x, hz⟩, x, hw⟩
· | Mathlib.Data.Set.Lattice.1954_0.5mONj49h3SYSDwc | theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x | Mathlib_Data_Set_Lattice |
case h.mk.right
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : SemilatticeSup α
s : α → Set β
t : α → Set γ
hs : Monotone s
ht : Monotone t
z : β
w : γ
x : α
hz : z ∈ s x
x' : α
hw : w ∈ t x'
⊢ ∃ i, z ∈ s i... | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | exact ⟨x ⊔ x', hs le_sup_left hz, ht le_sup_right hw⟩ | theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
ext ⟨z, w⟩; simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]; constructor
· intro x hz hw
exact ⟨⟨x, hz⟩, x, hw⟩
· intro x h... | Mathlib.Data.Set.Lattice.1954_0.5mONj49h3SYSDwc | theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
S : Set (Set α)
T : Set (Set β)
hS : Set.Nonempty S
hT : Set.Nonempty T
⊢ ⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | obtain ⟨s₁, h₁⟩ := hS | theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 := by
| Mathlib.Data.Set.Lattice.1974_0.5mONj49h3SYSDwc | theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 | Mathlib_Data_Set_Lattice |
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
S : Set (Set α)
T : Set (Set β)
hT : Set.Nonempty T
s₁ : Set α
h₁ : s₁ ∈ S
⊢ ⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | obtain ⟨s₂, h₂⟩ := hT | theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 := by
obtain ⟨s₁, h₁⟩ := hS
| Mathlib.Data.Set.Lattice.1974_0.5mONj49h3SYSDwc | theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 | Mathlib_Data_Set_Lattice |
case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
S : Set (Set α)
T : Set (Set β)
s₁ : Set α
h₁ : s₁ ∈ S
s₂ : Set β
h₂ : s₂ ∈ T
⊢ ⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | refine' Set.Subset.antisymm (sInter_prod_sInter_subset S T) fun x hx => _ | theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 := by
obtain ⟨s₁, h₁⟩ := hS
obtain ⟨s₂, h₂⟩ := hT
| Mathlib.Data.Set.Lattice.1974_0.5mONj49h3SYSDwc | theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 | Mathlib_Data_Set_Lattice |
case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
S : Set (Set α)
T : Set (Set β)
s₁ : Set α
h₁ : s₁ ∈ S
s₂ : Set β
h₂ : s₂ ∈ T
x : α × β
hx : x ∈ ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2
⊢ x ∈ ⋂₀ S ×ˢ ⋂₀ T | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [mem_iInter₂] at hx | theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 := by
obtain ⟨s₁, h₁⟩ := hS
obtain ⟨s₂, h₂⟩ := hT
refine' Set.Subset.antisymm (sInter_prod_sInter_subset S T) fun x hx => _
| Mathlib.Data.Set.Lattice.1974_0.5mONj49h3SYSDwc | theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 | Mathlib_Data_Set_Lattice |
case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
S : Set (Set α)
T : Set (Set β)
s₁ : Set α
h₁ : s₁ ∈ S
s₂ : Set β
h₂ : s₂ ∈ T
x : α × β
hx : ∀ i ∈ S ×ˢ T, x ∈ i.1 ×ˢ i.2
⊢ x ∈ ⋂₀ S ×ˢ ⋂₀ T | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | exact ⟨fun s₀ h₀ => (hx (s₀, s₂) ⟨h₀, h₂⟩).1, fun s₀ h₀ => (hx (s₁, s₀) ⟨h₁, h₀⟩).2⟩ | theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 := by
obtain ⟨s₁, h₁⟩ := hS
obtain ⟨s₂, h₂⟩ := hT
refine' Set.Subset.antisymm (sInter_prod_sInter_subset S T) fun x hx => _
rw [mem_iInter₂] at hx
| Mathlib.Data.Set.Lattice.1974_0.5mONj49h3SYSDwc | theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
S : Set (Set α)
hS : Set.Nonempty S
t : Set β
⊢ ⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [← sInter_singleton t, sInter_prod_sInter hS (singleton_nonempty t), sInter_singleton] | theorem sInter_prod {S : Set (Set α)} (hS : S.Nonempty) (t : Set β) :
⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t := by
| Mathlib.Data.Set.Lattice.1985_0.5mONj49h3SYSDwc | theorem sInter_prod {S : Set (Set α)} (hS : S.Nonempty) (t : Set β) :
⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
S : Set (Set α)
hS : Set.Nonempty S
t : Set β
⊢ ⋂ r ∈ S ×ˢ {t}, r.1 ×ˢ r.2 = ⋂ s ∈ S, s ×ˢ t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp_rw [prod_singleton, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right] | theorem sInter_prod {S : Set (Set α)} (hS : S.Nonempty) (t : Set β) :
⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t := by
rw [← sInter_singleton t, sInter_prod_sInter hS (singleton_nonempty t), sInter_singleton]
| Mathlib.Data.Set.Lattice.1985_0.5mONj49h3SYSDwc | theorem sInter_prod {S : Set (Set α)} (hS : S.Nonempty) (t : Set β) :
⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
T : Set (Set β)
hT : Set.Nonempty T
s : Set α
⊢ s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [← sInter_singleton s, sInter_prod_sInter (singleton_nonempty s) hT, sInter_singleton] | theorem prod_sInter {T : Set (Set β)} (hT : T.Nonempty) (s : Set α) :
s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t := by
| Mathlib.Data.Set.Lattice.1993_0.5mONj49h3SYSDwc | theorem prod_sInter {T : Set (Set β)} (hT : T.Nonempty) (s : Set α) :
s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
T : Set (Set β)
hT : Set.Nonempty T
s : Set α
⊢ ⋂ r ∈ {s} ×ˢ T, r.1 ×ˢ r.2 = ⋂ t ∈ T, s ×ˢ t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp_rw [singleton_prod, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right] | theorem prod_sInter {T : Set (Set β)} (hT : T.Nonempty) (s : Set α) :
s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t := by
rw [← sInter_singleton s, sInter_prod_sInter (singleton_nonempty s) hT, sInter_singleton]
| Mathlib.Data.Set.Lattice.1993_0.5mONj49h3SYSDwc | theorem prod_sInter {T : Set (Set β)} (hT : T.Nonempty) (s : Set α) :
s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β → γ
s✝ : Set α
t✝ : Set β
s : Set α
t : Set β
⊢ image2 f s t = ⋃ i ∈ s, ⋃ j ∈ t, {f i j} | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | ext | /-- The `Set.image2` version of `Set.image_eq_iUnion` -/
theorem image2_eq_iUnion (s : Set α) (t : Set β) : image2 f s t = ⋃ (i ∈ s) (j ∈ t), {f i j} := by
| Mathlib.Data.Set.Lattice.2005_0.5mONj49h3SYSDwc | /-- The `Set.image2` version of `Set.image_eq_iUnion` -/
theorem image2_eq_iUnion (s : Set α) (t : Set β) : image2 f s t = ⋃ (i ∈ s) (j ∈ t), {f i j} | Mathlib_Data_Set_Lattice |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β → γ
s✝ : Set α
t✝ : Set β
s : Set α
t : Set β
x✝ : γ
⊢ x✝ ∈ image2 f s t ↔ x✝ ∈ ⋃ i ∈ s, ⋃ j ∈ t, {f i j} | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp [eq_comm] | /-- The `Set.image2` version of `Set.image_eq_iUnion` -/
theorem image2_eq_iUnion (s : Set α) (t : Set β) : image2 f s t = ⋃ (i ∈ s) (j ∈ t), {f i j} := by
ext; | Mathlib.Data.Set.Lattice.2005_0.5mONj49h3SYSDwc | /-- The `Set.image2` version of `Set.image_eq_iUnion` -/
theorem image2_eq_iUnion (s : Set α) (t : Set β) : image2 f s t = ⋃ (i ∈ s) (j ∈ t), {f i j} | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β → γ
s : Set α
t : Set β
⊢ ⋃ a ∈ s, f a '' t = image2 f s t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp only [image2_eq_iUnion, image_eq_iUnion] | theorem iUnion_image_left : ⋃ a ∈ s, f a '' t = image2 f s t := by
| Mathlib.Data.Set.Lattice.2010_0.5mONj49h3SYSDwc | theorem iUnion_image_left : ⋃ a ∈ s, f a '' t = image2 f s t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β → γ
s : Set α
t : Set β
⊢ ⋃ b ∈ t, (fun x => f x b) '' s = image2 f s t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [image2_swap, iUnion_image_left] | theorem iUnion_image_right : ⋃ b ∈ t, (f · b) '' s = image2 f s t := by
| Mathlib.Data.Set.Lattice.2014_0.5mONj49h3SYSDwc | theorem iUnion_image_right : ⋃ b ∈ t, (f · b) '' s = image2 f s t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β → γ
s✝ : Set α
t✝ : Set β
s : ι → Set α
t : Set β
⊢ image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp only [← image_prod, iUnion_prod_const, image_iUnion] | theorem image2_iUnion_left (s : ι → Set α) (t : Set β) :
image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t := by
| Mathlib.Data.Set.Lattice.2018_0.5mONj49h3SYSDwc | theorem image2_iUnion_left (s : ι → Set α) (t : Set β) :
image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β → γ
s✝ : Set α
t✝ : Set β
s : Set α
t : ι → Set β
⊢ image2 f s (⋃ i, t i) = ⋃ i, image2 f s (t i) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp only [← image_prod, prod_iUnion, image_iUnion] | theorem image2_iUnion_right (s : Set α) (t : ι → Set β) :
image2 f s (⋃ i, t i) = ⋃ i, image2 f s (t i) := by
| Mathlib.Data.Set.Lattice.2023_0.5mONj49h3SYSDwc | theorem image2_iUnion_right (s : Set α) (t : ι → Set β) :
image2 f s (⋃ i, t i) = ⋃ i, image2 f s (t i) | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β → γ
s✝ : Set α
t✝ : Set β
s : (i : ι) → κ i → Set α
t : Set β
⊢ image2 f (⋃ i, ⋃ j, s i j) t = ⋃ i, ⋃ j, image2 f (s i j) t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp_rw [image2_iUnion_left] | theorem image2_iUnion₂_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋃ (i) (j), s i j) t = ⋃ (i) (j), image2 f (s i j) t := by | Mathlib.Data.Set.Lattice.2030_0.5mONj49h3SYSDwc | theorem image2_iUnion₂_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋃ (i) (j), s i j) t = ⋃ (i) (j), image2 f (s i j) t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β → γ
s✝ : Set α
t✝ : Set β
s : Set α
t : (i : ι) → κ i → Set β
⊢ image2 f s (⋃ i, ⋃ j, t i j) = ⋃ i, ⋃ j, image2 f s (t i j) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp_rw [image2_iUnion_right] | theorem image2_iUnion₂_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋃ (i) (j), t i j) = ⋃ (i) (j), image2 f s (t i j) :=
by | Mathlib.Data.Set.Lattice.2036_0.5mONj49h3SYSDwc | theorem image2_iUnion₂_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋃ (i) (j), t i j) = ⋃ (i) (j), image2 f s (t i j) | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β → γ
s✝ : Set α
t✝ : Set β
s : ι → Set α
t : Set β
⊢ image2 f (⋂ i, s i) t ⊆ ⋂ i, image2 f (s i) t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp_rw [image2_subset_iff, mem_iInter] | theorem image2_iInter_subset_left (s : ι → Set α) (t : Set β) :
image2 f (⋂ i, s i) t ⊆ ⋂ i, image2 f (s i) t := by
| Mathlib.Data.Set.Lattice.2041_0.5mONj49h3SYSDwc | theorem image2_iInter_subset_left (s : ι → Set α) (t : Set β) :
image2 f (⋂ i, s i) t ⊆ ⋂ i, image2 f (s i) t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β → γ
s✝ : Set α
t✝ : Set β
s : ι → Set α
t : Set β
⊢ ∀ (x : α), (∀ (i : ι), x ∈ s i) → ∀ y ∈ t, ∀ (i : ι), f x y ∈ image2 f (s i) t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | exact fun x hx y hy i => mem_image2_of_mem (hx _) hy | theorem image2_iInter_subset_left (s : ι → Set α) (t : Set β) :
image2 f (⋂ i, s i) t ⊆ ⋂ i, image2 f (s i) t := by
simp_rw [image2_subset_iff, mem_iInter]
| Mathlib.Data.Set.Lattice.2041_0.5mONj49h3SYSDwc | theorem image2_iInter_subset_left (s : ι → Set α) (t : Set β) :
image2 f (⋂ i, s i) t ⊆ ⋂ i, image2 f (s i) t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β → γ
s✝ : Set α
t✝ : Set β
s : Set α
t : ι → Set β
⊢ image2 f s (⋂ i, t i) ⊆ ⋂ i, image2 f s (t i) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp_rw [image2_subset_iff, mem_iInter] | theorem image2_iInter_subset_right (s : Set α) (t : ι → Set β) :
image2 f s (⋂ i, t i) ⊆ ⋂ i, image2 f s (t i) := by
| Mathlib.Data.Set.Lattice.2047_0.5mONj49h3SYSDwc | theorem image2_iInter_subset_right (s : Set α) (t : ι → Set β) :
image2 f s (⋂ i, t i) ⊆ ⋂ i, image2 f s (t i) | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β → γ
s✝ : Set α
t✝ : Set β
s : Set α
t : ι → Set β
⊢ ∀ x ∈ s, ∀ (y : β), (∀ (i : ι), y ∈ t i) → ∀ (i : ι), f x y ∈ image2 f s (t i) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | exact fun x hx y hy i => mem_image2_of_mem hx (hy _) | theorem image2_iInter_subset_right (s : Set α) (t : ι → Set β) :
image2 f s (⋂ i, t i) ⊆ ⋂ i, image2 f s (t i) := by
simp_rw [image2_subset_iff, mem_iInter]
| Mathlib.Data.Set.Lattice.2047_0.5mONj49h3SYSDwc | theorem image2_iInter_subset_right (s : Set α) (t : ι → Set β) :
image2 f s (⋂ i, t i) ⊆ ⋂ i, image2 f s (t i) | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β → γ
s✝ : Set α
t✝ : Set β
s : (i : ι) → κ i → Set α
t : Set β
⊢ image2 f (⋂ i, ⋂ j, s i j) t ⊆ ⋂ i, ⋂ j, image2 f (s i j) t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp_rw [image2_subset_iff, mem_iInter] | theorem image2_iInter₂_subset_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋂ (i) (j), s i j) t ⊆ ⋂ (i) (j), image2 f (s i j) t := by
| Mathlib.Data.Set.Lattice.2055_0.5mONj49h3SYSDwc | theorem image2_iInter₂_subset_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋂ (i) (j), s i j) t ⊆ ⋂ (i) (j), image2 f (s i j) t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β → γ
s✝ : Set α
t✝ : Set β
s : (i : ι) → κ i → Set α
t : Set β
⊢ ∀ (x : α), (∀ (i : ι) (i_1 : κ i), x ∈ s i i_1) → ∀ y ∈ t, ∀ (i : ι) (i_1 : κ i), f x y ∈ ima... | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | exact fun x hx y hy i j => mem_image2_of_mem (hx _ _) hy | theorem image2_iInter₂_subset_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋂ (i) (j), s i j) t ⊆ ⋂ (i) (j), image2 f (s i j) t := by
simp_rw [image2_subset_iff, mem_iInter]
| Mathlib.Data.Set.Lattice.2055_0.5mONj49h3SYSDwc | theorem image2_iInter₂_subset_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋂ (i) (j), s i j) t ⊆ ⋂ (i) (j), image2 f (s i j) t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β → γ
s✝ : Set α
t✝ : Set β
s : Set α
t : (i : ι) → κ i → Set β
⊢ image2 f s (⋂ i, ⋂ j, t i j) ⊆ ⋂ i, ⋂ j, image2 f s (t i j) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp_rw [image2_subset_iff, mem_iInter] | theorem image2_iInter₂_subset_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), image2 f s (t i j) := by
| Mathlib.Data.Set.Lattice.2063_0.5mONj49h3SYSDwc | theorem image2_iInter₂_subset_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), image2 f s (t i j) | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β → γ
s✝ : Set α
t✝ : Set β
s : Set α
t : (i : ι) → κ i → Set β
⊢ ∀ x ∈ s, ∀ (y : β), (∀ (i : ι) (i_1 : κ i), y ∈ t i i_1) → ∀ (i : ι) (i_1 : κ i), f x y ∈ ima... | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | exact fun x hx y hy i j => mem_image2_of_mem hx (hy _ _) | theorem image2_iInter₂_subset_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), image2 f s (t i j) := by
simp_rw [image2_subset_iff, mem_iInter]
| Mathlib.Data.Set.Lattice.2063_0.5mONj49h3SYSDwc | theorem image2_iInter₂_subset_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), image2 f s (t i j) | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β → γ
s : Set α
t : Set β
⊢ s ×ˢ t = ⋃ a ∈ s, (fun b => (a, b)) '' t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [iUnion_image_left, image2_mk_eq_prod] | theorem prod_eq_biUnion_left : s ×ˢ t = ⋃ a ∈ s, (fun b => (a, b)) '' t := by
| Mathlib.Data.Set.Lattice.2069_0.5mONj49h3SYSDwc | theorem prod_eq_biUnion_left : s ×ˢ t = ⋃ a ∈ s, (fun b => (a, b)) '' t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β → γ
s : Set α
t : Set β
⊢ s ×ˢ t = ⋃ b ∈ t, (fun a => (a, b)) '' s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [iUnion_image_right, image2_mk_eq_prod] | theorem prod_eq_biUnion_right : s ×ˢ t = ⋃ b ∈ t, (fun a => (a, b)) '' s := by
| Mathlib.Data.Set.Lattice.2073_0.5mONj49h3SYSDwc | theorem prod_eq_biUnion_right : s ×ˢ t = ⋃ b ∈ t, (fun a => (a, b)) '' s | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : Set (α → β)
t : Set α
⊢ seq s t = image2 (fun f a => f a) s t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | ext | lemma seq_eq_image2 (s : Set (α → β)) (t : Set α) : seq s t = image2 (fun f a ↦ f a) s t := by
| Mathlib.Data.Set.Lattice.2093_0.5mONj49h3SYSDwc | lemma seq_eq_image2 (s : Set (α → β)) (t : Set α) : seq s t = image2 (fun f a ↦ f a) s t | Mathlib_Data_Set_Lattice |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : Set (α → β)
t : Set α
x✝ : β
⊢ x✝ ∈ seq s t ↔ x✝ ∈ image2 (fun f a => f a) s t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp | lemma seq_eq_image2 (s : Set (α → β)) (t : Set α) : seq s t = image2 (fun f a ↦ f a) s t := by
ext; | Mathlib.Data.Set.Lattice.2093_0.5mONj49h3SYSDwc | lemma seq_eq_image2 (s : Set (α → β)) (t : Set α) : seq s t = image2 (fun f a ↦ f a) s t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : Set (α → β)
t : Set α
⊢ seq s t = ⋃ f ∈ s, f '' t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [seq_eq_image2, iUnion_image_left] | theorem seq_def {s : Set (α → β)} {t : Set α} : seq s t = ⋃ f ∈ s, f '' t := by
| Mathlib.Data.Set.Lattice.2096_0.5mONj49h3SYSDwc | theorem seq_def {s : Set (α → β)} {t : Set α} : seq s t = ⋃ f ∈ s, f '' t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : Set (α → β)
t : Set α
u : Set β
⊢ seq s t ⊆ u ↔ ∀ f ∈ s, ∀ a ∈ t, f a ∈ u | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [seq_eq_image2, image2_subset_iff] | theorem seq_subset {s : Set (α → β)} {t : Set α} {u : Set β} :
seq s t ⊆ u ↔ ∀ f ∈ s, ∀ a ∈ t, (f : α → β) a ∈ u := by
| Mathlib.Data.Set.Lattice.2100_0.5mONj49h3SYSDwc | theorem seq_subset {s : Set (α → β)} {t : Set α} {u : Set β} :
seq s t ⊆ u ↔ ∀ f ∈ s, ∀ a ∈ t, (f : α → β) a ∈ u | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β
t : Set α
⊢ seq {f} t = f '' t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [seq_eq_image2, image2_singleton_left] | theorem singleton_seq {f : α → β} {t : Set α} : Set.seq ({f} : Set (α → β)) t = f '' t := by
| Mathlib.Data.Set.Lattice.2110_0.5mONj49h3SYSDwc | theorem singleton_seq {f : α → β} {t : Set α} : Set.seq ({f} : Set (α → β)) t = f '' t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : Set (α → β)
a : α
⊢ seq s {a} = (fun f => f a) '' s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [seq_eq_image2, image2_singleton_right] | theorem seq_singleton {s : Set (α → β)} {a : α} : Set.seq s {a} = (fun f : α → β => f a) '' s := by
| Mathlib.Data.Set.Lattice.2114_0.5mONj49h3SYSDwc | theorem seq_singleton {s : Set (α → β)} {a : α} : Set.seq s {a} = (fun f : α → β => f a) '' s | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : Set (β → γ)
t : Set (α → β)
u : Set α
⊢ seq s (seq t u) = seq (seq ((fun x x_1 => x ∘ x_1) '' s) t) u | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp only [seq_eq_image2, image2_image_left, image2_image2_left, image2_image2_right, comp_apply] | theorem seq_seq {s : Set (β → γ)} {t : Set (α → β)} {u : Set α} :
seq s (seq t u) = seq (seq ((· ∘ ·) '' s) t) u := by
| Mathlib.Data.Set.Lattice.2118_0.5mONj49h3SYSDwc | theorem seq_seq {s : Set (β → γ)} {t : Set (α → β)} {u : Set α} :
seq s (seq t u) = seq (seq ((· ∘ ·) '' s) t) u | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : β → γ
s : Set (α → β)
t : Set α
⊢ f '' seq s t = seq ((fun x => f ∘ x) '' s) t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [← singleton_seq, ← singleton_seq, seq_seq, image_singleton] | theorem image_seq {f : β → γ} {s : Set (α → β)} {t : Set α} :
f '' seq s t = seq ((f ∘ ·) '' s) t := by
| Mathlib.Data.Set.Lattice.2123_0.5mONj49h3SYSDwc | theorem image_seq {f : β → γ} {s : Set (α → β)} {t : Set α} :
f '' seq s t = seq ((f ∘ ·) '' s) t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : Set α
t : Set β
⊢ s ×ˢ t = seq (Prod.mk '' s) t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [seq_eq_image2, image2_image_left, image2_mk_eq_prod] | theorem prod_eq_seq {s : Set α} {t : Set β} : s ×ˢ t = (Prod.mk '' s).seq t := by
| Mathlib.Data.Set.Lattice.2128_0.5mONj49h3SYSDwc | theorem prod_eq_seq {s : Set α} {t : Set β} : s ×ˢ t = (Prod.mk '' s).seq t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : Set α
t : Set β
⊢ seq (Prod.mk '' s) t = seq ((fun b a => (a, b)) '' t) s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [← prod_eq_seq, ← image_swap_prod, prod_eq_seq, image_seq, ← image_comp] | theorem prod_image_seq_comm (s : Set α) (t : Set β) :
(Prod.mk '' s).seq t = seq ((fun b a => (a, b)) '' t) s := by
| Mathlib.Data.Set.Lattice.2132_0.5mONj49h3SYSDwc | theorem prod_image_seq_comm (s : Set α) (t : Set β) :
(Prod.mk '' s).seq t = seq ((fun b a => (a, b)) '' t) s | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : Set α
t : Set β
⊢ seq ((fun x => Prod.swap ∘ x) ∘ Prod.mk '' t) s = seq ((fun b a => (a, b)) '' t) s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rfl | theorem prod_image_seq_comm (s : Set α) (t : Set β) :
(Prod.mk '' s).seq t = seq ((fun b a => (a, b)) '' t) s := by
rw [← prod_eq_seq, ← image_swap_prod, prod_eq_seq, image_seq, ← image_comp]; | Mathlib.Data.Set.Lattice.2132_0.5mONj49h3SYSDwc | theorem prod_image_seq_comm (s : Set α) (t : Set β) :
(Prod.mk '' s).seq t = seq ((fun b a => (a, b)) '' t) s | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β → γ
s : Set α
t : Set β
⊢ image2 f s t = seq (f '' s) t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [seq_eq_image2, image2_image_left] | theorem image2_eq_seq (f : α → β → γ) (s : Set α) (t : Set β) : image2 f s t = seq (f '' s) t := by
| Mathlib.Data.Set.Lattice.2137_0.5mONj49h3SYSDwc | theorem image2_eq_seq (f : α → β → γ) (s : Set α) (t : Set β) : image2 f s t = seq (f '' s) t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
π : α → Type u_11
i : Set α
s : (a : α) → Set (π a)
⊢ pi i s = ⋂ a ∈ i, eval a ⁻¹' s a | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | ext | theorem pi_def (i : Set α) (s : ∀ a, Set (π a)) : pi i s = ⋂ a ∈ i, eval a ⁻¹' s a := by
| Mathlib.Data.Set.Lattice.2147_0.5mONj49h3SYSDwc | theorem pi_def (i : Set α) (s : ∀ a, Set (π a)) : pi i s = ⋂ a ∈ i, eval a ⁻¹' s a | Mathlib_Data_Set_Lattice |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
π : α → Type u_11
i : Set α
s : (a : α) → Set (π a)
x✝ : (i : α) → π i
⊢ x✝ ∈ pi i s ↔ x✝ ∈ ⋂ a ∈ i, eval a ⁻¹' s a | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp | theorem pi_def (i : Set α) (s : ∀ a, Set (π a)) : pi i s = ⋂ a ∈ i, eval a ⁻¹' s a := by
ext
| Mathlib.Data.Set.Lattice.2147_0.5mONj49h3SYSDwc | theorem pi_def (i : Set α) (s : ∀ a, Set (π a)) : pi i s = ⋂ a ∈ i, eval a ⁻¹' s a | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
π : α → Type u_11
t : (i : α) → Set (π i)
⊢ pi univ t = ⋂ i, eval i ⁻¹' t i | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp only [pi_def, iInter_true, mem_univ] | theorem univ_pi_eq_iInter (t : ∀ i, Set (π i)) : pi univ t = ⋂ i, eval i ⁻¹' t i := by
| Mathlib.Data.Set.Lattice.2152_0.5mONj49h3SYSDwc | theorem univ_pi_eq_iInter (t : ∀ i, Set (π i)) : pi univ t = ⋂ i, eval i ⁻¹' t i | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
π : α → Type u_11
i : Set α
s t : (a : α) → Set (π a)
⊢ pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | refine' diff_subset_comm.2 fun x hx a ha => _ | theorem pi_diff_pi_subset (i : Set α) (s t : ∀ a, Set (π a)) :
pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) := by
| Mathlib.Data.Set.Lattice.2156_0.5mONj49h3SYSDwc | theorem pi_diff_pi_subset (i : Set α) (s t : ∀ a, Set (π a)) :
pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
π : α → Type u_11
i : Set α
s t : (a : α) → Set (π a)
x : (i : α) → π i
hx : x ∈ pi i s \ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a)
a : α
ha : a ∈ i
⊢ x a ∈ t a | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp only [mem_diff, mem_pi, mem_iUnion, not_exists, mem_preimage, not_and, not_not,
eval_apply] at hx | theorem pi_diff_pi_subset (i : Set α) (s t : ∀ a, Set (π a)) :
pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) := by
refine' diff_subset_comm.2 fun x hx a ha => _
| Mathlib.Data.Set.Lattice.2156_0.5mONj49h3SYSDwc | theorem pi_diff_pi_subset (i : Set α) (s t : ∀ a, Set (π a)) :
pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
π : α → Type u_11
i : Set α
s t : (a : α) → Set (π a)
x : (i : α) → π i
a : α
ha : a ∈ i
hx : (∀ i_1 ∈ i, x i_1 ∈ s i_1) ∧ ∀ x_1 ∈ i, eval x_1 x ∈ s x_1 → eval x_1 x ∈... | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | exact hx.2 _ ha (hx.1 _ ha) | theorem pi_diff_pi_subset (i : Set α) (s t : ∀ a, Set (π a)) :
pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) := by
refine' diff_subset_comm.2 fun x hx a ha => _
simp only [mem_diff, mem_pi, mem_iUnion, not_exists, mem_preimage, not_and, not_not,
eval_apply] at hx
| Mathlib.Data.Set.Lattice.2156_0.5mONj49h3SYSDwc | theorem pi_diff_pi_subset (i : Set α) (s t : ∀ a, Set (π a)) :
pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι✝ → Sort u_7
κ₁ : ι✝ → Sort u_8
κ₂ : ι✝ → Sort u_9
κ' : ι' → Sort u_10
π : α → Type u_11
ι : α → Type u_12
t : (a : α) → ι a → Set (π a)
⊢ (⋃ x, pi univ fun a => t a (x a)) = pi univ fun a => ⋃ j, t a j | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | ext | theorem iUnion_univ_pi {ι : α → Type*} (t : (a : α) → ι a → Set (π a)) :
⋃ x : (a : α) → ι a, pi univ (fun a => t a (x a)) = pi univ fun a => ⋃ j : ι a, t a j := by
| Mathlib.Data.Set.Lattice.2164_0.5mONj49h3SYSDwc | theorem iUnion_univ_pi {ι : α → Type*} (t : (a : α) → ι a → Set (π a)) :
⋃ x : (a : α) → ι a, pi univ (fun a => t a (x a)) = pi univ fun a => ⋃ j : ι a, t a j | Mathlib_Data_Set_Lattice |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι✝ → Sort u_7
κ₁ : ι✝ → Sort u_8
κ₂ : ι✝ → Sort u_9
κ' : ι' → Sort u_10
π : α → Type u_11
ι : α → Type u_12
t : (a : α) → ι a → Set (π a)
x✝ : (i : α) → π i
⊢ (x✝ ∈ ⋃ x, pi univ fun a => t a (x a)) ↔ x✝ ∈ pi univ fun a => ⋃ j, t... | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp [Classical.skolem] | theorem iUnion_univ_pi {ι : α → Type*} (t : (a : α) → ι a → Set (π a)) :
⋃ x : (a : α) → ι a, pi univ (fun a => t a (x a)) = pi univ fun a => ⋃ j : ι a, t a j := by
ext
| Mathlib.Data.Set.Lattice.2164_0.5mONj49h3SYSDwc | theorem iUnion_univ_pi {ι : α → Type*} (t : (a : α) → ι a → Set (π a)) :
⋃ x : (a : α) → ι a, pi univ (fun a => t a (x a)) = pi univ fun a => ⋃ j : ι a, t a j | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : Preorder α
f : ι → α
⊢ Set.Nonempty (⋂ i, Iic (f i)) ↔ BddBelow (range f) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | have : (⋂ (i : ι), Iic (f i)) = lowerBounds (range f) := by
ext c; simp [lowerBounds] | lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) := by
| Mathlib.Data.Set.Lattice.2247_0.5mONj49h3SYSDwc | lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : Preorder α
f : ι → α
⊢ ⋂ i, Iic (f i) = lowerBounds (range f) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | ext c | lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) := by
have : (⋂ (i : ι), Iic (f i)) = lowerBounds (range f) := by
| Mathlib.Data.Set.Lattice.2247_0.5mONj49h3SYSDwc | lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) | Mathlib_Data_Set_Lattice |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : Preorder α
f : ι → α
c : α
⊢ c ∈ ⋂ i, Iic (f i) ↔ c ∈ lowerBounds (range f) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp [lowerBounds] | lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) := by
have : (⋂ (i : ι), Iic (f i)) = lowerBounds (range f) := by
ext c; | Mathlib.Data.Set.Lattice.2247_0.5mONj49h3SYSDwc | lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : Preorder α
f : ι → α
this : ⋂ i, Iic (f i) = lowerBounds (range f)
⊢ Set.Nonempty (⋂ i, Iic (f i)) ↔ BddBelow (range f) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp [this, BddBelow] | lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) := by
have : (⋂ (i : ι), Iic (f i)) = lowerBounds (range f) := by
ext c; simp [lowerBounds]
| Mathlib.Data.Set.Lattice.2247_0.5mONj49h3SYSDwc | lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice α
f : ι → α
x✝ : α
⊢ x✝ ∈ Ici (⨆ i, f i) ↔ x✝ ∈ ⋂ i, Ici (f i) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp only [mem_Ici, iSup_le_iff, mem_iInter] | theorem Ici_iSup (f : ι → α) : Ici (⨆ i, f i) = ⋂ i, Ici (f i) :=
ext fun _ => by | Mathlib.Data.Set.Lattice.2259_0.5mONj49h3SYSDwc | theorem Ici_iSup (f : ι → α) : Ici (⨆ i, f i) = ⋂ i, Ici (f i) | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice α
f : ι → α
x✝ : α
⊢ x✝ ∈ Iic (⨅ i, f i) ↔ x✝ ∈ ⋂ i, Iic (f i) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp only [mem_Iic, le_iInf_iff, mem_iInter] | theorem Iic_iInf (f : ι → α) : Iic (⨅ i, f i) = ⋂ i, Iic (f i) :=
ext fun _ => by | Mathlib.Data.Set.Lattice.2263_0.5mONj49h3SYSDwc | theorem Iic_iInf (f : ι → α) : Iic (⨅ i, f i) = ⋂ i, Iic (f i) | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice α
f : (i : ι) → κ i → α
⊢ Ici (⨆ i, ⨆ j, f i j) = ⋂ i, ⋂ j, Ici (f i j) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp_rw [Ici_iSup] | theorem Ici_iSup₂ (f : ∀ i, κ i → α) : Ici (⨆ (i) (j), f i j) = ⋂ (i) (j), Ici (f i j) := by
| Mathlib.Data.Set.Lattice.2269_0.5mONj49h3SYSDwc | theorem Ici_iSup₂ (f : ∀ i, κ i → α) : Ici (⨆ (i) (j), f i j) = ⋂ (i) (j), Ici (f i j) | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice α
f : (i : ι) → κ i → α
⊢ Iic (⨅ i, ⨅ j, f i j) = ⋂ i, ⋂ j, Iic (f i j) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp_rw [Iic_iInf] | theorem Iic_iInf₂ (f : ∀ i, κ i → α) : Iic (⨅ (i) (j), f i j) = ⋂ (i) (j), Iic (f i j) := by
| Mathlib.Data.Set.Lattice.2275_0.5mONj49h3SYSDwc | theorem Iic_iInf₂ (f : ∀ i, κ i → α) : Iic (⨅ (i) (j), f i j) = ⋂ (i) (j), Iic (f i j) | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice α
s : Set α
⊢ Ici (sSup s) = ⋂ a ∈ s, Ici a | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [sSup_eq_iSup, Ici_iSup₂] | theorem Ici_sSup (s : Set α) : Ici (sSup s) = ⋂ a ∈ s, Ici a := by | Mathlib.Data.Set.Lattice.2279_0.5mONj49h3SYSDwc | theorem Ici_sSup (s : Set α) : Ici (sSup s) = ⋂ a ∈ s, Ici a | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice α
s : Set α
⊢ Iic (sInf s) = ⋂ a ∈ s, Iic a | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [sInf_eq_iInf, Iic_iInf₂] | theorem Iic_sInf (s : Set α) : Iic (sInf s) = ⋂ a ∈ s, Iic a := by | Mathlib.Data.Set.Lattice.2282_0.5mONj49h3SYSDwc | theorem Iic_sInf (s : Set α) : Iic (sInf s) = ⋂ a ∈ s, Iic a | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
t : α → Set β
s₁ s₂ : Set α
⊢ (⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x ⊆ ⋃ x ∈ s₁ \ s₂, t x | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp only [diff_subset_iff, ← biUnion_union] | theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
| Mathlib.Data.Set.Lattice.2291_0.5mONj49h3SYSDwc | theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x | Mathlib_Data_Set_Lattice |
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