mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Ring.Center

Goal state before:
pp: M : Type u_1
inst✝ : NonAssocRing M
n : ℤ
x✝ : M
⊢ Commute (↑n) x✝

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Center

Goal state before:
pp: M : Type u_1
inst✝ : NonAssocRing M
n✝ : ℤ
x✝¹ x✝ : M
n : ℕ
⊢ ↑↑n * (x✝¹ * x✝) = ↑↑n * x✝¹ * x✝

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Membership

Goal state before:
pp: M : Type u_2
inst✝ : Mul M
p : Prop
S : p → Subsemigroup M
x : M
⊢ x ∈ ⨆ (h : p), S h ↔ ∃ (h : p), x ∈ S h

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Membership

Goal state before:
pp: ι : Sort u_1
M : Type u_2
inst✝ : Mul M
S : ι → Subsemigroup M
hS : Directed (fun x1 x2 ↦ x1 ≤ x2) S
x : M
⊢ x ∈ ↑(⨆ i, S i) ↔ x ∈ ⋃ i, ↑(S i)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Center

Goal state before:
pp: M : Type u_1
inst✝ : NonAssocRing M
n✝ : ℤ
x✝¹ x✝ : M
n : ℕ
⊢ ↑(Int.negSucc n) * (x✝¹ * x✝) = ↑(Int.negSucc n) * x✝¹ * x✝

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Center

Goal state before:
pp: M : Type u_1
inst✝ : NonAssocRing M
n✝ : ℤ
x✝¹ x✝ : M
n : ℕ
⊢ x✝¹ * x✝ * ↑↑n = x✝¹ * (x✝ * ↑↑n)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Membership

Goal state before:
pp: ι : Sort u_1
M : Type u_2
inst✝ : Mul M
S : ι → Subsemigroup M
hS : ∀ (i : ι), IsMulCommutative ↥(S i)
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) S
⊢ IsMulCommutative ↥(⨆ i, S i)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Membership

Goal state before:
pp: M : Type u_2
inst✝ : Mul M
S : Set (Subsemigroup M)
hS : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) S
x : M
⊢ x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Membership

Goal state before:
pp: M : Type u_2
inst✝ : Mul M
S : Set (Subsemigroup M)
hS : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) S
x : M
⊢ x ∈ ↑(sSup S) ↔ x ∈ ⋃ s ∈ S, ↑s

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Membership

Goal state before:
pp: M : Type u_2
inst✝ : Mul M
S T : Subsemigroup M
⊢ ∀ {x : M}, x ∈ S → x ∈ S ⊔ T

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Membership

Goal state before:
pp: M : Type u_2
inst✝ : Mul M
S T : Subsemigroup M
⊢ ∀ {x : M}, x ∈ T → x ∈ S ⊔ T

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Center

Goal state before:
pp: M : Type u_1
inst✝ : NonAssocRing M
n✝ : ℤ
x✝¹ x✝ : M
n : ℕ
⊢ x✝¹ * x✝ * ↑(Int.negSucc n) = x✝¹ * (x✝ * ↑(Int.negSucc n))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Membership

Goal state before:
pp: ι : Sort u_1
M : Type u_2
inst✝ : Mul M
S : ι → Subsemigroup M
i : ι
⊢ ∀ {x : M}, x ∈ S i → x ∈ iSup S

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Membership

Goal state before:
pp: M : Type u_2
inst✝ : Mul M
S : Set (Subsemigroup M)
s : Subsemigroup M
hs : s ∈ S
⊢ ∀ {x : M}, x ∈ s → x ∈ sSup S

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Center

Goal state before:
pp: M : Type u_1
inst✝ : Distrib M
a b : M
ha : a ∈ center M
hb : b ∈ center M
x✝ : M
⊢ Commute (a + b) x✝

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Center

Goal state before:
pp: M : Type u_1
inst✝ : Distrib M
a b : M
ha : a ∈ center M
hb : b ∈ center M
x✝¹ x✝ : M
⊢ (a + b) * (x✝¹ * x✝) = (a + b) * x✝¹ * x✝

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Membership

Goal state before:
pp: ι : Sort u_1
M : Type u_2
inst✝ : Mul M
S : ι → Subsemigroup M
C : M → Prop
x₁ : M
hx₁ : x₁ ∈ ⨆ i, S i
mem : ∀ (i : ι), ∀ x₂ ∈ S i, C x₂
mul : ∀ (x y : M), C x → C y → C (x * y)
⊢ C x₁

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Center

Goal state before:
pp: M : Type u_1
inst✝ : Distrib M
a b : M
ha : a ∈ center M
hb : b ∈ center M
x✝¹ x✝ : M
⊢ x✝¹ * x✝ * (a + b) = x✝¹ * (x✝ * (a + b))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Center

Goal state before:
pp: M : Type u_1
inst✝ : NonUnitalNonAssocRing M
a : M
ha : a ∈ center M
x✝ : M
⊢ Commute (-a) x✝

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Center

Goal state before:
pp: M : Type u_1
inst✝ : NonUnitalNonAssocRing M
a : M
ha : a ∈ center M
x✝¹ x✝ : M
⊢ -a * (x✝¹ * x✝) = -a * x✝¹ * x✝

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Center

Goal state before:
pp: M : Type u_1
inst✝ : NonUnitalNonAssocRing M
a : M
ha : a ∈ center M
x✝¹ x✝ : M
⊢ x✝¹ * x✝ * -a = x✝¹ * (x✝ * -a)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Centralizer

Goal state before:
pp: M : Type u_1
S : Set M
a b : M
inst✝ : Distrib M
ha : a ∈ S.centralizer
hb : b ∈ S.centralizer
c : M
hc : c ∈ S
⊢ c * (a + b) = (a + b) * c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Centralizer

Goal state before:
pp: M : Type u_1
S : Set M
a : M
inst✝¹ : Mul M
inst✝ : HasDistribNeg M
ha : a ∈ S.centralizer
c : M
hc : c ∈ S
⊢ c * -a = -a * c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Submonoid.Basic

Goal state before:
pp: M : Type u_1
R : Type u_2
inst✝² : NonUnitalNonAssocSemiring R
inst✝¹ : SetLike M R
inst✝ : MulMemClass M R
S : M
a b : R
ha : a ∈ closure ↑S
hb : b ∈ S
⊢ a * b ∈ closure ↑S

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Submonoid.Basic

Goal state before:
pp: M : Type u_1
R : Type u_2
inst✝² : NonUnitalNonAssocSemiring R
inst✝¹ : SetLike M R
inst✝ : MulMemClass M R
S : M
a b : R
ha : a ∈ closure ↑S
hb : b ∈ closure ↑S
⊢ a * b ∈ closure ↑S

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Membership

Goal state before:
pp: ι : Sort u_1
M : Type u_2
inst✝ : Mul M
S : ι → Subsemigroup M
C : (x : M) → x ∈ ⨆ i, S i → Prop
mem : ∀ (i : ι) (x : M) (hxS : x ∈ S i), C x ⋯
mul : ∀ (x y : M) (hx : x ∈ ⨆ i, S i) (hy : y ∈ ⨆ i, S i), C x hx → C y hy → C (x * y) ⋯
x₁ : M
hx₁ : x₁ ∈ ⨆ i, S i
⊢ C x₁ hx₁

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Prod

Goal state before:
pp: R : Type u_1
R' : Type u_2
S : Type u_3
S' : Type u_4
T : Type u_5
inst✝¹ : Distrib R
inst✝ : Distrib S
x✝² x✝¹ x✝ : R × S
⊢ x✝² * (x✝¹ + x✝) = x✝² * x✝¹ + x✝² * x✝

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Prod

Goal state before:
pp: R : Type u_1
R' : Type u_2
S : Type u_3
S' : Type u_4
T : Type u_5
inst✝¹ : Distrib R
inst✝ : Distrib S
x✝² x✝¹ x✝ : R × S
⊢ (x✝² + x✝¹) * x✝ = x✝² * x✝ + x✝¹ * x✝

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Prod

Goal state before:
pp: R : Type u_1
S : Type u_3
T : Type u_5
inst✝² : NonUnitalNonAssocSemiring R
inst✝¹ : NonUnitalNonAssocSemiring S
inst✝ : NonUnitalNonAssocSemiring T
f : R →ₙ+* S × T
x : R
⊢ (((fst S T).comp f).prod ((snd S T).comp f)) x = f x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Prod

Goal state before:
pp: R : Type u_1
S : Type u_3
T : Type u_5
inst✝² : NonAssocSemiring R
inst✝¹ : NonAssocSemiring S
inst✝ : NonAssocSemiring T
f : R →+* S × T
x : R
⊢ (((fst S T).comp f).prod ((snd S T).comp f)) x = f x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Prod

Goal state before:
pp: R : Type u_1
R' : Type u_2
S : Type u_3
S' : Type u_4
T : Type u_5
inst✝⁴ : NonAssocSemiring R
inst✝³ : NonAssocSemiring S
inst✝² : NonAssocSemiring R'
inst✝¹ : NonAssocSemiring S'
inst✝ : Subsingleton S
x : R × S
⊢ (fun x ↦ (x, 0)) x.1 = x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Prod

Goal state before:
pp: R : Type u_1
R' : Type u_2
S : Type u_3
S' : Type u_4
T : Type u_5
inst✝⁴ : NonAssocSemiring R
inst✝³ : NonAssocSemiring S
inst✝² : NonAssocSemiring R'
inst✝¹ : NonAssocSemiring S'
inst✝ : Subsingleton S
⊢ ∀ (x y : R), (x * y, 0) = (x, 0) * (y, 0)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Prod

Goal state before:
pp: R : Type u_1
R' : Type u_2
S : Type u_3
S' : Type u_4
T : Type u_5
inst✝⁴ : NonAssocSemiring R
inst✝³ : NonAssocSemiring S
inst✝² : NonAssocSemiring R'
inst✝¹ : NonAssocSemiring S'
inst✝ : Subsingleton S
⊢ ∀ (x y : R), (x + y, 0) = (x, 0) + (y, 0)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Prod

Goal state before:
pp: R : Type u_1
R' : Type u_2
S : Type u_3
S' : Type u_4
T : Type u_5
inst✝⁴ : NonAssocSemiring R
inst✝³ : NonAssocSemiring S
inst✝² : NonAssocSemiring R'
inst✝¹ : NonAssocSemiring S'
inst✝ : Subsingleton S
x : S × R
⊢ (fun x ↦ (0, x)) x.2 = x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Prod

Goal state before:
pp: R : Type u_1
R' : Type u_2
S : Type u_3
S' : Type u_4
T : Type u_5
inst✝⁴ : NonAssocSemiring R
inst✝³ : NonAssocSemiring S
inst✝² : NonAssocSemiring R'
inst✝¹ : NonAssocSemiring S'
inst✝ : Subsingleton S
⊢ ∀ (x y : R), (0, x * y) = (0, x) * (0, y)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Prod

Goal state before:
pp: R : Type u_1
R' : Type u_2
S : Type u_3
S' : Type u_4
T : Type u_5
inst✝⁴ : NonAssocSemiring R
inst✝³ : NonAssocSemiring S
inst✝² : NonAssocSemiring R'
inst✝¹ : NonAssocSemiring S'
inst✝ : Subsingleton S
⊢ ∀ (x y : R), (0, x + y) = (0, x) + (0, y)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Prod

Goal state before:
pp: R : Type u_6
S : Type u_7
inst✝⁴ : Semiring R
inst✝³ : Semiring S
inst✝² : IsDomain (R × S)
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
⊢ False

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Operations

Goal state before:
pp: P : Type u_3
inst✝ : Mul P
S : Subsemigroup P
⊢ ∀ (x : P), x ∈ comap (MulHom.id P) S ↔ x ∈ S

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Operations

Goal state before:
pp: M : Type u_1
N : Type u_2
P : Type u_3
σ : Type u_4
inst✝² : Mul M
inst✝¹ : Mul N
inst✝ : Mul P
S✝ : Subsemigroup M
f : M →ₙ* N
S : Subsemigroup M
x : M
hx : x ∈ ↑S
y : M
hy : y ∈ ↑S
⊢ f (x * y) = f x * f y

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Operations

Goal state before:
pp: M : Type u_1
N : Type u_2
P : Type u_3
σ : Type u_4
inst✝² : Mul M
inst✝¹ : Mul N
inst✝ : Mul P
S✝ : Subsemigroup M
f : M →ₙ* N
S : Subsemigroup M
⊢ ∀ {a b : N}, a ∈ ⇑f '' ↑S → b ∈ ⇑f '' ↑S → a * b ∈ ⇑f '' ↑S

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Operations

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝² : Mul M
inst✝¹ : Mul N
ι : Sort u_5
inst✝ : Nonempty ι
f : M →ₙ* N
hf : Function.Injective ⇑f
s : ι → Subsemigroup M
⊢ map f (iInf s) = ⨅ i, map f (s i)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Operations

Goal state before:
pp: M : Type u_1
N : Type u_2
P : Type u_3
σ : Type u_4
inst✝² : Mul M
inst✝¹ : Mul N
inst✝ : Mul P
S✝ : Subsemigroup M
ι : Type u_5
f : M →ₙ* N
hf : Function.Injective ⇑f
S : Subsemigroup M
x : M
⊢ x ∈ comap f (map f S) → x ∈ S

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Operations

Goal state before:
pp: M : Type u_1
N : Type u_2
P : Type u_3
σ : Type u_4
inst✝² : Mul M
inst✝¹ : Mul N
inst✝ : Mul P
S✝ : Subsemigroup M
ι : Type u_5
f : M →ₙ* N
hf : Function.Surjective ⇑f
S : Subsemigroup N
x : N
h : x ∈ S
y : M
hy : f y = x
⊢ y ∈ comap f S ∧ f y = x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Operations

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝¹ : Mul M
inst✝ : Mul N
s : Subsemigroup M
x : M × N
⊢ x ∈ s.prod ⊤ ↔ x ∈ comap (MulHom.fst M N) s

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Operations

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝¹ : Mul M
inst✝ : Mul N
s : Subsemigroup N
x : M × N
⊢ x ∈ ⊤.prod s ↔ x ∈ comap (MulHom.snd M N) s

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Operations

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝¹ : Mul M
inst✝ : Mul N
⊢ ↑(⊥.prod ⊥) = ↑⊥

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Operations

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝¹ : Mul M
inst✝ : Mul N
s : Subsemigroup M
t : Subsemigroup N
u : Subsemigroup (M × N)
⊢ u ≤ s.prod t ↔ map (fst M N) u ≤ s ∧ map (snd M N) u ≤ t

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Operations

Goal state before:
pp: M : Type u_1
N : Type u_2
P : Type u_3
σ : Type u_4
inst✝² : Mul M
inst✝¹ : Mul N
inst✝ : Mul P
S : Subsemigroup M
f : M → N
hf : ∀ (x y : M), f (x * y) = f x * f y
⊢ ∀ {a b : N}, a ∈ Set.range f → b ∈ Set.range f → a * b ∈ Set.range f

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Operations

Goal state before:
pp: M : Type u_1
N : Type u_2
P : Type u_3
inst✝² : Mul M
inst✝¹ : Mul N
inst✝ : Mul P
g : N →ₙ* P
f : M →ₙ* N
⊢ map g f.srange = (g.comp f).srange

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Operations

Goal state before:
pp: M : Type u_1
inst✝¹ : Mul M
N : Type u_5
inst✝ : Mul N
f : M →ₙ* N
⊢ ↑f.srange = ↑⊤ ↔ Set.range ⇑f = Set.univ

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Operations

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝¹ : Mul M
inst✝ : Mul N
f : M →ₙ* N
M' : Subsemigroup M
⊢ Function.Surjective ⇑(f.subsemigroupMap M')

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Operations

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝³ : Mul M
inst✝² : Mul N
inst✝¹ : Nonempty M
inst✝ : Nonempty N
s : Subsemigroup M
t : Subsemigroup N
⊢ s.prod t = ⊤ ↔ s = ⊤ ∧ t = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Subsemigroup.Operations

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝¹ : Mul M
inst✝ : Mul N
f : M →ₙ* N
S : Subsemigroup N
h : S ≤ f.srange
⊢ map f (comap f S) = S

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
S : Type v
inst✝³ : NonUnitalNonAssocSemiring R
inst✝² : NonUnitalNonAssocSemiring S
F : Type u_1
inst✝¹ : FunLike F R S
inst✝ : NonUnitalRingHomClass F R S
f : F
⊢ srange f = map f ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
S : Type v
T : Type w
inst✝² : NonUnitalNonAssocSemiring R
inst✝¹ : NonUnitalNonAssocSemiring S
inst✝ : NonUnitalNonAssocSemiring T
g : S →ₙ+* T
f : R →ₙ+* S
⊢ map g (srange f) = srange (g.comp f)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
S : Type v
T : Type w
inst✝ : NonUnitalNonAssocSemiring R
M : Subsemigroup R
s : Set (NonUnitalSubsemiring R)
⊢ ↑(⨅ t ∈ s, t.toSubsemigroup) = ⋂ t ∈ s, ↑t

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
S : Type v
T : Type w
inst✝ : NonUnitalNonAssocSemiring R
M : Subsemigroup R
s : Set (NonUnitalSubsemiring R)
⊢ ↑(⨅ t ∈ s, t.toAddSubmonoid) = ⋂ t ∈ s, ↑t

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonUnitalNonAssocSemiring R
ι : Sort u_1
S : ι → NonUnitalSubsemiring R
⊢ ↑(⨅ i, S i) = ⋂ i, ↑(S i)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonUnitalNonAssocSemiring R
ι : Sort u_1
S : ι → NonUnitalSubsemiring R
x : R
⊢ x ∈ ⨅ i, S i ↔ ∀ (i : ι), x ∈ S i

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonUnitalNonAssocSemiring R
a : R
⊢ a ∈ Set.center R ↔
    AddMonoidHom.mulLeft a = AddMonoidHom.mulRight a ∧
      AddMonoidHom.mul.compr₂ (AddMonoidHom.mulLeft a) = AddMonoidHom.mul.comp (AddMonoidHom.mulLeft a) ∧
        AddMonoidHom.mul.compr₂ (AddMonoidHom.mulRight a) = AddMonoidHom.mul.compl₂ (AddMonoidHom.mulRight a)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
S : Type v
T : Type w
inst✝¹ : NonUnitalNonAssocSemiring R
M : Subsemigroup R
inst✝ : NonUnitalNonAssocSemiring S
e : R ≃+* S
x✝¹ x✝ : ↥(center R)
⊢ ↑((Subsemigroup.centerCongr ↑e).toFun (x✝¹ + x✝)) =
    ↑((Subsemigroup.centerCongr ↑e).toFun x✝¹ + (Subsemigroup.centerCongr ↑e).toFun x✝)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R✝ : Type u
S : Type v
T : Type w
inst✝¹ : NonUnitalNonAssocSemiring R✝
M : Subsemigroup R✝
R : Type u_1
inst✝ : NonUnitalSemiring R
⊢ (center.instNonUnitalCommSemiring R).toNonUnitalSemiring = NonUnitalSubsemiringClass.toNonUnitalSemiring (center R)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u_1
inst✝ : NonUnitalSemiring R
z : R
⊢ z ∈ center R ↔ ∀ (g : R), g * z = z * g

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
S : Type v
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f : R ≃+* S
K : NonUnitalSubsemiring R
x : S
⊢ x ∈ map (↑f) K ↔ f.symm x ∈ K

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonUnitalNonAssocSemiring R
M : Subsemigroup R
⊢ M.nonUnitalSubsemiringClosure = NonUnitalSubsemiring.closure ↑M

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonUnitalNonAssocSemiring R
s : Set R
⊢ ↑(closure s) = ↑(AddSubmonoid.closure ↑(Subsemigroup.closure s))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonUnitalNonAssocSemiring R
s : Set R
⊢ closure ↑(AddSubmonoid.closure s) = closure s

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonUnitalNonAssocSemiring R
s : Set R
p : (x y : R) → x ∈ closure s → y ∈ closure s → Prop
mem_mem : ∀ (x : R) (hx : x ∈ s) (y : R) (hy : y ∈ s), p x y ⋯ ⋯
zero_left : ∀ (x : R) (hx : x ∈ closure s), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : x ∈ closure s), p x 0 hx ⋯
add_left :
  ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s),
    p x z hx hz → p y z hy hz → p (x + y) z ⋯ hz
add_right :
  ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s),
    p x y hx hy → p x z hx hz → p x (y + z) hx ⋯
mul_left :
  ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s),
    p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s),
    p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y : R
hx : x ∈ closure s
hy : y ∈ closure s
⊢ p x y hx hy

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
S : Type v
inst✝⁴ : NonUnitalNonAssocSemiring R
inst✝³ : NonUnitalNonAssocSemiring S
F : Type u_1
inst✝² : FunLike F R S
inst✝¹ : NonUnitalRingHomClass F R S
ι : Sort u_2
inst✝ : Nonempty ι
f : F
hf : Function.Injective ⇑f
s : ι → NonUnitalSubsemiring R
⊢ map f (iInf s) = ⨅ i, map f (s i)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
S : Type v
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
s : NonUnitalSubsemiring R
x : R × S
⊢ x ∈ s.prod ⊤ ↔ x ∈ comap (NonUnitalRingHom.fst R S) s

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
S : Type v
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
s : NonUnitalSubsemiring S
x : R × S
⊢ x ∈ ⊤.prod s ↔ x ∈ comap (NonUnitalRingHom.snd R S) s

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonUnitalNonAssocSemiring R
ι : Sort u_2
hι : Nonempty ι
S : ι → NonUnitalSubsemiring R
hS : Directed (fun x1 x2 ↦ x1 ≤ x2) S
x : R
U : NonUnitalSubsemiring R :=
  NonUnitalSubsemiring.mk' (⋃ i, ↑(S i)) (⨆ i, (S i).toSubsemigroup) ⋯ (⨆ i, (S i).toAddSubmonoid) ⋯
this : ⨆ i, S i ≤ U
⊢ x ∈ ⨆ i, S i → ∃ i, x ∈ S i

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonUnitalNonAssocSemiring R
ι : Sort u_2
hι : Nonempty ι
S : ι → NonUnitalSubsemiring R
hS : Directed (fun x1 x2 ↦ x1 ≤ x2) S
x : R
⊢ x ∈ ⨆ i, S i ↔ ∃ i, x ∈ S i

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonUnitalNonAssocSemiring R
ι : Sort u_2
hι : Nonempty ι
S : ι → NonUnitalSubsemiring R
hS : Directed (fun x1 x2 ↦ x1 ≤ x2) S
x : R
⊢ x ∈ ↑(⨆ i, S i) ↔ x ∈ ⋃ i, ↑(S i)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonUnitalNonAssocSemiring R
S : Set (NonUnitalSubsemiring R)
Sne : S.Nonempty
hS : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) S
x : R
⊢ x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonUnitalNonAssocSemiring R
S : Set (NonUnitalSubsemiring R)
Sne : S.Nonempty
hS : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) S
x : R
⊢ x ∈ ↑(sSup S) ↔ x ∈ ⋃ s ∈ S, ↑s

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
inst✝¹ : NonUnitalNonAssocSemiring R
ι : Sort u_2
inst✝ : Nonempty ι
S : ι → NonUnitalSubsemiring R
hS : ∀ (i : ι), IsMulCommutative ↥(S i)
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) S
⊢ IsMulCommutative ↥(⨆ i, S i)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
S : Type v
inst✝³ : NonUnitalNonAssocSemiring R
F : Type u_1
inst✝² : FunLike F R S
inst✝¹ : NonUnitalNonAssocSemiring S
inst✝ : NonUnitalRingHomClass F R S
f : F
⊢ ↑(srange f) = ↑⊤ ↔ Set.range ⇑f = Set.univ

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonUnitalNonAssocRing R
S : NonUnitalSubring R
⊢ S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonUnitalNonAssocRing R
S : NonUnitalSubring R
⊢ S.toAddSubgroup = ⊤ ↔ S = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubring.Basic

Goal state before:
pp: R : Type u
S : Type v
inst✝¹ : NonUnitalNonAssocRing R
inst✝ : NonUnitalNonAssocRing S
f : R →ₙ+* S
⊢ f.range = NonUnitalSubring.map f ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubring.Basic

Goal state before:
pp: R : Type u
S : Type v
T : Type u_1
inst✝² : NonUnitalNonAssocRing R
inst✝¹ : NonUnitalNonAssocRing S
inst✝ : NonUnitalNonAssocRing T
g : S →ₙ+* T
f : R →ₙ+* S
⊢ NonUnitalSubring.map g f.range = (g.comp f).range

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonUnitalNonAssocRing R
s : Set (NonUnitalSubring R)
⊢ ↑(⨅ t ∈ s, t.toSubsemigroup) = ⋂ t ∈ s, ↑t

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonUnitalNonAssocRing R
s : Set (NonUnitalSubring R)
⊢ ↑(⨅ t ∈ s, t.toAddSubgroup) = ⋂ t ∈ s, ↑t

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonUnitalNonAssocRing R
ι : Sort u_1
S : ι → NonUnitalSubring R
⊢ ↑(⨅ i, S i) = ⋂ i, ↑(S i)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonUnitalNonAssocRing R
ι : Sort u_1
S : ι → NonUnitalSubring R
x : R
⊢ x ∈ ⨅ i, S i ↔ ∀ (i : ι), x ∈ S i

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonUnitalRing R
⊢ (center.instNonUnitalCommRing R).toNonUnitalRing = NonUnitalSubringClass.toNonUnitalRing (center R)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Algebra.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
S : Type u_4
inst✝¹ : Semiring S
inst✝ : Algebra R S
r : R
⊢ algebraMapSubmonoid S (Submonoid.powers r) = Submonoid.powers ((algebraMap R S) r)

Tactic:
by

Goal state after: