mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Star.StarProjection

Goal state before:
pp: R : Type u_1
inst✝² : NonUnitalRing R
inst✝¹ : StarRing R
inst✝ : IsAddTorsionFree R
p q : R
hp : IsStarProjection p
hq : IsStarProjection q
⊢ p * q = p ∧ star q * star p = p ↔ p * q = p

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : NonUnitalRing R
inst✝¹ : StarRing R
inst✝ : IsAddTorsionFree R
p q : R
hp : IsStarProjection p
hq : IsStarProjection q
⊢ p * q = p ∧ star q * star p = p ↔ p * q = p
Module: Mathlib.Algebra.Star.StarProjection

Goal state before:
pp: R : Type u_1
inst✝² : NonUnitalRing R
inst✝¹ : StarRing R
inst✝ : IsAddTorsionFree R
p q : R
hp : IsStarProjection p
hq : IsStarProjection q
⊢ p = p * q ∧ p = p * q ↔ p = p * q

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : NonUnitalRing R
inst✝¹ : StarRing R
inst✝ : IsAddTorsionFree R
p q : R
hp : IsStarProjection p
hq : IsStarProjection q
⊢ p = p * q ∧ p = p * q ↔ p = p * q
Module: Mathlib.Algebra.Star.StarProjection

Goal state before:
pp: R : Type u_1
inst✝² : NonUnitalRing R
inst✝¹ : StarRing R
inst✝ : IsAddTorsionFree R
p q : R
hp : IsStarProjection p
hq : IsStarProjection q
⊢ IsStarProjection (q - p) ↔ star (q * p) = star p

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : NonUnitalRing R
inst✝¹ : StarRing R
inst✝ : IsAddTorsionFree R
p q : R
hp : IsStarProjection p
hq : IsStarProjection q
⊢ IsStarProjection (q - p) ↔ star (q * p) = star p
Module: Mathlib.Algebra.Star.Center

Goal state before:
pp: R : Type u_1
inst✝¹ : Mul R
inst✝ : StarMul R
s : Set R
hcomm : ∀ x ∈ s, ∀ y ∈ s, y * x = x * y
hcomm_star : ∀ x ∈ s, ∀ y ∈ s, y * star x = star x * y
⊢ (s ⊆ s.centralizer ∧ s ⊆ star s.centralizer) ∧ star s ⊆ s.centralizer ∧ s ⊆ s.centralizer

Tactic:
star_subset

Goal state after:
R : Type u_1
inst✝¹ : Mul R
inst✝ : StarMul R
s : Set R
hcomm : ∀ x ∈ s, ∀ y ∈ s, y * x = x * y
hcomm_star : ∀ x ∈ s, ∀ y ∈ s, y * star x = star x * y
⊢ (s ⊆ s.centralizer ∧ s ⊆ star s.centralizer) ∧ s ⊆ star s.centralizer ∧ s ⊆ s.centralizer
Module: Mathlib.Algebra.Star.Center

Goal state before:
pp: R : Type u_1
inst✝¹ : Mul R
inst✝ : StarMul R
s : Set R
hcomm : ∀ x ∈ s, ∀ y ∈ s, y * x = x * y
hcomm_star : ∀ x ∈ s, ∀ y ∈ s, y * star x = star x * y
⊢ ∀ x ∈ s ∪ star s, ∀ y ∈ s ∪ star s, y * x = x * y

Tactic:
by
  change s ∪ star s ⊆ (s ∪ star s).centralizer
  simp_rw [centralizer_union, ← star_centralizer, union_subset_iff, subset_inter_iff,
    star_subset_star, star_subset]
  exact ⟨⟨hcomm, hcomm_star⟩, ⟨hcomm_star, hcomm⟩⟩

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

Goal state before:
pp: case a.tmul
R : Type u_1
M : Type u_2
N : Type u_3
A : Type u_4
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring A
inst✝⁶ : Algebra R A
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Module A N
inst✝ : IsScalarTower R A N
l : M →ₗ[R] N
x✝ : A
y✝ : M
⊢ x✝ • l y✝ ∈ Submodule.span A ↑l.range

Tactic:
]

Goal state after:
case a.tmul
R : Type u_1
M : Type u_2
N : Type u_3
A : Type u_4
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring A
inst✝⁶ : Algebra R A
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Module A N
inst✝ : IsScalarTower R A N
l : M →ₗ[R] N
x✝ : A
y✝ : M
⊢ x✝ • l y✝ ∈ Submodule.span A ↑l.range
Module: Mathlib.RingTheory.TensorProduct.Basic

Goal state before:
pp: case a.add
R : Type u_1
M : Type u_2
N : Type u_3
A : Type u_4
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring A
inst✝⁶ : Algebra R A
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Module A N
inst✝ : IsScalarTower R A N
l : M →ₗ[R] N
x✝ y✝ : A ⊗[R] M
a✝¹ : (liftBaseChange A l) x✝ ∈ Submodule.span A ↑l.range
a✝ : (liftBaseChange A l) y✝ ∈ Submodule.span A ↑l.range
⊢ (liftBaseChange A l) x✝ + (liftBaseChange A l) y✝ ∈ Submodule.span A ↑l.range

Tactic:
]

Goal state after:
case a.add
R : Type u_1
M : Type u_2
N : Type u_3
A : Type u_4
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring A
inst✝⁶ : Algebra R A
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Module A N
inst✝ : IsScalarTower R A N
l : M →ₗ[R] N
x✝ y✝ : A ⊗[R] M
a✝¹ : (liftBaseChange A l) x✝ ∈ Submodule.span A ↑l.range
a✝ : (liftBaseChange A l) y✝ ∈ Submodule.span A ↑l.range
⊢ (liftBaseChange A l) x✝ + (liftBaseChange A l) y✝ ∈ Submodule.span A ↑l.range
Module: Mathlib.RingTheory.TensorProduct.Basic

Goal state before:
pp: case a
R : Type u_1
M : Type u_2
N : Type u_3
A : Type u_4
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring A
inst✝⁶ : Algebra R A
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Module A N
inst✝ : IsScalarTower R A N
l : M →ₗ[R] N
⊢ ↑l.range ⊆ ↑(liftBaseChange A l).range

Tactic:
]

Goal state after:
case a
R : Type u_1
M : Type u_2
N : Type u_3
A : Type u_4
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring A
inst✝⁶ : Algebra R A
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Module A N
inst✝ : IsScalarTower R A N
l : M →ₗ[R] N
⊢ ↑l.range ⊆ ↑(liftBaseChange A l).range
Module: Mathlib.RingTheory.TensorProduct.Basic

Goal state before:
pp: R : Type uR
A : Type uA
B : Type uB
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoidWithOne A
inst✝² : Module R A
inst✝¹ : AddCommMonoidWithOne B
inst✝ : Module R B
n : ℕ
⊢ 1 ⊗ₜ[R] ↑n = 1 ⊗ₜ[R] ↑n

Tactic:
]

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

Goal state before:
pp: case tmul.tmul
R : Type uR
R' : Type u_1
S : Type uS
T : Type u_2
A : Type uA
B : Type uB
C : Type uC
D : Type uD
E : Type uE
F : Type uF
inst✝¹² : CommSemiring R
inst✝¹¹ : NonUnitalNonAssocSemiring A
inst✝¹⁰ : Module R A
inst✝⁹ : SMulCommClass R A A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : NonUnitalNonAssocSemiring B
inst✝⁶ : Module R B
inst✝⁵ : SMulCommClass R B B
inst✝⁴ : IsScalarTower R B B
inst✝³ : Monoid S
inst✝² : DistribMulAction S A
inst✝¹ : IsScalarTower S A A
inst✝ : SMulCommClass R S A
r : S
a : A
b : B
a' : A
b' : B
⊢ (r • (a' * a)) ⊗ₜ[R] (b' * b) = (r • (a' * a)) ⊗ₜ[R] (b' * b)

Tactic:
]

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

Goal state before:
pp: case tmul.tmul
R : Type uR
R' : Type u_1
S : Type uS
T : Type u_2
A : Type uA
B : Type uB
C : Type uC
D : Type uD
E : Type uE
F : Type uF
inst✝¹² : CommSemiring R
inst✝¹¹ : NonUnitalNonAssocSemiring A
inst✝¹⁰ : Module R A
inst✝⁹ : SMulCommClass R A A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : NonUnitalNonAssocSemiring B
inst✝⁶ : Module R B
inst✝⁵ : SMulCommClass R B B
inst✝⁴ : IsScalarTower R B B
inst✝³ : Monoid S
inst✝² : DistribMulAction S A
inst✝¹ : SMulCommClass S A A
inst✝ : SMulCommClass R S A
r : S
a : A
b : B
a' : A
b' : B
⊢ (r • (a' * a)) ⊗ₜ[R] (b' * b) = (r • (a' * a)) ⊗ₜ[R] (b' * b)

Tactic:
]

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

Goal state before:
pp: R : Type uR
R' : Type u_1
S : Type uS
T : Type u_2
A : Type uA
B : Type uB
C : Type uC
D : Type uD
E : Type uE
F : Type uF
inst✝⁹ : CommSemiring R
inst✝⁸ : Semiring A
inst✝⁷ : Algebra R A
inst✝⁶ : Semiring B
inst✝⁵ : Algebra R B
inst✝⁴ : Semiring C
inst✝³ : Algebra R C
inst✝² : CommSemiring S
inst✝¹ : Algebra S A
inst✝ : SMulCommClass R S A
r : S
x : A ⊗[R] B
⊢ r • x = r • x

Tactic:
]

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

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : Rˣ
s : R
a : A
h : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
⊢ ¬IsUnit a ↔ ¬IsUnit a

Tactic:
]

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

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
⊢ IsUnit a ↔ IsUnit a

Tactic:
]

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

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : R
a b c : A
h₁ : (↑ₐ r - a) * b = 1
h₂ : c * (↑ₐ r - a) = 1
⊢ c * (↑ₐ r - a) = 1

Tactic:
]

Goal state after:
R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : R
a b c : A
h₁ : (↑ₐ r - a) * b = 1
h₂ : c * (↑ₐ r - a) = 1
⊢ c * (↑ₐ r - a) = 1
Module: Mathlib.RingTheory.TensorProduct.Basic

Goal state before:
pp: R : Type uR
A : Type uA
B : Type uB
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : Semiring B
inst✝ : Algebra R B
r : R
⊢ 1 ⊗ₜ[R] (r • 1) = 1 ⊗ₜ[R] (r • 1)

Tactic:
]

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

Goal state before:
pp: R : Type u
A : Type v
inst✝³ : CommSemiring R
inst✝² : Ring A
inst✝¹ : Algebra R A
inst✝ : Subsingleton A
a : A
⊢ ∅ = ∅

Tactic:
]

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

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
a b : A
r : R
ha : r ∈ resolventSet R a
hb : r ∈ resolventSet R b
⊢ ↑ₐ r - b - (↑ₐ r - a) = a - b

Tactic:
]

Goal state after:
R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
a b : A
r : R
ha : r ∈ resolventSet R a
hb : r ∈ resolventSet R b
⊢ ↑ₐ r - b - (↑ₐ r - a) = a - b
Module: Mathlib.Algebra.Algebra.Spectrum.Basic

Goal state before:
pp: case pos
R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : Rˣ
s : R
a : A
h : ¬IsUnit (↑ₐ s - a)
⊢ r • resolvent a s = resolvent (r⁻¹ • a) (r⁻¹ • s)

Tactic:
]

Goal state after:
case pos
R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : Rˣ
s : R
a : A
h : ¬IsUnit (↑ₐ s - a)
⊢ r • resolvent a s = resolvent (r⁻¹ • a) (r⁻¹ • s)
Module: Mathlib.Algebra.Algebra.Spectrum.Basic

Goal state before:
pp: case pos
R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : Rˣ
s : R
a : A
h : ¬IsUnit (s • 1 - a)
⊢ r • (s • 1 - a)⁻¹ʳ = (r⁻¹ • (s • 1 - a))⁻¹ʳ

Tactic:
]

Goal state after:
case pos
R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : Rˣ
s : R
a : A
h : ¬IsUnit (s • 1 - a)
⊢ r • (s • 1 - a)⁻¹ʳ = (r⁻¹ • (s • 1 - a))⁻¹ʳ
Module: Mathlib.RingTheory.TensorProduct.Basic

Goal state before:
pp: R : Type uR
S : Type uS
A : Type uA
B : Type uB
C : Type uC
inst✝¹² : CommSemiring R
inst✝¹¹ : Semiring A
inst✝¹⁰ : Algebra R A
inst✝⁹ : Semiring B
inst✝⁸ : Algebra R B
inst✝⁷ : Semiring C
inst✝⁶ : Algebra R C
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra S A
inst✝³ : Algebra R S
inst✝² : Algebra S C
inst✝¹ : IsScalarTower R S A
inst✝ : IsScalarTower R S C
f g : A ⊗[R] B →ₐ[S] C
ha : f.comp includeLeft = g.comp includeLeft
hb : (AlgHom.restrictScalars R f).comp includeRight = (AlgHom.restrictScalars R g).comp includeRight
a : A
b : B
this : f (a ⊗ₜ[R] b) = g (a ⊗ₜ[R] b)
⊢ ((AlgebraTensorModule.curry f.toLinearMap) a) b = ((AlgebraTensorModule.curry g.toLinearMap) a) b

Tactic:
]

Goal state after:
R : Type uR
S : Type uS
A : Type uA
B : Type uB
C : Type uC
inst✝¹² : CommSemiring R
inst✝¹¹ : Semiring A
inst✝¹⁰ : Algebra R A
inst✝⁹ : Semiring B
inst✝⁸ : Algebra R B
inst✝⁷ : Semiring C
inst✝⁶ : Algebra R C
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra S A
inst✝³ : Algebra R S
inst✝² : Algebra S C
inst✝¹ : IsScalarTower R S A
inst✝ : IsScalarTower R S C
f g : A ⊗[R] B →ₐ[S] C
ha : f.comp includeLeft = g.comp includeLeft
hb : (AlgHom.restrictScalars R f).comp includeRight = (AlgHom.restrictScalars R g).comp includeRight
a : A
b : B
this : f (a ⊗ₜ[R] b) = g (a ⊗ₜ[R] b)
⊢ ((AlgebraTensorModule.curry f.toLinearMap) a) b = ((AlgebraTensorModule.curry g.toLinearMap) a) b
Module: Mathlib.Algebra.Algebra.Spectrum.Basic

Goal state before:
pp: case neg
R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : Rˣ
s : R
a : A
h : s ∉ σ a
h' : IsUnit (r • ↑ₐ (r⁻¹ • s) - a)
⊢ r • ↑⋯.unit⁻¹ = r • ↑h'.unit⁻¹

Tactic:
]

Goal state after:
case neg
R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : Rˣ
s : R
a : A
h : s ∉ σ a
h' : IsUnit (r • ↑ₐ (r⁻¹ • s) - a)
⊢ r • ↑⋯.unit⁻¹ = r • ↑h'.unit⁻¹
Module: Mathlib.Algebra.Algebra.Spectrum.Basic

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : Rˣ
a : Aˣ
h : IsUnit (r • 1 - ↑a)
⊢ IsUnit (r⁻¹ • 1 - ↑a⁻¹)

Tactic:
]

Goal state after:
R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : Rˣ
a : Aˣ
h : IsUnit (r • 1 - ↑a)
⊢ IsUnit (r⁻¹ • 1 - ↑a⁻¹)
Module: Mathlib.Algebra.Algebra.Spectrum.Basic

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : Rˣ
a : Aˣ
h : IsUnit (r • 1 - ↑a)
⊢ IsUnit (r • ↑a⁻¹ - 1)

Tactic:
]

Goal state after:
R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : Rˣ
a : Aˣ
h : IsUnit (r • 1 - ↑a)
⊢ IsUnit (r • ↑a⁻¹ - 1)
Module: Mathlib.Algebra.Algebra.Spectrum.Basic

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : Rˣ
a : Aˣ
h : IsUnit (r • 1 - ↑a)
⊢ r • 1 - ↑a = r • 1 - ↑a

Tactic:
]

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

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : Rˣ
a : Aˣ
h : IsUnit (r • 1 - ↑a)
h₁ : ↑a * (r • ↑a⁻¹ - 1) = r • 1 - ↑a
⊢ r • 1 - ↑a = r • 1 - ↑a

Tactic:
]

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

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : Rˣ
a : Aˣ
h : IsUnit (r • 1 - ↑a)
h₁ : ↑a * (r • ↑a⁻¹ - 1) = (r • ↑a⁻¹ - 1) * ↑a
h₂ : (r • ↑a⁻¹ - 1) * ↑a = r • 1 - ↑a
⊢ Commute (↑a) (r • ↑a⁻¹ - 1)

Tactic:
]

Goal state after:
R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : Rˣ
a : Aˣ
h : IsUnit (r • 1 - ↑a)
h₁ : ↑a * (r • ↑a⁻¹ - 1) = (r • ↑a⁻¹ - 1) * ↑a
h₂ : (r • ↑a⁻¹ - 1) * ↑a = r • 1 - ↑a
⊢ Commute (↑a) (r • ↑a⁻¹ - 1)
Module: Mathlib.Algebra.Algebra.Spectrum.Basic

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
r s : R
⊢ r ∈ σ a ↔ r ∈ σ a

Tactic:
]

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

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
r : Rˣ
x : R
x_eq : x = r • r⁻¹ • x
⊢ r • r⁻¹ • x ∈ σ (r • a) ↔ x ∈ r • σ a

Tactic:
]

Goal state after:
R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
r : Rˣ
x : R
x_eq : x = r • r⁻¹ • x
⊢ r • r⁻¹ • x ∈ σ (r • a) ↔ x ∈ r • σ a
Module: Mathlib.Algebra.Algebra.Spectrum.Basic

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
r : Rˣ
x : R
x_eq : x = r • r⁻¹ • x
⊢ r⁻¹ • x ∈ σ a ↔ x ∈ r • σ a

Tactic:
]

Goal state after:
R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
r : Rˣ
x : R
x_eq : x = r • r⁻¹ • x
⊢ r⁻¹ • x ∈ σ a ↔ x ∈ r • σ a
Module: Mathlib.Algebra.Algebra.Spectrum.Basic

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
a b : A
r : Rˣ
h₁ : ∀ (x y : A), IsUnit (1 - x * y) → IsUnit (1 - y * x)
this : IsUnit (1 - r⁻¹ • a * b) ↔ IsUnit (1 - r⁻¹ • (b * a))
⊢ ↑r ∈ σ (a * b) ↔ ↑r ∈ σ (b * a)

Tactic:
]

Goal state after:
R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
a b : A
r : Rˣ
h₁ : ∀ (x y : A), IsUnit (1 - x * y) → IsUnit (1 - y * x)
this : IsUnit (1 - r⁻¹ • a * b) ↔ IsUnit (1 - r⁻¹ • (b * a))
⊢ ↑r ∈ σ (a * b) ↔ ↑r ∈ σ (b * a)
Module: Mathlib.Algebra.Algebra.Spectrum.Basic

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
r x : R
⊢ x ∈ σ (↑ₐ r + a) ↔ x ∈ σ (↑ₐ r + a)

Tactic:
]

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

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
⊢ (-resolventSet R a)ᶜ = (-resolventSet R a)ᶜ

Tactic:
]

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

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
r : R
⊢ σ (↑ₐ r + -a) = σ (↑ₐ r + -a)

Tactic:
]

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

Goal state before:
pp: 𝕜 : Type u
A : Type v
inst✝³ : Field 𝕜
inst✝² : Ring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial A
⊢ {0}ᶜ ⊆ resolventSet 𝕜 0

Tactic:
]

Goal state after:
𝕜 : Type u
A : Type v
inst✝³ : Field 𝕜
inst✝² : Ring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial A
⊢ {0}ᶜ ⊆ resolventSet 𝕜 0
Module: Mathlib.Algebra.Algebra.Spectrum.Basic

Goal state before:
pp: 𝕜 : Type u
A : Type v
inst✝³ : Field 𝕜
inst✝² : Ring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial A
k : 𝕜
hk : k ≠ 0
⊢ k ∈ resolventSet 𝕜 0

Tactic:
]

Goal state after:
𝕜 : Type u
A : Type v
inst✝³ : Field 𝕜
inst✝² : Ring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial A
k : 𝕜
hk : k ≠ 0
⊢ k ∈ resolventSet 𝕜 0
Module: Mathlib.Algebra.Algebra.Spectrum.Basic

Goal state before:
pp: 𝕜 : Type u
A : Type v
inst✝³ : Field 𝕜
inst✝² : Ring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial A
k : 𝕜
⊢ {k} = {k}

Tactic:
]

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

Goal state before:
pp: 𝕜 : Type u
A : Type v
inst✝³ : Field 𝕜
inst✝² : Ring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial A
⊢ σ 1 = σ 1

Tactic:
]

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

Goal state before:
pp: R : Type uR
A : Type uA
inst✝⁴ : CommSemiring R
inst✝³ : Ring A
inst✝² : Algebra R A
B : Type u_3
inst✝¹ : Ring B
inst✝ : Algebra R B
z : ℤ
⊢ 1 ⊗ₜ[R] ↑z = 1 ⊗ₜ[R] ↑z

Tactic:
]

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

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
a✝ : A
u✝ : Aˣ
a : A
u : Aˣ
μ : R
hμ : ¬IsUnit ((algebraMap R A) μ - ↑u * a * ↑u⁻¹)
⊢ ¬IsUnit ((algebraMap R A) μ - a)

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
a✝ : A
u✝ : Aˣ
a : A
u : Aˣ
μ : R
hμ : ¬IsUnit ((algebraMap R A) μ - ↑u * a * ↑u⁻¹)
⊢ ¬IsUnit ((algebraMap R A) μ - a)
Module: Mathlib.RingTheory.TensorProduct.Basic

Goal state before:
pp: R : Type uR
A : Type uA
B : Type uB
inst✝⁴ : CommRing R
inst✝³ : Ring A
inst✝² : Ring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
⊢ ⊤ ≤ Subring.closure (Set.range ⇑includeLeft ∪ Set.range ⇑includeRight)

Tactic:
]

Goal state after:
R : Type uR
A : Type uA
B : Type uB
inst✝⁴ : CommRing R
inst✝³ : Ring A
inst✝² : Ring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
⊢ ⊤ ≤ Subring.closure (Set.range ⇑includeLeft ∪ Set.range ⇑includeRight)
Module: Mathlib.RingTheory.TensorProduct.Basic

Goal state before:
pp: R : Type uR
A : Type uA
B : Type uB
inst✝⁴ : CommSemiring R
inst✝³ : CommSemiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
s : Set B
hs : adjoin R s = ⊤
⊢ Submodule.span A ↑((TensorProduct.mk R A B) 1).range ≤ Subalgebra.toSubmodule (adjoin A ↑includeRight.range)

Tactic:
]

Goal state after:
R : Type uR
A : Type uA
B : Type uB
inst✝⁴ : CommSemiring R
inst✝³ : CommSemiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
s : Set B
hs : adjoin R s = ⊤
⊢ Submodule.span A ↑((TensorProduct.mk R A B) 1).range ≤ Subalgebra.toSubmodule (adjoin A ↑includeRight.range)
Module: Mathlib.RingTheory.TensorProduct.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
M : Type u_4
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : Semiring A
inst✝⁷ : Semiring B
inst✝⁶ : Module A M
inst✝⁵ : Module B M
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
inst✝² : IsScalarTower R A M
inst✝¹ : IsScalarTower R B M
inst✝ : SMulCommClass A B M
m : M
⊢ 1 • m = m

Tactic:
by
    -- Porting note: was one `simp only`, not two
    simp only [(· • ·), Algebra.TensorProduct.one_def]
    simp only [moduleAux_apply, one_smul]

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

Goal state before:
pp: R : Type u_1
Q : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid Q
inst✝ : Module R Q
I : Submodule R R
⊢ (↑(TensorProduct.lid R Q) ∘ₗ rTensor Q I.subtype).range = I • ⊤

Tactic:
]

Goal state after:
R : Type u_1
Q : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid Q
inst✝ : Module R Q
I : Submodule R R
⊢ (↑(TensorProduct.lid R Q) ∘ₗ rTensor Q I.subtype).range = I • ⊤
Module: Mathlib.RingTheory.TensorProduct.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
S : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Semiring S
inst✝ : Algebra R S
h : Function.Surjective ⇑(algebraMap R S)
⊢ {t | ∃ m n, m ⊗ₜ[R] n = t} ⊆ ↑((mk R S M) 1).range

Tactic:
]

Goal state after:
R : Type u_1
M : Type u_2
S : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Semiring S
inst✝ : Algebra R S
h : Function.Surjective ⇑(algebraMap R S)
⊢ {t | ∃ m n, m ⊗ₜ[R] n = t} ⊆ ↑((mk R S M) 1).range
Module: Mathlib.RingTheory.TensorProduct.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
S : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Semiring S
inst✝ : Algebra R S
h : Function.Surjective ⇑(algebraMap R S)
y : M
x : R
⊢ 1 ⊗ₜ[R] (x • y) ∈ ↑((mk R S M) 1).range

Tactic:
]

Goal state after:
R : Type u_1
M : Type u_2
S : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Semiring S
inst✝ : Algebra R S
h : Function.Surjective ⇑(algebraMap R S)
y : M
x : R
⊢ 1 ⊗ₜ[R] (x • y) ∈ ↑((mk R S M) 1).range
Module: Mathlib.RingTheory.TensorProduct.Basic

Goal state before:
pp: R : Type u_1
S : Type u_3
T : Type u_4
inst✝⁴ : CommSemiring R
inst✝³ : Semiring S
inst✝² : Algebra R S
inst✝¹ : Ring T
inst✝ : Algebra R T
h : Function.Surjective ⇑(algebraMap R T)
⊢ {t | ∃ m n, m ⊗ₜ[R] n = t} ⊆ ↑((mk R S T).flip 1).range

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_3
T : Type u_4
inst✝⁴ : CommSemiring R
inst✝³ : Semiring S
inst✝² : Algebra R S
inst✝¹ : Ring T
inst✝ : Algebra R T
h : Function.Surjective ⇑(algebraMap R T)
⊢ {t | ∃ m n, m ⊗ₜ[R] n = t} ⊆ ↑((mk R S T).flip 1).range
Module: Mathlib.RingTheory.TensorProduct.Basic

Goal state before:
pp: R : Type u_1
S : Type u_3
T : Type u_4
inst✝⁴ : CommSemiring R
inst✝³ : Semiring S
inst✝² : Algebra R S
inst✝¹ : Ring T
inst✝ : Algebra R T
h : Function.Surjective ⇑(algebraMap R T)
s : S
r : R
⊢ (r • s) ⊗ₜ[R] 1 ∈ ↑((mk R S T).flip 1).range

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_3
T : Type u_4
inst✝⁴ : CommSemiring R
inst✝³ : Semiring S
inst✝² : Algebra R S
inst✝¹ : Ring T
inst✝ : Algebra R T
h : Function.Surjective ⇑(algebraMap R T)
s : S
r : R
⊢ (r • s) ⊗ₜ[R] 1 ∈ ↑((mk R S T).flip 1).range