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 |
|---|---|---|---|---|---|---|
M : Type u_1
inst✝ : MulOneClass M
x✝¹ x✝ : M
⊢ 1 * (x✝¹ * x✝) = 1 * x✝¹ * x✝ | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [one_mul, one_mul] | @[to_additive (attr := simp) zero_mem_addCenter]
theorem one_mem_center [MulOneClass M] : (1 : M) ∈ Set.center M where
comm _ := by rw [one_mul, mul_one]
left_assoc _ _ := by | Mathlib.GroupTheory.Subsemigroup.Center.160_0.vKbtzx3rREtft3E | @[to_additive (attr | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : MulOneClass M
x✝¹ x✝ : M
⊢ x✝¹ * 1 * x✝ = x✝¹ * (1 * x✝) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [mul_one, one_mul] | @[to_additive (attr := simp) zero_mem_addCenter]
theorem one_mem_center [MulOneClass M] : (1 : M) ∈ Set.center M where
comm _ := by rw [one_mul, mul_one]
left_assoc _ _ := by rw [one_mul, one_mul]
mid_assoc _ _ := by | Mathlib.GroupTheory.Subsemigroup.Center.160_0.vKbtzx3rREtft3E | @[to_additive (attr | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : MulOneClass M
x✝¹ x✝ : M
⊢ x✝¹ * x✝ * 1 = x✝¹ * (x✝ * 1) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [mul_one, mul_one] | @[to_additive (attr := simp) zero_mem_addCenter]
theorem one_mem_center [MulOneClass M] : (1 : M) ∈ Set.center M where
comm _ := by rw [one_mul, mul_one]
left_assoc _ _ := by rw [one_mul, one_mul]
mid_assoc _ _ := by rw [mul_one, one_mul]
right_assoc _ _ := by | Mathlib.GroupTheory.Subsemigroup.Center.160_0.vKbtzx3rREtft3E | @[to_additive (attr | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : MulZeroClass M
x✝ : M
⊢ 0 * x✝ = x✝ * 0 | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [zero_mul, mul_zero] | @[simp]
theorem zero_mem_center [MulZeroClass M] : (0 : M) ∈ Set.center M where
comm _ := by | Mathlib.GroupTheory.Subsemigroup.Center.169_0.vKbtzx3rREtft3E | @[simp]
theorem zero_mem_center [MulZeroClass M] : (0 : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : MulZeroClass M
x✝¹ x✝ : M
⊢ 0 * (x✝¹ * x✝) = 0 * x✝¹ * x✝ | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [zero_mul, zero_mul, zero_mul] | @[simp]
theorem zero_mem_center [MulZeroClass M] : (0 : M) ∈ Set.center M where
comm _ := by rw [zero_mul, mul_zero]
left_assoc _ _ := by | Mathlib.GroupTheory.Subsemigroup.Center.169_0.vKbtzx3rREtft3E | @[simp]
theorem zero_mem_center [MulZeroClass M] : (0 : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : MulZeroClass M
x✝¹ x✝ : M
⊢ x✝¹ * 0 * x✝ = x✝¹ * (0 * x✝) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [mul_zero, zero_mul, mul_zero] | @[simp]
theorem zero_mem_center [MulZeroClass M] : (0 : M) ∈ Set.center M where
comm _ := by rw [zero_mul, mul_zero]
left_assoc _ _ := by rw [zero_mul, zero_mul, zero_mul]
mid_assoc _ _ := by | Mathlib.GroupTheory.Subsemigroup.Center.169_0.vKbtzx3rREtft3E | @[simp]
theorem zero_mem_center [MulZeroClass M] : (0 : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : MulZeroClass M
x✝¹ x✝ : M
⊢ x✝¹ * x✝ * 0 = x✝¹ * (x✝ * 0) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [mul_zero, mul_zero, mul_zero] | @[simp]
theorem zero_mem_center [MulZeroClass M] : (0 : M) ∈ Set.center M where
comm _ := by rw [zero_mul, mul_zero]
left_assoc _ _ := by rw [zero_mul, zero_mul, zero_mul]
mid_assoc _ _ := by rw [mul_zero, zero_mul, mul_zero]
right_assoc _ _ := by | Mathlib.GroupTheory.Subsemigroup.Center.169_0.vKbtzx3rREtft3E | @[simp]
theorem zero_mem_center [MulZeroClass M] : (0 : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : NonAssocSemiring M
n : ℕ
x✝ : M
⊢ ↑n * x✝ = x✝ * ↑n | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [Nat.commute_cast] | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by | Mathlib.GroupTheory.Subsemigroup.Center.177_0.vKbtzx3rREtft3E | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : NonAssocSemiring M
n : ℕ
x✝¹ x✝ : M
⊢ ↑n * (x✝¹ * x✝) = ↑n * x✝¹ * x✝ | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | induction n with
| zero => rw [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul, add_mul, one_mul] | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by rw [Nat.commute_cast]
left_assoc _ _ := by
| Mathlib.GroupTheory.Subsemigroup.Center.177_0.vKbtzx3rREtft3E | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : NonAssocSemiring M
n : ℕ
x✝¹ x✝ : M
⊢ ↑n * (x✝¹ * x✝) = ↑n * x✝¹ * x✝ | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | induction n with
| zero => rw [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul, add_mul, one_mul] | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by rw [Nat.commute_cast]
left_assoc _ _ := by
| Mathlib.GroupTheory.Subsemigroup.Center.177_0.vKbtzx3rREtft3E | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
case zero
M : Type u_1
inst✝ : NonAssocSemiring M
x✝¹ x✝ : M
⊢ ↑Nat.zero * (x✝¹ * x✝) = ↑Nat.zero * x✝¹ * x✝ | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | | zero => rw [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_mul, zero_mul] | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| Mathlib.GroupTheory.Subsemigroup.Center.177_0.vKbtzx3rREtft3E | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
case zero
M : Type u_1
inst✝ : NonAssocSemiring M
x✝¹ x✝ : M
⊢ ↑Nat.zero * (x✝¹ * x✝) = ↑Nat.zero * x✝¹ * x✝ | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_mul, zero_mul] | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| zero => | Mathlib.GroupTheory.Subsemigroup.Center.177_0.vKbtzx3rREtft3E | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
case succ
M : Type u_1
inst✝ : NonAssocSemiring M
x✝¹ x✝ : M
n : ℕ
ihn : ↑n * (x✝¹ * x✝) = ↑n * x✝¹ * x✝
⊢ ↑(Nat.succ n) * (x✝¹ * x✝) = ↑(Nat.succ n) * x✝¹ * x✝ | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | | succ n ihn => rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul, add_mul, one_mul] | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| zero => rw [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| Mathlib.GroupTheory.Subsemigroup.Center.177_0.vKbtzx3rREtft3E | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
case succ
M : Type u_1
inst✝ : NonAssocSemiring M
x✝¹ x✝ : M
n : ℕ
ihn : ↑n * (x✝¹ * x✝) = ↑n * x✝¹ * x✝
⊢ ↑(Nat.succ n) * (x✝¹ * x✝) = ↑(Nat.succ n) * x✝¹ * x✝ | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul, add_mul, one_mul] | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| zero => rw [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| succ n ihn => | Mathlib.GroupTheory.Subsemigroup.Center.177_0.vKbtzx3rREtft3E | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : NonAssocSemiring M
n : ℕ
x✝¹ x✝ : M
⊢ x✝¹ * ↑n * x✝ = x✝¹ * (↑n * x✝) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | induction n with
| zero => rw [Nat.zero_eq, Nat.cast_zero, zero_mul, mul_zero, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, mul_add, add_mul, ihn, mul_add, one_mul, mul_one] | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| zero => rw [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul... | Mathlib.GroupTheory.Subsemigroup.Center.177_0.vKbtzx3rREtft3E | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : NonAssocSemiring M
n : ℕ
x✝¹ x✝ : M
⊢ x✝¹ * ↑n * x✝ = x✝¹ * (↑n * x✝) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | induction n with
| zero => rw [Nat.zero_eq, Nat.cast_zero, zero_mul, mul_zero, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, mul_add, add_mul, ihn, mul_add, one_mul, mul_one] | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| zero => rw [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul... | Mathlib.GroupTheory.Subsemigroup.Center.177_0.vKbtzx3rREtft3E | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
case zero
M : Type u_1
inst✝ : NonAssocSemiring M
x✝¹ x✝ : M
⊢ x✝¹ * ↑Nat.zero * x✝ = x✝¹ * (↑Nat.zero * x✝) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | | zero => rw [Nat.zero_eq, Nat.cast_zero, zero_mul, mul_zero, zero_mul] | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| zero => rw [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul... | Mathlib.GroupTheory.Subsemigroup.Center.177_0.vKbtzx3rREtft3E | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
case zero
M : Type u_1
inst✝ : NonAssocSemiring M
x✝¹ x✝ : M
⊢ x✝¹ * ↑Nat.zero * x✝ = x✝¹ * (↑Nat.zero * x✝) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [Nat.zero_eq, Nat.cast_zero, zero_mul, mul_zero, zero_mul] | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| zero => rw [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul... | Mathlib.GroupTheory.Subsemigroup.Center.177_0.vKbtzx3rREtft3E | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
case succ
M : Type u_1
inst✝ : NonAssocSemiring M
x✝¹ x✝ : M
n : ℕ
ihn : x✝¹ * ↑n * x✝ = x✝¹ * (↑n * x✝)
⊢ x✝¹ * ↑(Nat.succ n) * x✝ = x✝¹ * (↑(Nat.succ n) * x✝) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | | succ n ihn => rw [Nat.cast_succ, add_mul, mul_add, add_mul, ihn, mul_add, one_mul, mul_one] | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| zero => rw [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul... | Mathlib.GroupTheory.Subsemigroup.Center.177_0.vKbtzx3rREtft3E | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
case succ
M : Type u_1
inst✝ : NonAssocSemiring M
x✝¹ x✝ : M
n : ℕ
ihn : x✝¹ * ↑n * x✝ = x✝¹ * (↑n * x✝)
⊢ x✝¹ * ↑(Nat.succ n) * x✝ = x✝¹ * (↑(Nat.succ n) * x✝) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [Nat.cast_succ, add_mul, mul_add, add_mul, ihn, mul_add, one_mul, mul_one] | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| zero => rw [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul... | Mathlib.GroupTheory.Subsemigroup.Center.177_0.vKbtzx3rREtft3E | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : NonAssocSemiring M
n : ℕ
x✝¹ x✝ : M
⊢ x✝¹ * x✝ * ↑n = x✝¹ * (x✝ * ↑n) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | induction n with
| zero => rw [Nat.zero_eq, Nat.cast_zero, mul_zero, mul_zero, mul_zero]
| succ n ihn => rw [Nat.cast_succ, mul_add, ihn, mul_add, mul_add, mul_one, mul_one] | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| zero => rw [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul... | Mathlib.GroupTheory.Subsemigroup.Center.177_0.vKbtzx3rREtft3E | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : NonAssocSemiring M
n : ℕ
x✝¹ x✝ : M
⊢ x✝¹ * x✝ * ↑n = x✝¹ * (x✝ * ↑n) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | induction n with
| zero => rw [Nat.zero_eq, Nat.cast_zero, mul_zero, mul_zero, mul_zero]
| succ n ihn => rw [Nat.cast_succ, mul_add, ihn, mul_add, mul_add, mul_one, mul_one] | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| zero => rw [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul... | Mathlib.GroupTheory.Subsemigroup.Center.177_0.vKbtzx3rREtft3E | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
case zero
M : Type u_1
inst✝ : NonAssocSemiring M
x✝¹ x✝ : M
⊢ x✝¹ * x✝ * ↑Nat.zero = x✝¹ * (x✝ * ↑Nat.zero) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | | zero => rw [Nat.zero_eq, Nat.cast_zero, mul_zero, mul_zero, mul_zero] | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| zero => rw [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul... | Mathlib.GroupTheory.Subsemigroup.Center.177_0.vKbtzx3rREtft3E | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
case zero
M : Type u_1
inst✝ : NonAssocSemiring M
x✝¹ x✝ : M
⊢ x✝¹ * x✝ * ↑Nat.zero = x✝¹ * (x✝ * ↑Nat.zero) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [Nat.zero_eq, Nat.cast_zero, mul_zero, mul_zero, mul_zero] | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| zero => rw [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul... | Mathlib.GroupTheory.Subsemigroup.Center.177_0.vKbtzx3rREtft3E | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
case succ
M : Type u_1
inst✝ : NonAssocSemiring M
x✝¹ x✝ : M
n : ℕ
ihn : x✝¹ * x✝ * ↑n = x✝¹ * (x✝ * ↑n)
⊢ x✝¹ * x✝ * ↑(Nat.succ n) = x✝¹ * (x✝ * ↑(Nat.succ n)) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | | succ n ihn => rw [Nat.cast_succ, mul_add, ihn, mul_add, mul_add, mul_one, mul_one] | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| zero => rw [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul... | Mathlib.GroupTheory.Subsemigroup.Center.177_0.vKbtzx3rREtft3E | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
case succ
M : Type u_1
inst✝ : NonAssocSemiring M
x✝¹ x✝ : M
n : ℕ
ihn : x✝¹ * x✝ * ↑n = x✝¹ * (x✝ * ↑n)
⊢ x✝¹ * x✝ * ↑(Nat.succ n) = x✝¹ * (x✝ * ↑(Nat.succ n)) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [Nat.cast_succ, mul_add, ihn, mul_add, mul_add, mul_one, mul_one] | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| zero => rw [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul... | Mathlib.GroupTheory.Subsemigroup.Center.177_0.vKbtzx3rREtft3E | @[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : NonAssocRing M
n : ℤ
x✝ : M
⊢ ↑n * x✝ = x✝ * ↑n | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [Int.commute_cast] | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ := by | Mathlib.GroupTheory.Subsemigroup.Center.199_0.vKbtzx3rREtft3E | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : NonAssocRing M
n✝ : ℤ
x✝¹ x✝ : M
n : ℕ
⊢ ↑↑n * (x✝¹ * x✝) = ↑↑n * x✝¹ * x✝ | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [Int.cast_ofNat, (natCast_mem_center _ n).left_assoc _ _] | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ := by rw [Int.commute_cast]
left_assoc _ _ := match n with
| (n : ℕ) => by | Mathlib.GroupTheory.Subsemigroup.Center.199_0.vKbtzx3rREtft3E | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
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✝ | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [Int.cast_negSucc, Nat.cast_add, Nat.cast_one, neg_add_rev, add_mul, add_mul, add_mul,
neg_mul, one_mul, neg_mul 1, one_mul, ← neg_mul, add_right_inj, neg_mul,
(natCast_mem_center _ n).left_assoc _ _, neg_mul, neg_mul] | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ := by rw [Int.commute_cast]
left_assoc _ _ := match n with
| (n : ℕ) => by rw [Int.cast_ofNat, (natCast_mem_center _ n).left_assoc _ _]
| Int.negSucc n => by
| Mathlib.GroupTheory.Subsemigroup.Center.199_0.vKbtzx3rREtft3E | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : NonAssocRing M
n✝ : ℤ
x✝¹ x✝ : M
n : ℕ
⊢ x✝¹ * ↑↑n * x✝ = x✝¹ * (↑↑n * x✝) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [Int.cast_ofNat, (natCast_mem_center _ n).mid_assoc _ _] | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ := by rw [Int.commute_cast]
left_assoc _ _ := match n with
| (n : ℕ) => by rw [Int.cast_ofNat, (natCast_mem_center _ n).left_assoc _ _]
| Int.negSucc n => by
rw [Int.cast_negSucc, Nat.cast_add, Nat.cast_o... | Mathlib.GroupTheory.Subsemigroup.Center.199_0.vKbtzx3rREtft3E | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : NonAssocRing M
n✝ : ℤ
x✝¹ x✝ : M
n : ℕ
⊢ x✝¹ * ↑(Int.negSucc n) * x✝ = x✝¹ * (↑(Int.negSucc n) * x✝) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | simp only [Int.cast_negSucc, Nat.cast_add, Nat.cast_one, neg_add_rev] | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ := by rw [Int.commute_cast]
left_assoc _ _ := match n with
| (n : ℕ) => by rw [Int.cast_ofNat, (natCast_mem_center _ n).left_assoc _ _]
| Int.negSucc n => by
rw [Int.cast_negSucc, Nat.cast_add, Nat.cast_o... | Mathlib.GroupTheory.Subsemigroup.Center.199_0.vKbtzx3rREtft3E | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : NonAssocRing M
n✝ : ℤ
x✝¹ x✝ : M
n : ℕ
⊢ x✝¹ * (-1 + -↑n) * x✝ = x✝¹ * ((-1 + -↑n) * x✝) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [add_mul, mul_add, add_mul, mul_add, neg_mul, one_mul] | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ := by rw [Int.commute_cast]
left_assoc _ _ := match n with
| (n : ℕ) => by rw [Int.cast_ofNat, (natCast_mem_center _ n).left_assoc _ _]
| Int.negSucc n => by
rw [Int.cast_negSucc, Nat.cast_add, Nat.cast_o... | Mathlib.GroupTheory.Subsemigroup.Center.199_0.vKbtzx3rREtft3E | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : NonAssocRing M
n✝ : ℤ
x✝¹ x✝ : M
n : ℕ
⊢ x✝¹ * -1 * x✝ + x✝¹ * -↑n * x✝ = x✝¹ * -x✝ + x✝¹ * (-↑n * x✝) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [neg_mul, mul_neg, mul_one, mul_neg, neg_mul, neg_mul] | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ := by rw [Int.commute_cast]
left_assoc _ _ := match n with
| (n : ℕ) => by rw [Int.cast_ofNat, (natCast_mem_center _ n).left_assoc _ _]
| Int.negSucc n => by
rw [Int.cast_negSucc, Nat.cast_add, Nat.cast_o... | Mathlib.GroupTheory.Subsemigroup.Center.199_0.vKbtzx3rREtft3E | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : NonAssocRing M
n✝ : ℤ
x✝¹ x✝ : M
n : ℕ
⊢ -(x✝¹ * x✝) + -(x✝¹ * ↑n * x✝) = x✝¹ * -x✝ + x✝¹ * -(↑n * x✝) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [(natCast_mem_center _ n).mid_assoc _ _] | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ := by rw [Int.commute_cast]
left_assoc _ _ := match n with
| (n : ℕ) => by rw [Int.cast_ofNat, (natCast_mem_center _ n).left_assoc _ _]
| Int.negSucc n => by
rw [Int.cast_negSucc, Nat.cast_add, Nat.cast_o... | Mathlib.GroupTheory.Subsemigroup.Center.199_0.vKbtzx3rREtft3E | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : NonAssocRing M
n✝ : ℤ
x✝¹ x✝ : M
n : ℕ
⊢ -(x✝¹ * x✝) + -(x✝¹ * (↑n * x✝)) = x✝¹ * -x✝ + x✝¹ * -(↑n * x✝) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | simp only [mul_neg] | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ := by rw [Int.commute_cast]
left_assoc _ _ := match n with
| (n : ℕ) => by rw [Int.cast_ofNat, (natCast_mem_center _ n).left_assoc _ _]
| Int.negSucc n => by
rw [Int.cast_negSucc, Nat.cast_add, Nat.cast_o... | Mathlib.GroupTheory.Subsemigroup.Center.199_0.vKbtzx3rREtft3E | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : NonAssocRing M
n✝ : ℤ
x✝¹ x✝ : M
n : ℕ
⊢ x✝¹ * x✝ * ↑↑n = x✝¹ * (x✝ * ↑↑n) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [Int.cast_ofNat, (natCast_mem_center _ n).right_assoc _ _] | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ := by rw [Int.commute_cast]
left_assoc _ _ := match n with
| (n : ℕ) => by rw [Int.cast_ofNat, (natCast_mem_center _ n).left_assoc _ _]
| Int.negSucc n => by
rw [Int.cast_negSucc, Nat.cast_add, Nat.cast_o... | Mathlib.GroupTheory.Subsemigroup.Center.199_0.vKbtzx3rREtft3E | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
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)) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | simp only [Int.cast_negSucc, Nat.cast_add, Nat.cast_one, neg_add_rev] | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ := by rw [Int.commute_cast]
left_assoc _ _ := match n with
| (n : ℕ) => by rw [Int.cast_ofNat, (natCast_mem_center _ n).left_assoc _ _]
| Int.negSucc n => by
rw [Int.cast_negSucc, Nat.cast_add, Nat.cast_o... | Mathlib.GroupTheory.Subsemigroup.Center.199_0.vKbtzx3rREtft3E | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : NonAssocRing M
n✝ : ℤ
x✝¹ x✝ : M
n : ℕ
⊢ x✝¹ * x✝ * (-1 + -↑n) = x✝¹ * (x✝ * (-1 + -↑n)) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [mul_add, mul_add, mul_add, mul_neg, mul_one, mul_neg, mul_neg, mul_one, mul_neg,
add_right_inj, (natCast_mem_center _ n).right_assoc _ _, mul_neg, mul_neg] | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ := by rw [Int.commute_cast]
left_assoc _ _ := match n with
| (n : ℕ) => by rw [Int.cast_ofNat, (natCast_mem_center _ n).left_assoc _ _]
| Int.negSucc n => by
rw [Int.cast_negSucc, Nat.cast_add, Nat.cast_o... | Mathlib.GroupTheory.Subsemigroup.Center.199_0.vKbtzx3rREtft3E | @[simp]
theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : Group M
a : M
ha : a ∈ center M
⊢ a⁻¹ ∈ center M | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [_root_.Semigroup.mem_center_iff] | @[to_additive (attr := simp) neg_mem_addCenter]
theorem inv_mem_center [Group M] {a : M} (ha : a ∈ Set.center M) : a⁻¹ ∈ Set.center M := by
| Mathlib.GroupTheory.Subsemigroup.Center.225_0.vKbtzx3rREtft3E | @[to_additive (attr | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : Group M
a : M
ha : a ∈ center M
⊢ ∀ (g : M), g * a⁻¹ = a⁻¹ * g | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | intro _ | @[to_additive (attr := simp) neg_mem_addCenter]
theorem inv_mem_center [Group M] {a : M} (ha : a ∈ Set.center M) : a⁻¹ ∈ Set.center M := by
rw [_root_.Semigroup.mem_center_iff]
| Mathlib.GroupTheory.Subsemigroup.Center.225_0.vKbtzx3rREtft3E | @[to_additive (attr | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : Group M
a : M
ha : a ∈ center M
g✝ : M
⊢ g✝ * a⁻¹ = a⁻¹ * g✝ | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [← inv_inj, mul_inv_rev, inv_inv, ha.comm, mul_inv_rev, inv_inv] | @[to_additive (attr := simp) neg_mem_addCenter]
theorem inv_mem_center [Group M] {a : M} (ha : a ∈ Set.center M) : a⁻¹ ∈ Set.center M := by
rw [_root_.Semigroup.mem_center_iff]
intro _
| Mathlib.GroupTheory.Subsemigroup.Center.225_0.vKbtzx3rREtft3E | @[to_additive (attr | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : Distrib M
a b : M
ha : a ∈ center M
hb : b ∈ center M
x✝ : M
⊢ (a + b) * x✝ = x✝ * (a + b) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [add_mul, mul_add, ha.comm, hb.comm] | @[simp]
theorem add_mem_center [Distrib M] {a b : M} (ha : a ∈ Set.center M) (hb : b ∈ Set.center M) :
a + b ∈ Set.center M where
comm _ := by | Mathlib.GroupTheory.Subsemigroup.Center.233_0.vKbtzx3rREtft3E | @[simp]
theorem add_mem_center [Distrib M] {a b : M} (ha : a ∈ Set.center M) (hb : b ∈ Set.center M) :
a + b ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
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✝ | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [add_mul, ha.left_assoc, hb.left_assoc, ← add_mul, ← add_mul] | @[simp]
theorem add_mem_center [Distrib M] {a b : M} (ha : a ∈ Set.center M) (hb : b ∈ Set.center M) :
a + b ∈ Set.center M where
comm _ := by rw [add_mul, mul_add, ha.comm, hb.comm]
left_assoc _ _ := by | Mathlib.GroupTheory.Subsemigroup.Center.233_0.vKbtzx3rREtft3E | @[simp]
theorem add_mem_center [Distrib M] {a b : M} (ha : a ∈ Set.center M) (hb : b ∈ Set.center M) :
a + b ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : Distrib M
a b : M
ha : a ∈ center M
hb : b ∈ center M
x✝¹ x✝ : M
⊢ x✝¹ * (a + b) * x✝ = x✝¹ * ((a + b) * x✝) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [mul_add, add_mul, ha.mid_assoc, hb.mid_assoc, ← mul_add, ← add_mul] | @[simp]
theorem add_mem_center [Distrib M] {a b : M} (ha : a ∈ Set.center M) (hb : b ∈ Set.center M) :
a + b ∈ Set.center M where
comm _ := by rw [add_mul, mul_add, ha.comm, hb.comm]
left_assoc _ _ := by rw [add_mul, ha.left_assoc, hb.left_assoc, ← add_mul, ← add_mul]
mid_assoc _ _ := by | Mathlib.GroupTheory.Subsemigroup.Center.233_0.vKbtzx3rREtft3E | @[simp]
theorem add_mem_center [Distrib M] {a b : M} (ha : a ∈ Set.center M) (hb : b ∈ Set.center M) :
a + b ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
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)) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [mul_add, ha.right_assoc, hb.right_assoc, ← mul_add, ← mul_add] | @[simp]
theorem add_mem_center [Distrib M] {a b : M} (ha : a ∈ Set.center M) (hb : b ∈ Set.center M) :
a + b ∈ Set.center M where
comm _ := by rw [add_mul, mul_add, ha.comm, hb.comm]
left_assoc _ _ := by rw [add_mul, ha.left_assoc, hb.left_assoc, ← add_mul, ← add_mul]
mid_assoc _ _ := by rw [mul_add, add_mul... | Mathlib.GroupTheory.Subsemigroup.Center.233_0.vKbtzx3rREtft3E | @[simp]
theorem add_mem_center [Distrib M] {a b : M} (ha : a ∈ Set.center M) (hb : b ∈ Set.center M) :
a + b ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : NonUnitalNonAssocRing M
a : M
ha : a ∈ center M
x✝ : M
⊢ -a * x✝ = x✝ * -a | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [← neg_mul_comm, ← ha.comm, neg_mul_comm] | @[simp]
theorem neg_mem_center [NonUnitalNonAssocRing M] {a : M} (ha : a ∈ Set.center M) :
-a ∈ Set.center M where
comm _ := by | Mathlib.GroupTheory.Subsemigroup.Center.242_0.vKbtzx3rREtft3E | @[simp]
theorem neg_mem_center [NonUnitalNonAssocRing M] {a : M} (ha : a ∈ Set.center M) :
-a ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
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✝ | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [neg_mul, ha.left_assoc, neg_mul, neg_mul] | @[simp]
theorem neg_mem_center [NonUnitalNonAssocRing M] {a : M} (ha : a ∈ Set.center M) :
-a ∈ Set.center M where
comm _ := by rw [← neg_mul_comm, ← ha.comm, neg_mul_comm]
left_assoc _ _ := by | Mathlib.GroupTheory.Subsemigroup.Center.242_0.vKbtzx3rREtft3E | @[simp]
theorem neg_mem_center [NonUnitalNonAssocRing M] {a : M} (ha : a ∈ Set.center M) :
-a ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : NonUnitalNonAssocRing M
a : M
ha : a ∈ center M
x✝¹ x✝ : M
⊢ x✝¹ * -a * x✝ = x✝¹ * (-a * x✝) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [← neg_mul_comm, ha.mid_assoc, neg_mul_comm, neg_mul] | @[simp]
theorem neg_mem_center [NonUnitalNonAssocRing M] {a : M} (ha : a ∈ Set.center M) :
-a ∈ Set.center M where
comm _ := by rw [← neg_mul_comm, ← ha.comm, neg_mul_comm]
left_assoc _ _ := by rw [neg_mul, ha.left_assoc, neg_mul, neg_mul]
mid_assoc _ _ := by | Mathlib.GroupTheory.Subsemigroup.Center.242_0.vKbtzx3rREtft3E | @[simp]
theorem neg_mem_center [NonUnitalNonAssocRing M] {a : M} (ha : a ∈ Set.center M) :
-a ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
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) | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [mul_neg, ha.right_assoc, mul_neg, mul_neg] | @[simp]
theorem neg_mem_center [NonUnitalNonAssocRing M] {a : M} (ha : a ∈ Set.center M) :
-a ∈ Set.center M where
comm _ := by rw [← neg_mul_comm, ← ha.comm, neg_mul_comm]
left_assoc _ _ := by rw [neg_mul, ha.left_assoc, neg_mul, neg_mul]
mid_assoc _ _ := by rw [← neg_mul_comm, ha.mid_assoc, neg_mul_comm, ne... | Mathlib.GroupTheory.Subsemigroup.Center.242_0.vKbtzx3rREtft3E | @[simp]
theorem neg_mem_center [NonUnitalNonAssocRing M] {a : M} (ha : a ∈ Set.center M) :
-a ∈ Set.center M where
comm _ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : Monoid M
x✝ : Mˣ
ha : x✝ ∈ Units.val ⁻¹' center M
⊢ x✝ ∈ center Mˣ | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [_root_.Semigroup.mem_center_iff] | @[to_additive subset_addCenter_add_units]
theorem subset_center_units [Monoid M] : ((↑) : Mˣ → M) ⁻¹' center M ⊆ Set.center Mˣ :=
fun _ ha => by
| Mathlib.GroupTheory.Subsemigroup.Center.251_0.vKbtzx3rREtft3E | @[to_additive subset_addCenter_add_units]
theorem subset_center_units [Monoid M] : ((↑) : Mˣ → M) ⁻¹' center M ⊆ Set.center Mˣ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : Monoid M
x✝ : Mˣ
ha : x✝ ∈ Units.val ⁻¹' center M
⊢ ∀ (g : Mˣ), g * x✝ = x✝ * g | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | intro _ | @[to_additive subset_addCenter_add_units]
theorem subset_center_units [Monoid M] : ((↑) : Mˣ → M) ⁻¹' center M ⊆ Set.center Mˣ :=
fun _ ha => by
rw [_root_.Semigroup.mem_center_iff]
| Mathlib.GroupTheory.Subsemigroup.Center.251_0.vKbtzx3rREtft3E | @[to_additive subset_addCenter_add_units]
theorem subset_center_units [Monoid M] : ((↑) : Mˣ → M) ⁻¹' center M ⊆ Set.center Mˣ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : Monoid M
x✝ : Mˣ
ha : x✝ ∈ Units.val ⁻¹' center M
g✝ : Mˣ
⊢ g✝ * x✝ = x✝ * g✝ | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [← Units.eq_iff, Units.val_mul, Units.val_mul, ha.comm] | @[to_additive subset_addCenter_add_units]
theorem subset_center_units [Monoid M] : ((↑) : Mˣ → M) ⁻¹' center M ⊆ Set.center Mˣ :=
fun _ ha => by
rw [_root_.Semigroup.mem_center_iff]
intro _
| Mathlib.GroupTheory.Subsemigroup.Center.251_0.vKbtzx3rREtft3E | @[to_additive subset_addCenter_add_units]
theorem subset_center_units [Monoid M] : ((↑) : Mˣ → M) ⁻¹' center M ⊆ Set.center Mˣ | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : GroupWithZero M
x✝ : Mˣ
ha : x✝ ∈ center Mˣ
⊢ x✝ ∈ Units.val ⁻¹' center M | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [mem_preimage, _root_.Semigroup.mem_center_iff] | theorem center_units_subset [GroupWithZero M] : Set.center Mˣ ⊆ ((↑) : Mˣ → M) ⁻¹' center M :=
fun _ ha => by
| Mathlib.GroupTheory.Subsemigroup.Center.260_0.vKbtzx3rREtft3E | theorem center_units_subset [GroupWithZero M] : Set.center Mˣ ⊆ ((↑) : Mˣ → M) ⁻¹' center M | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : GroupWithZero M
x✝ : Mˣ
ha : x✝ ∈ center Mˣ
⊢ ∀ (g : M), g * ↑x✝ = ↑x✝ * g | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | intro b | theorem center_units_subset [GroupWithZero M] : Set.center Mˣ ⊆ ((↑) : Mˣ → M) ⁻¹' center M :=
fun _ ha => by
rw [mem_preimage, _root_.Semigroup.mem_center_iff]
| Mathlib.GroupTheory.Subsemigroup.Center.260_0.vKbtzx3rREtft3E | theorem center_units_subset [GroupWithZero M] : Set.center Mˣ ⊆ ((↑) : Mˣ → M) ⁻¹' center M | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : GroupWithZero M
x✝ : Mˣ
ha : x✝ ∈ center Mˣ
b : M
⊢ b * ↑x✝ = ↑x✝ * b | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | obtain rfl | hb := eq_or_ne b 0 | theorem center_units_subset [GroupWithZero M] : Set.center Mˣ ⊆ ((↑) : Mˣ → M) ⁻¹' center M :=
fun _ ha => by
rw [mem_preimage, _root_.Semigroup.mem_center_iff]
intro b
| Mathlib.GroupTheory.Subsemigroup.Center.260_0.vKbtzx3rREtft3E | theorem center_units_subset [GroupWithZero M] : Set.center Mˣ ⊆ ((↑) : Mˣ → M) ⁻¹' center M | Mathlib_GroupTheory_Subsemigroup_Center |
case inl
M : Type u_1
inst✝ : GroupWithZero M
x✝ : Mˣ
ha : x✝ ∈ center Mˣ
⊢ 0 * ↑x✝ = ↑x✝ * 0 | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [zero_mul, mul_zero] | theorem center_units_subset [GroupWithZero M] : Set.center Mˣ ⊆ ((↑) : Mˣ → M) ⁻¹' center M :=
fun _ ha => by
rw [mem_preimage, _root_.Semigroup.mem_center_iff]
intro b
obtain rfl | hb := eq_or_ne b 0
· | Mathlib.GroupTheory.Subsemigroup.Center.260_0.vKbtzx3rREtft3E | theorem center_units_subset [GroupWithZero M] : Set.center Mˣ ⊆ ((↑) : Mˣ → M) ⁻¹' center M | Mathlib_GroupTheory_Subsemigroup_Center |
case inr
M : Type u_1
inst✝ : GroupWithZero M
x✝ : Mˣ
ha : x✝ ∈ center Mˣ
b : M
hb : b ≠ 0
⊢ b * ↑x✝ = ↑x✝ * b | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | exact Units.ext_iff.mp (ha.comm (Units.mk0 b hb)).symm | theorem center_units_subset [GroupWithZero M] : Set.center Mˣ ⊆ ((↑) : Mˣ → M) ⁻¹' center M :=
fun _ ha => by
rw [mem_preimage, _root_.Semigroup.mem_center_iff]
intro b
obtain rfl | hb := eq_or_ne b 0
· rw [zero_mul, mul_zero]
· | Mathlib.GroupTheory.Subsemigroup.Center.260_0.vKbtzx3rREtft3E | theorem center_units_subset [GroupWithZero M] : Set.center Mˣ ⊆ ((↑) : Mˣ → M) ⁻¹' center M | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : Monoid M
a : Mˣ
ha : ↑a ∈ center M
⊢ ↑a⁻¹ ∈ center M | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [Semigroup.mem_center_iff] at * | @[simp]
theorem units_inv_mem_center [Monoid M] {a : Mˣ} (ha : ↑a ∈ Set.center M) :
↑a⁻¹ ∈ Set.center M := by
| Mathlib.GroupTheory.Subsemigroup.Center.274_0.vKbtzx3rREtft3E | @[simp]
theorem units_inv_mem_center [Monoid M] {a : Mˣ} (ha : ↑a ∈ Set.center M) :
↑a⁻¹ ∈ Set.center M | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : Monoid M
a : Mˣ
ha : ∀ (g : M), g * ↑a = ↑a * g
⊢ ∀ (g : M), g * ↑a⁻¹ = ↑a⁻¹ * g | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | exact (Commute.units_inv_right <| ha ·) | @[simp]
theorem units_inv_mem_center [Monoid M] {a : Mˣ} (ha : ↑a ∈ Set.center M) :
↑a⁻¹ ∈ Set.center M := by
rw [Semigroup.mem_center_iff] at *
| Mathlib.GroupTheory.Subsemigroup.Center.274_0.vKbtzx3rREtft3E | @[simp]
theorem units_inv_mem_center [Monoid M] {a : Mˣ} (ha : ↑a ∈ Set.center M) :
↑a⁻¹ ∈ Set.center M | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝¹ : Monoid M
a : M
inst✝ : Invertible a
ha : a ∈ center M
⊢ ⅟a ∈ center M | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [Semigroup.mem_center_iff] at * | @[simp]
theorem invOf_mem_center [Monoid M] {a : M} [Invertible a] (ha : a ∈ Set.center M) :
⅟a ∈ Set.center M := by
| Mathlib.GroupTheory.Subsemigroup.Center.280_0.vKbtzx3rREtft3E | @[simp]
theorem invOf_mem_center [Monoid M] {a : M} [Invertible a] (ha : a ∈ Set.center M) :
⅟a ∈ Set.center M | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝¹ : Monoid M
a : M
inst✝ : Invertible a
ha : ∀ (g : M), g * a = a * g
⊢ ∀ (g : M), g * ⅟a = ⅟a * g | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | exact (Commute.invOf_right <| ha ·) | @[simp]
theorem invOf_mem_center [Monoid M] {a : M} [Invertible a] (ha : a ∈ Set.center M) :
⅟a ∈ Set.center M := by
rw [Semigroup.mem_center_iff] at *
| Mathlib.GroupTheory.Subsemigroup.Center.280_0.vKbtzx3rREtft3E | @[simp]
theorem invOf_mem_center [Monoid M] {a : M} [Invertible a] (ha : a ∈ Set.center M) :
⅟a ∈ Set.center M | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : GroupWithZero M
a : M
ha : a ∈ center M
⊢ a⁻¹ ∈ center M | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | obtain rfl | ha0 := eq_or_ne a 0 | @[simp]
theorem inv_mem_center₀ [GroupWithZero M] {a : M} (ha : a ∈ Set.center M) : a⁻¹ ∈ Set.center M := by
| Mathlib.GroupTheory.Subsemigroup.Center.286_0.vKbtzx3rREtft3E | @[simp]
theorem inv_mem_center₀ [GroupWithZero M] {a : M} (ha : a ∈ Set.center M) : a⁻¹ ∈ Set.center M | Mathlib_GroupTheory_Subsemigroup_Center |
case inl
M : Type u_1
inst✝ : GroupWithZero M
ha : 0 ∈ center M
⊢ 0⁻¹ ∈ center M | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [inv_zero] | @[simp]
theorem inv_mem_center₀ [GroupWithZero M] {a : M} (ha : a ∈ Set.center M) : a⁻¹ ∈ Set.center M := by
obtain rfl | ha0 := eq_or_ne a 0
· | Mathlib.GroupTheory.Subsemigroup.Center.286_0.vKbtzx3rREtft3E | @[simp]
theorem inv_mem_center₀ [GroupWithZero M] {a : M} (ha : a ∈ Set.center M) : a⁻¹ ∈ Set.center M | Mathlib_GroupTheory_Subsemigroup_Center |
case inl
M : Type u_1
inst✝ : GroupWithZero M
ha : 0 ∈ center M
⊢ 0 ∈ center M | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | exact zero_mem_center M | @[simp]
theorem inv_mem_center₀ [GroupWithZero M] {a : M} (ha : a ∈ Set.center M) : a⁻¹ ∈ Set.center M := by
obtain rfl | ha0 := eq_or_ne a 0
· rw [inv_zero]
| Mathlib.GroupTheory.Subsemigroup.Center.286_0.vKbtzx3rREtft3E | @[simp]
theorem inv_mem_center₀ [GroupWithZero M] {a : M} (ha : a ∈ Set.center M) : a⁻¹ ∈ Set.center M | Mathlib_GroupTheory_Subsemigroup_Center |
case inr
M : Type u_1
inst✝ : GroupWithZero M
a : M
ha : a ∈ center M
ha0 : a ≠ 0
⊢ a⁻¹ ∈ center M | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | lift a to Mˣ using IsUnit.mk0 _ ha0 | @[simp]
theorem inv_mem_center₀ [GroupWithZero M] {a : M} (ha : a ∈ Set.center M) : a⁻¹ ∈ Set.center M := by
obtain rfl | ha0 := eq_or_ne a 0
· rw [inv_zero]
exact zero_mem_center M
· | Mathlib.GroupTheory.Subsemigroup.Center.286_0.vKbtzx3rREtft3E | @[simp]
theorem inv_mem_center₀ [GroupWithZero M] {a : M} (ha : a ∈ Set.center M) : a⁻¹ ∈ Set.center M | Mathlib_GroupTheory_Subsemigroup_Center |
case inr.intro
M : Type u_1
inst✝ : GroupWithZero M
a : Mˣ
ha : ↑a ∈ center M
ha0 : ↑a ≠ 0
⊢ (↑a)⁻¹ ∈ center M | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | simpa only [Units.val_inv_eq_inv_val] using units_inv_mem_center ha | @[simp]
theorem inv_mem_center₀ [GroupWithZero M] {a : M} (ha : a ∈ Set.center M) : a⁻¹ ∈ Set.center M := by
obtain rfl | ha0 := eq_or_ne a 0
· rw [inv_zero]
exact zero_mem_center M
· lift a to Mˣ using IsUnit.mk0 _ ha0
| Mathlib.GroupTheory.Subsemigroup.Center.286_0.vKbtzx3rREtft3E | @[simp]
theorem inv_mem_center₀ [GroupWithZero M] {a : M} (ha : a ∈ Set.center M) : a⁻¹ ∈ Set.center M | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : Group M
a b : M
ha : a ∈ center M
hb : b ∈ center M
⊢ a / b ∈ center M | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [div_eq_mul_inv] | @[to_additive (attr := simp) sub_mem_addCenter]
theorem div_mem_center [Group M] {a b : M} (ha : a ∈ Set.center M) (hb : b ∈ Set.center M) :
a / b ∈ Set.center M := by
| Mathlib.GroupTheory.Subsemigroup.Center.295_0.vKbtzx3rREtft3E | @[to_additive (attr | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : Group M
a b : M
ha : a ∈ center M
hb : b ∈ center M
⊢ a * b⁻¹ ∈ center M | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | exact mul_mem_center ha (inv_mem_center hb) | @[to_additive (attr := simp) sub_mem_addCenter]
theorem div_mem_center [Group M] {a b : M} (ha : a ∈ Set.center M) (hb : b ∈ Set.center M) :
a / b ∈ Set.center M := by
rw [div_eq_mul_inv]
| Mathlib.GroupTheory.Subsemigroup.Center.295_0.vKbtzx3rREtft3E | @[to_additive (attr | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : GroupWithZero M
a b : M
ha : a ∈ center M
hb : b ∈ center M
⊢ a / b ∈ center M | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [div_eq_mul_inv] | @[simp]
theorem div_mem_center₀ [GroupWithZero M] {a b : M} (ha : a ∈ Set.center M)
(hb : b ∈ Set.center M) : a / b ∈ Set.center M := by
| Mathlib.GroupTheory.Subsemigroup.Center.303_0.vKbtzx3rREtft3E | @[simp]
theorem div_mem_center₀ [GroupWithZero M] {a b : M} (ha : a ∈ Set.center M)
(hb : b ∈ Set.center M) : a / b ∈ Set.center M | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : GroupWithZero M
a b : M
ha : a ∈ center M
hb : b ∈ center M
⊢ a * b⁻¹ ∈ center M | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | exact mul_mem_center ha (inv_mem_center₀ hb) | @[simp]
theorem div_mem_center₀ [GroupWithZero M] {a b : M} (ha : a ∈ Set.center M)
(hb : b ∈ Set.center M) : a / b ∈ Set.center M := by
rw [div_eq_mul_inv]
| Mathlib.GroupTheory.Subsemigroup.Center.303_0.vKbtzx3rREtft3E | @[simp]
theorem div_mem_center₀ [GroupWithZero M] {a b : M} (ha : a ∈ Set.center M)
(hb : b ∈ Set.center M) : a / b ∈ Set.center M | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : Semigroup M
z : M
⊢ z ∈ center M ↔ ∀ (g : M), g * z = z * g | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | rw [← Semigroup.mem_center_iff] | @[to_additive]
theorem mem_center_iff {z : M} : z ∈ center M ↔ ∀ g, g * z = z * g := by
| Mathlib.GroupTheory.Subsemigroup.Center.347_0.vKbtzx3rREtft3E | @[to_additive]
theorem mem_center_iff {z : M} : z ∈ center M ↔ ∀ g, g * z = z * g | Mathlib_GroupTheory_Subsemigroup_Center |
M : Type u_1
inst✝ : Semigroup M
z : M
⊢ z ∈ center M ↔ z ∈ Set.center M | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Invertible.Basic
import Mathlib.GroupTheory.Subsemigroup.O... | exact Iff.rfl | @[to_additive]
theorem mem_center_iff {z : M} : z ∈ center M ↔ ∀ g, g * z = z * g := by
rw [← Semigroup.mem_center_iff]
| Mathlib.GroupTheory.Subsemigroup.Center.347_0.vKbtzx3rREtft3E | @[to_additive]
theorem mem_center_iff {z : M} : z ∈ center M ↔ ∀ g, g * z = z * g | Mathlib_GroupTheory_Subsemigroup_Center |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ SemiRingCatMax
j : J
⊢ Semiring ((F ⋙ forget SemiRingCat).obj j) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | change Semiring (F.obj j) | instance semiringObj (F : J ⥤ SemiRingCatMax.{v, u}) (j) :
Semiring ((F ⋙ forget SemiRingCat).obj j) := by
| Mathlib.Algebra.Category.Ring.Limits.42_0.VxjNIkMLPSqe2rX | instance semiringObj (F : J ⥤ SemiRingCatMax.{v, u}) (j) :
Semiring ((F ⋙ forget SemiRingCat).obj j) | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ SemiRingCatMax
j : J
⊢ Semiring ↑(F.obj j) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | infer_instance | instance semiringObj (F : J ⥤ SemiRingCatMax.{v, u}) (j) :
Semiring ((F ⋙ forget SemiRingCat).obj j) := by
change Semiring (F.obj j)
| Mathlib.Algebra.Category.Ring.Limits.42_0.VxjNIkMLPSqe2rX | instance semiringObj (F : J ⥤ SemiRingCatMax.{v, u}) (j) :
Semiring ((F ⋙ forget SemiRingCat).obj j) | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ SemiRingCatMax
⊢ IsLimit (limitCone F) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | refine IsLimit.ofFaithful (forget SemiRingCatMax.{v, u}) (Types.limitConeIsLimit.{v, u} _)
(fun s : Cone F => ofHom
{ toFun := _
map_one' := Subtype.ext <| funext fun j => by exact (s.π.app j).map_one
map_mul' := fun x y => Subtype.ext <| funext fun j => by exact (s.π.app j).map_mul x y
... | /-- Witness that the limit cone in `SemiRingCat` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ SemiRingCatMax.{v, u}) : IsLimit (limitCone F) := by
| Mathlib.Algebra.Category.Ring.Limits.97_0.VxjNIkMLPSqe2rX | /-- Witness that the limit cone in `SemiRingCat` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ SemiRingCatMax.{v, u}) : IsLimit (limitCone F) | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ SemiRingCatMax
s : Cone F
j : J
⊢ ↑{ val := fun j => ((forget SemiRingCatMax).mapCone s).π.app j 1,
property :=
(_ :
∀ {j j' : J} (f : j ⟶ j'),
(((forget SemiRingCatMax).mapCone s).π.app j ≫ (F ⋙ forget SemiRingCatMax).map f)... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | exact (s.π.app j).map_one | /-- Witness that the limit cone in `SemiRingCat` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ SemiRingCatMax.{v, u}) : IsLimit (limitCone F) := by
refine IsLimit.ofFaithful (forget SemiRingCatMax.{v, u}) (Types.limitConeIsLimit.{v, u} _)
(fun s : Cone F => ofHom
... | Mathlib.Algebra.Category.Ring.Limits.97_0.VxjNIkMLPSqe2rX | /-- Witness that the limit cone in `SemiRingCat` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ SemiRingCatMax.{v, u}) : IsLimit (limitCone F) | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ SemiRingCatMax
s : Cone F
x y : ↑s.1
j : J
⊢ ↑(OneHom.toFun
{
toFun := fun v =>
{ val := fun j => ((forget SemiRingCatMax).mapCone s).π.app j v,
property :=
(_ :
∀ {j j' : J} (f : j ⟶ j')... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | exact (s.π.app j).map_mul x y | /-- Witness that the limit cone in `SemiRingCat` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ SemiRingCatMax.{v, u}) : IsLimit (limitCone F) := by
refine IsLimit.ofFaithful (forget SemiRingCatMax.{v, u}) (Types.limitConeIsLimit.{v, u} _)
(fun s : Cone F => ofHom
... | Mathlib.Algebra.Category.Ring.Limits.97_0.VxjNIkMLPSqe2rX | /-- Witness that the limit cone in `SemiRingCat` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ SemiRingCatMax.{v, u}) : IsLimit (limitCone F) | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ SemiRingCatMax
s : Cone F
j : J
⊢ ↑(OneHom.toFun
(↑{
toOneHom :=
{
toFun := fun v =>
{ val := fun j => ((forget SemiRingCatMax).mapCone s).π.app j v,
property :=
... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | exact (s.π.app j).map_zero | /-- Witness that the limit cone in `SemiRingCat` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ SemiRingCatMax.{v, u}) : IsLimit (limitCone F) := by
refine IsLimit.ofFaithful (forget SemiRingCatMax.{v, u}) (Types.limitConeIsLimit.{v, u} _)
(fun s : Cone F => ofHom
... | Mathlib.Algebra.Category.Ring.Limits.97_0.VxjNIkMLPSqe2rX | /-- Witness that the limit cone in `SemiRingCat` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ SemiRingCatMax.{v, u}) : IsLimit (limitCone F) | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ SemiRingCatMax
s : Cone F
x y : ↑s.1
j : J
⊢ ↑(OneHom.toFun
(↑{
toOneHom :=
{
toFun := fun v =>
{ val := fun j => ((forget SemiRingCatMax).mapCone s).π.app j v,
property :=
... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | exact (s.π.app j).map_add x y | /-- Witness that the limit cone in `SemiRingCat` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ SemiRingCatMax.{v, u}) : IsLimit (limitCone F) := by
refine IsLimit.ofFaithful (forget SemiRingCatMax.{v, u}) (Types.limitConeIsLimit.{v, u} _)
(fun s : Cone F => ofHom
... | Mathlib.Algebra.Category.Ring.Limits.97_0.VxjNIkMLPSqe2rX | /-- Witness that the limit cone in `SemiRingCat` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ SemiRingCatMax.{v, u}) : IsLimit (limitCone F) | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ SemiRingCatMax
⊢ IsLimit ((forget₂ SemiRingCat AddCommMonCat).mapCone (limitCone F)) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | apply AddCommMonCat.limitConeIsLimit.{v, u} | /--
Auxiliary lemma to prove the cone induced by `limitCone` is a limit cone.
-/
def forget₂AddCommMonPreservesLimitsAux (F : J ⥤ SemiRingCatMax.{v, u}) :
IsLimit ((forget₂ SemiRingCat AddCommMonCat).mapCone (limitCone F)) := by
| Mathlib.Algebra.Category.Ring.Limits.129_0.VxjNIkMLPSqe2rX | /--
Auxiliary lemma to prove the cone induced by `limitCone` is a limit cone.
-/
def forget₂AddCommMonPreservesLimitsAux (F : J ⥤ SemiRingCatMax.{v, u}) :
IsLimit ((forget₂ SemiRingCat AddCommMonCat).mapCone (limitCone F)) | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ SemiRingCatMax
⊢ IsLimit ((forget₂ SemiRingCat MonCat).mapCone (limitCone F)) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | apply MonCat.HasLimits.limitConeIsLimit (F ⋙ forget₂ SemiRingCat MonCat.{max v u}) | /-- An auxiliary declaration to speed up typechecking.
-/
def forget₂MonPreservesLimitsAux (F : J ⥤ SemiRingCatMax.{v, u}) :
IsLimit ((forget₂ SemiRingCat MonCat).mapCone (limitCone F)) := by
| Mathlib.Algebra.Category.Ring.Limits.155_0.VxjNIkMLPSqe2rX | /-- An auxiliary declaration to speed up typechecking.
-/
def forget₂MonPreservesLimitsAux (F : J ⥤ SemiRingCatMax.{v, u}) :
IsLimit ((forget₂ SemiRingCat MonCat).mapCone (limitCone F)) | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommSemiRingCatMax
j : J
⊢ CommSemiring ((F ⋙ forget CommSemiRingCat).obj j) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | change CommSemiring (F.obj j) | instance commSemiringObj (F : J ⥤ CommSemiRingCatMax.{v, u}) (j) :
CommSemiring ((F ⋙ forget CommSemiRingCat).obj j) := by
| Mathlib.Algebra.Category.Ring.Limits.206_0.VxjNIkMLPSqe2rX | instance commSemiringObj (F : J ⥤ CommSemiRingCatMax.{v, u}) (j) :
CommSemiring ((F ⋙ forget CommSemiRingCat).obj j) | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommSemiRingCatMax
j : J
⊢ CommSemiring ↑(F.obj j) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | infer_instance | instance commSemiringObj (F : J ⥤ CommSemiRingCatMax.{v, u}) (j) :
CommSemiring ((F ⋙ forget CommSemiRingCat).obj j) := by
change CommSemiring (F.obj j)
| Mathlib.Algebra.Category.Ring.Limits.206_0.VxjNIkMLPSqe2rX | instance commSemiringObj (F : J ⥤ CommSemiRingCatMax.{v, u}) (j) :
CommSemiring ((F ⋙ forget CommSemiRingCat).obj j) | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommSemiRingCatMax
this✝ : ReflectsIsomorphisms (forget CommSemiRingCatMax) := forgetReflectIsos
this : ReflectsIsomorphisms (forget₂ CommSemiRingCatMax SemiRingCatMax) :=
reflectsIsomorphisms_forget₂ CommSemiRingCatMax SemiRingCatMax
c : Cone F :=
{ pt := of (Types.limitC... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | refine IsLimit.ofFaithful (forget₂ CommSemiRingCatMax.{v, u} SemiRingCatMax.{v, u})
(SemiRingCat.HasLimits.limitConeIsLimit.{v, u} _)
(fun s : Cone F => CommSemiRingCat.ofHom
{ toFun := _
map_one' := Subtype.ext <| funext fun j => by exact (s.π.app j).map_one
... | /-- We show that the forgetful functor `CommSemiRingCat ⥤ SemiRingCat` creates limits.
All we need to do is notice that the limit point has a `CommSemiring` instance available,
and then reuse the existing limit.
-/
instance (F : J ⥤ CommSemiRingCatMax.{v, u}) :
CreatesLimit F (forget₂ CommSemiRingCatMax.{v, u} Sem... | Mathlib.Algebra.Category.Ring.Limits.220_0.VxjNIkMLPSqe2rX | /-- We show that the forgetful functor `CommSemiRingCat ⥤ SemiRingCat` creates limits.
All we need to do is notice that the limit point has a `CommSemiring` instance available,
and then reuse the existing limit.
-/
instance (F : J ⥤ CommSemiRingCatMax.{v, u}) :
CreatesLimit F (forget₂ CommSemiRingCatMax.{v, u} Sem... | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommSemiRingCatMax
this✝ : ReflectsIsomorphisms (forget CommSemiRingCatMax) := forgetReflectIsos
this : ReflectsIsomorphisms (forget₂ CommSemiRingCatMax SemiRingCatMax) :=
reflectsIsomorphisms_forget₂ CommSemiRingCatMax SemiRingCatMax
c : Cone F :=
{ pt := of (Types.limitC... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | exact (s.π.app j).map_one | /-- We show that the forgetful functor `CommSemiRingCat ⥤ SemiRingCat` creates limits.
All we need to do is notice that the limit point has a `CommSemiring` instance available,
and then reuse the existing limit.
-/
instance (F : J ⥤ CommSemiRingCatMax.{v, u}) :
CreatesLimit F (forget₂ CommSemiRingCatMax.{v, u} Sem... | Mathlib.Algebra.Category.Ring.Limits.220_0.VxjNIkMLPSqe2rX | /-- We show that the forgetful functor `CommSemiRingCat ⥤ SemiRingCat` creates limits.
All we need to do is notice that the limit point has a `CommSemiring` instance available,
and then reuse the existing limit.
-/
instance (F : J ⥤ CommSemiRingCatMax.{v, u}) :
CreatesLimit F (forget₂ CommSemiRingCatMax.{v, u} Sem... | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommSemiRingCatMax
this✝ : ReflectsIsomorphisms (forget CommSemiRingCatMax) := forgetReflectIsos
this : ReflectsIsomorphisms (forget₂ CommSemiRingCatMax SemiRingCatMax) :=
reflectsIsomorphisms_forget₂ CommSemiRingCatMax SemiRingCatMax
c : Cone F :=
{ pt := of (Types.limitC... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | exact (s.π.app j).map_mul x y | /-- We show that the forgetful functor `CommSemiRingCat ⥤ SemiRingCat` creates limits.
All we need to do is notice that the limit point has a `CommSemiring` instance available,
and then reuse the existing limit.
-/
instance (F : J ⥤ CommSemiRingCatMax.{v, u}) :
CreatesLimit F (forget₂ CommSemiRingCatMax.{v, u} Sem... | Mathlib.Algebra.Category.Ring.Limits.220_0.VxjNIkMLPSqe2rX | /-- We show that the forgetful functor `CommSemiRingCat ⥤ SemiRingCat` creates limits.
All we need to do is notice that the limit point has a `CommSemiring` instance available,
and then reuse the existing limit.
-/
instance (F : J ⥤ CommSemiRingCatMax.{v, u}) :
CreatesLimit F (forget₂ CommSemiRingCatMax.{v, u} Sem... | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommSemiRingCatMax
this✝ : ReflectsIsomorphisms (forget CommSemiRingCatMax) := forgetReflectIsos
this : ReflectsIsomorphisms (forget₂ CommSemiRingCatMax SemiRingCatMax) :=
reflectsIsomorphisms_forget₂ CommSemiRingCatMax SemiRingCatMax
c : Cone F :=
{ pt := of (Types.limitC... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | exact (s.π.app j).map_zero | /-- We show that the forgetful functor `CommSemiRingCat ⥤ SemiRingCat` creates limits.
All we need to do is notice that the limit point has a `CommSemiring` instance available,
and then reuse the existing limit.
-/
instance (F : J ⥤ CommSemiRingCatMax.{v, u}) :
CreatesLimit F (forget₂ CommSemiRingCatMax.{v, u} Sem... | Mathlib.Algebra.Category.Ring.Limits.220_0.VxjNIkMLPSqe2rX | /-- We show that the forgetful functor `CommSemiRingCat ⥤ SemiRingCat` creates limits.
All we need to do is notice that the limit point has a `CommSemiring` instance available,
and then reuse the existing limit.
-/
instance (F : J ⥤ CommSemiRingCatMax.{v, u}) :
CreatesLimit F (forget₂ CommSemiRingCatMax.{v, u} Sem... | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommSemiRingCatMax
this✝ : ReflectsIsomorphisms (forget CommSemiRingCatMax) := forgetReflectIsos
this : ReflectsIsomorphisms (forget₂ CommSemiRingCatMax SemiRingCatMax) :=
reflectsIsomorphisms_forget₂ CommSemiRingCatMax SemiRingCatMax
c : Cone F :=
{ pt := of (Types.limitC... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | exact (s.π.app j).map_add x y | /-- We show that the forgetful functor `CommSemiRingCat ⥤ SemiRingCat` creates limits.
All we need to do is notice that the limit point has a `CommSemiring` instance available,
and then reuse the existing limit.
-/
instance (F : J ⥤ CommSemiRingCatMax.{v, u}) :
CreatesLimit F (forget₂ CommSemiRingCatMax.{v, u} Sem... | Mathlib.Algebra.Category.Ring.Limits.220_0.VxjNIkMLPSqe2rX | /-- We show that the forgetful functor `CommSemiRingCat ⥤ SemiRingCat` creates limits.
All we need to do is notice that the limit point has a `CommSemiring` instance available,
and then reuse the existing limit.
-/
instance (F : J ⥤ CommSemiRingCatMax.{v, u}) :
CreatesLimit F (forget₂ CommSemiRingCatMax.{v, u} Sem... | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ RingCatMax
j : J
⊢ Ring ((F ⋙ forget RingCat).obj j) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | change Ring (F.obj j) | instance ringObj (F : J ⥤ RingCatMax.{v, u}) (j) : Ring ((F ⋙ forget RingCat).obj j) := by
| Mathlib.Algebra.Category.Ring.Limits.333_0.VxjNIkMLPSqe2rX | instance ringObj (F : J ⥤ RingCatMax.{v, u}) (j) : Ring ((F ⋙ forget RingCat).obj j) | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ RingCatMax
j : J
⊢ Ring ↑(F.obj j) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | infer_instance | instance ringObj (F : J ⥤ RingCatMax.{v, u}) (j) : Ring ((F ⋙ forget RingCat).obj j) := by
change Ring (F.obj j)
| Mathlib.Algebra.Category.Ring.Limits.333_0.VxjNIkMLPSqe2rX | instance ringObj (F : J ⥤ RingCatMax.{v, u}) (j) : Ring ((F ⋙ forget RingCat).obj j) | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ RingCatMax
this : ReflectsIsomorphisms (forget₂ RingCatMax SemiRingCatMax) :=
reflectsIsomorphisms_forget₂ RingCatMax SemiRingCatMax
c : Cone F :=
{ pt := of (Types.limitCone (F ⋙ forget RingCatMax)).pt,
π :=
NatTrans.mk fun x =>
SemiRingCat.ofHom ((SemiR... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | apply IsLimit.uniqueUpToIso (SemiRingCat.HasLimits.limitConeIsLimit _) t | /-- We show that the forgetful functor `CommRingCat ⥤ RingCat` creates limits.
All we need to do is notice that the limit point has a `Ring` instance available,
and then reuse the existing limit.
-/
instance (F : J ⥤ RingCatMax.{v, u}) :
CreatesLimit F (forget₂ RingCatMax.{v, u} SemiRingCatMax.{v, u}) :=
letI : ... | Mathlib.Algebra.Category.Ring.Limits.358_0.VxjNIkMLPSqe2rX | /-- We show that the forgetful functor `CommRingCat ⥤ RingCat` creates limits.
All we need to do is notice that the limit point has a `Ring` instance available,
and then reuse the existing limit.
-/
instance (F : J ⥤ RingCatMax.{v, u}) :
CreatesLimit F (forget₂ RingCatMax.{v, u} SemiRingCatMax.{v, u}) | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ RingCatMax
this : ReflectsIsomorphisms (forget₂ RingCatMax SemiRingCatMax) :=
reflectsIsomorphisms_forget₂ RingCatMax SemiRingCatMax
c : Cone F :=
{ pt := of (Types.limitCone (F ⋙ forget RingCatMax)).pt,
π :=
NatTrans.mk fun x =>
SemiRingCat.ofHom ((SemiR... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | apply SemiRingCat.HasLimits.limitConeIsLimit _ | /-- We show that the forgetful functor `CommRingCat ⥤ RingCat` creates limits.
All we need to do is notice that the limit point has a `Ring` instance available,
and then reuse the existing limit.
-/
instance (F : J ⥤ RingCatMax.{v, u}) :
CreatesLimit F (forget₂ RingCatMax.{v, u} SemiRingCatMax.{v, u}) :=
letI : ... | Mathlib.Algebra.Category.Ring.Limits.358_0.VxjNIkMLPSqe2rX | /-- We show that the forgetful functor `CommRingCat ⥤ RingCat` creates limits.
All we need to do is notice that the limit point has a `Ring` instance available,
and then reuse the existing limit.
-/
instance (F : J ⥤ RingCatMax.{v, u}) :
CreatesLimit F (forget₂ RingCatMax.{v, u} SemiRingCatMax.{v, u}) | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ RingCatMax
⊢ IsLimit ((forget₂ RingCatMax AddCommGroupCat).mapCone (limitCone F)) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | letI f := (F ⋙ forget₂ RingCatMax.{v, u} AddCommGroupCat.{max v u}) | /-- An auxiliary declaration to speed up typechecking.
-/
def forget₂AddCommGroupPreservesLimitsAux (F : J ⥤ RingCatMax.{v, u}) :
IsLimit ((forget₂ RingCatMax.{v, u} AddCommGroupCat).mapCone (limitCone.{v, u} F)) := by
-- Porting note : inline `f` would not compile
| Mathlib.Algebra.Category.Ring.Limits.426_0.VxjNIkMLPSqe2rX | /-- An auxiliary declaration to speed up typechecking.
-/
def forget₂AddCommGroupPreservesLimitsAux (F : J ⥤ RingCatMax.{v, u}) :
IsLimit ((forget₂ RingCatMax.{v, u} AddCommGroupCat).mapCone (limitCone.{v, u} F)) | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ RingCatMax
f : J ⥤ AddCommGroupCat := F ⋙ forget₂ RingCatMax AddCommGroupCat
⊢ IsLimit ((forget₂ RingCatMax AddCommGroupCat).mapCone (limitCone F)) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | apply AddCommGroupCat.limitConeIsLimit.{v, u} f | /-- An auxiliary declaration to speed up typechecking.
-/
def forget₂AddCommGroupPreservesLimitsAux (F : J ⥤ RingCatMax.{v, u}) :
IsLimit ((forget₂ RingCatMax.{v, u} AddCommGroupCat).mapCone (limitCone.{v, u} F)) := by
-- Porting note : inline `f` would not compile
letI f := (F ⋙ forget₂ RingCatMax.{v, u} AddCo... | Mathlib.Algebra.Category.Ring.Limits.426_0.VxjNIkMLPSqe2rX | /-- An auxiliary declaration to speed up typechecking.
-/
def forget₂AddCommGroupPreservesLimitsAux (F : J ⥤ RingCatMax.{v, u}) :
IsLimit ((forget₂ RingCatMax.{v, u} AddCommGroupCat).mapCone (limitCone.{v, u} F)) | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommRingCatMax
j : J
⊢ CommRing ((F ⋙ forget CommRingCat).obj j) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | change CommRing (F.obj j) | instance commRingObj (F : J ⥤ CommRingCatMax.{v, u}) (j) :
CommRing ((F ⋙ forget CommRingCat).obj j) := by
| Mathlib.Algebra.Category.Ring.Limits.480_0.VxjNIkMLPSqe2rX | instance commRingObj (F : J ⥤ CommRingCatMax.{v, u}) (j) :
CommRing ((F ⋙ forget CommRingCat).obj j) | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommRingCatMax
j : J
⊢ CommRing ↑(F.obj j) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | infer_instance | instance commRingObj (F : J ⥤ CommRingCatMax.{v, u}) (j) :
CommRing ((F ⋙ forget CommRingCat).obj j) := by
change CommRing (F.obj j)
| Mathlib.Algebra.Category.Ring.Limits.480_0.VxjNIkMLPSqe2rX | instance commRingObj (F : J ⥤ CommRingCatMax.{v, u}) (j) :
CommRing ((F ⋙ forget CommRingCat).obj j) | Mathlib_Algebra_Category_Ring_Limits |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommRingCatMax
⊢ IsLimit ((forget₂ CommRingCat CommSemiRingCat).mapCone (limitCone F)) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.RingTheory.Subring.Basic
#al... | apply CommSemiRingCat.limitConeIsLimit (F ⋙ forget₂ CommRingCat CommSemiRingCat.{max v u}) | /-- An auxiliary declaration to speed up typechecking.
-/
def forget₂CommSemiRingPreservesLimitsAux (F : J ⥤ CommRingCatMax.{v, u}) :
IsLimit ((forget₂ CommRingCat CommSemiRingCat).mapCone (limitCone F)) := by
| Mathlib.Algebra.Category.Ring.Limits.586_0.VxjNIkMLPSqe2rX | /-- An auxiliary declaration to speed up typechecking.
-/
def forget₂CommSemiRingPreservesLimitsAux (F : J ⥤ CommRingCatMax.{v, u}) :
IsLimit ((forget₂ CommRingCat CommSemiRingCat).mapCone (limitCone F)) | Mathlib_Algebra_Category_Ring_Limits |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
⊢ mirror 0 = 0 | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | simp [mirror] | @[simp]
theorem mirror_zero : (0 : R[X]).mirror = 0 := by | Mathlib.Data.Polynomial.Mirror.43_0.jRYkh9xLrkM32QX | @[simp]
theorem mirror_zero : (0 : R[X]).mirror = 0 | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
a : R
⊢ mirror ((monomial n) a) = (monomial n) a | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | classical
by_cases ha : a = 0
· rw [ha, monomial_zero_right, mirror_zero]
· rw [mirror, reverse, natDegree_monomial n a, if_neg ha, natTrailingDegree_monomial ha, ←
C_mul_X_pow_eq_monomial, reflect_C_mul_X_pow, revAt_le (le_refl n), tsub_self, pow_zero,
mul_one] | theorem mirror_monomial (n : ℕ) (a : R) : (monomial n a).mirror = monomial n a := by
| Mathlib.Data.Polynomial.Mirror.47_0.jRYkh9xLrkM32QX | theorem mirror_monomial (n : ℕ) (a : R) : (monomial n a).mirror = monomial n a | Mathlib_Data_Polynomial_Mirror |
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
a : R
⊢ mirror ((monomial n) a) = (monomial n) a | /-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mat... | by_cases ha : a = 0 | theorem mirror_monomial (n : ℕ) (a : R) : (monomial n a).mirror = monomial n a := by
classical
| Mathlib.Data.Polynomial.Mirror.47_0.jRYkh9xLrkM32QX | theorem mirror_monomial (n : ℕ) (a : R) : (monomial n a).mirror = monomial n a | Mathlib_Data_Polynomial_Mirror |
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