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 |
|---|---|---|---|---|---|---|
case refine'_1
x z✝ z : ℂ
hre : z.re < 0
him : z.im = 0
this : arg =ᶠ[𝓝[{z | 0 ≤ z.im}] z] fun x => arcsin ((-x).im / abs x) + π
⊢ abs z ≠ 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | lift z to ℝ using him | theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) :
ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z := by
have : arg =ᶠ[𝓝[{ z : ℂ | 0 ≤ z.im }] z] fun x => Real.arcsin ((-x).im / abs x) + π := by
have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nh... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.597_0.CflASCTDE9UCom5 | theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) :
ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case refine'_1.intro
x z✝ : ℂ
z : ℝ
hre : (↑z).re < 0
this : arg =ᶠ[𝓝[{z | 0 ≤ z.im}] ↑z] fun x => arcsin ((-x).im / abs x) + π
⊢ abs ↑z ≠ 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simpa using hre.ne | theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) :
ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z := by
have : arg =ᶠ[𝓝[{ z : ℂ | 0 ≤ z.im }] z] fun x => Real.arcsin ((-x).im / abs x) + π := by
have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nh... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.597_0.CflASCTDE9UCom5 | theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) :
ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case refine'_2
x z✝ z : ℂ
hre : z.re < 0
him : z.im = 0
this : arg =ᶠ[𝓝[{z | 0 ≤ z.im}] z] fun x => arcsin ((-x).im / abs x) + π
⊢ arg z = arcsin ((-z).im / abs z) + π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [arg, if_neg hre.not_le, if_pos him.ge] | theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) :
ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z := by
have : arg =ᶠ[𝓝[{ z : ℂ | 0 ≤ z.im }] z] fun x => Real.arcsin ((-x).im / abs x) + π := by
have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nh... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.597_0.CflASCTDE9UCom5 | theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) :
ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x z✝ z : ℂ
hre : z.re < 0
him : z.im = 0
⊢ Tendsto arg (𝓝[{z | 0 ≤ z.im}] z) (𝓝 π) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simpa only [arg_eq_pi_iff.2 ⟨hre, him⟩] using
(continuousWithinAt_arg_of_re_neg_of_im_zero hre him).tendsto | theorem tendsto_arg_nhdsWithin_im_nonneg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0)
(him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | 0 ≤ z.im }] z) (𝓝 π) := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.615_0.CflASCTDE9UCom5 | theorem tendsto_arg_nhdsWithin_im_nonneg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0)
(him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | 0 ≤ z.im }] z) (𝓝 π) | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x z : ℂ
h : x ≠ 0
⊢ ContinuousAt (Angle.coe ∘ arg) x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | by_cases hs : 0 < x.re ∨ x.im ≠ 0 | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.621_0.CflASCTDE9UCom5 | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
x z : ℂ
h : x ≠ 0
hs : 0 < x.re ∨ x.im ≠ 0
⊢ ContinuousAt (Angle.coe ∘ arg) x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | exact Real.Angle.continuous_coe.continuousAt.comp (continuousAt_arg hs) | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x := by
by_cases hs : 0 < x.re ∨ x.im ≠ 0
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.621_0.CflASCTDE9UCom5 | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
x z : ℂ
h : x ≠ 0
hs : ¬(0 < x.re ∨ x.im ≠ 0)
⊢ ContinuousAt (Angle.coe ∘ arg) x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [← Function.comp.right_id (((↑) : ℝ → Real.Angle) ∘ arg),
(Function.funext_iff.2 fun _ => (neg_neg _).symm : (id : ℂ → ℂ) = Neg.neg ∘ Neg.neg), ←
Function.comp.assoc] | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x := by
by_cases hs : 0 < x.re ∨ x.im ≠ 0
· exact Real.Angle.continuous_coe.continuousAt.comp (continuousAt_arg hs)
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.621_0.CflASCTDE9UCom5 | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
x z : ℂ
h : x ≠ 0
hs : ¬(0 < x.re ∨ x.im ≠ 0)
⊢ ContinuousAt (((Angle.coe ∘ arg) ∘ Neg.neg) ∘ Neg.neg) x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | refine' ContinuousAt.comp _ continuous_neg.continuousAt | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x := by
by_cases hs : 0 < x.re ∨ x.im ≠ 0
· exact Real.Angle.continuous_coe.continuousAt.comp (continuousAt_arg hs)
· rw [← Function.comp.right_id (((↑) : ℝ → Real.Angle) ∘ arg),
(Function.funext_iff.2 fun _ => (neg_n... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.621_0.CflASCTDE9UCom5 | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
x z : ℂ
h : x ≠ 0
hs : ¬(0 < x.re ∨ x.im ≠ 0)
⊢ ContinuousAt ((Angle.coe ∘ arg) ∘ Neg.neg) (-x) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | suffices ContinuousAt (Function.update (((↑) ∘ arg) ∘ Neg.neg : ℂ → Real.Angle) 0 π) (-x) by
rwa [continuousAt_update_of_ne (neg_ne_zero.2 h)] at this | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x := by
by_cases hs : 0 < x.re ∨ x.im ≠ 0
· exact Real.Angle.continuous_coe.continuousAt.comp (continuousAt_arg hs)
· rw [← Function.comp.right_id (((↑) : ℝ → Real.Angle) ∘ arg),
(Function.funext_iff.2 fun _ => (neg_n... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.621_0.CflASCTDE9UCom5 | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x z : ℂ
h : x ≠ 0
hs : ¬(0 < x.re ∨ x.im ≠ 0)
this : ContinuousAt (Function.update ((Angle.coe ∘ arg) ∘ Neg.neg) 0 ↑π) (-x)
⊢ ContinuousAt ((Angle.coe ∘ arg) ∘ Neg.neg) (-x) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rwa [continuousAt_update_of_ne (neg_ne_zero.2 h)] at this | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x := by
by_cases hs : 0 < x.re ∨ x.im ≠ 0
· exact Real.Angle.continuous_coe.continuousAt.comp (continuousAt_arg hs)
· rw [← Function.comp.right_id (((↑) : ℝ → Real.Angle) ∘ arg),
(Function.funext_iff.2 fun _ => (neg_n... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.621_0.CflASCTDE9UCom5 | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
x z : ℂ
h : x ≠ 0
hs : ¬(0 < x.re ∨ x.im ≠ 0)
⊢ ContinuousAt (Function.update ((Angle.coe ∘ arg) ∘ Neg.neg) 0 ↑π) (-x) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | have ha :
Function.update (((↑) ∘ arg) ∘ Neg.neg : ℂ → Real.Angle) 0 π = fun z =>
(arg z : Real.Angle) + π := by
rw [Function.update_eq_iff]
exact ⟨by simp, fun z hz => arg_neg_coe_angle hz⟩ | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x := by
by_cases hs : 0 < x.re ∨ x.im ≠ 0
· exact Real.Angle.continuous_coe.continuousAt.comp (continuousAt_arg hs)
· rw [← Function.comp.right_id (((↑) : ℝ → Real.Angle) ∘ arg),
(Function.funext_iff.2 fun _ => (neg_n... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.621_0.CflASCTDE9UCom5 | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x z : ℂ
h : x ≠ 0
hs : ¬(0 < x.re ∨ x.im ≠ 0)
⊢ Function.update ((Angle.coe ∘ arg) ∘ Neg.neg) 0 ↑π = fun z => ↑(arg z) + ↑π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [Function.update_eq_iff] | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x := by
by_cases hs : 0 < x.re ∨ x.im ≠ 0
· exact Real.Angle.continuous_coe.continuousAt.comp (continuousAt_arg hs)
· rw [← Function.comp.right_id (((↑) : ℝ → Real.Angle) ∘ arg),
(Function.funext_iff.2 fun _ => (neg_n... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.621_0.CflASCTDE9UCom5 | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x z : ℂ
h : x ≠ 0
hs : ¬(0 < x.re ∨ x.im ≠ 0)
⊢ ↑π = ↑(arg 0) + ↑π ∧ ∀ (x : ℂ), x ≠ 0 → ((Angle.coe ∘ arg) ∘ Neg.neg) x = ↑(arg x) + ↑π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | exact ⟨by simp, fun z hz => arg_neg_coe_angle hz⟩ | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x := by
by_cases hs : 0 < x.re ∨ x.im ≠ 0
· exact Real.Angle.continuous_coe.continuousAt.comp (continuousAt_arg hs)
· rw [← Function.comp.right_id (((↑) : ℝ → Real.Angle) ∘ arg),
(Function.funext_iff.2 fun _ => (neg_n... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.621_0.CflASCTDE9UCom5 | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x z : ℂ
h : x ≠ 0
hs : ¬(0 < x.re ∨ x.im ≠ 0)
⊢ ↑π = ↑(arg 0) + ↑π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x := by
by_cases hs : 0 < x.re ∨ x.im ≠ 0
· exact Real.Angle.continuous_coe.continuousAt.comp (continuousAt_arg hs)
· rw [← Function.comp.right_id (((↑) : ℝ → Real.Angle) ∘ arg),
(Function.funext_iff.2 fun _ => (neg_n... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.621_0.CflASCTDE9UCom5 | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
x z : ℂ
h : x ≠ 0
hs : ¬(0 < x.re ∨ x.im ≠ 0)
ha : Function.update ((Angle.coe ∘ arg) ∘ Neg.neg) 0 ↑π = fun z => ↑(arg z) + ↑π
⊢ ContinuousAt (Function.update ((Angle.coe ∘ arg) ∘ Neg.neg) 0 ↑π) (-x) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [ha] | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x := by
by_cases hs : 0 < x.re ∨ x.im ≠ 0
· exact Real.Angle.continuous_coe.continuousAt.comp (continuousAt_arg hs)
· rw [← Function.comp.right_id (((↑) : ℝ → Real.Angle) ∘ arg),
(Function.funext_iff.2 fun _ => (neg_n... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.621_0.CflASCTDE9UCom5 | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
x z : ℂ
h : x ≠ 0
hs : ¬(0 < x.re ∨ x.im ≠ 0)
ha : Function.update ((Angle.coe ∘ arg) ∘ Neg.neg) 0 ↑π = fun z => ↑(arg z) + ↑π
⊢ ContinuousAt (fun z => ↑(arg z) + ↑π) (-x) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | push_neg at hs | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x := by
by_cases hs : 0 < x.re ∨ x.im ≠ 0
· exact Real.Angle.continuous_coe.continuousAt.comp (continuousAt_arg hs)
· rw [← Function.comp.right_id (((↑) : ℝ → Real.Angle) ∘ arg),
(Function.funext_iff.2 fun _ => (neg_n... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.621_0.CflASCTDE9UCom5 | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
x z : ℂ
h : x ≠ 0
ha : Function.update ((Angle.coe ∘ arg) ∘ Neg.neg) 0 ↑π = fun z => ↑(arg z) + ↑π
hs : x.re ≤ 0 ∧ x.im = 0
⊢ ContinuousAt (fun z => ↑(arg z) + ↑π) (-x) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | refine'
(Real.Angle.continuous_coe.continuousAt.comp (continuousAt_arg (Or.inl _))).add
continuousAt_const | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x := by
by_cases hs : 0 < x.re ∨ x.im ≠ 0
· exact Real.Angle.continuous_coe.continuousAt.comp (continuousAt_arg hs)
· rw [← Function.comp.right_id (((↑) : ℝ → Real.Angle) ∘ arg),
(Function.funext_iff.2 fun _ => (neg_n... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.621_0.CflASCTDE9UCom5 | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
x z : ℂ
h : x ≠ 0
ha : Function.update ((Angle.coe ∘ arg) ∘ Neg.neg) 0 ↑π = fun z => ↑(arg z) + ↑π
hs : x.re ≤ 0 ∧ x.im = 0
⊢ 0 < (-x).re | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [neg_re, neg_pos] | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x := by
by_cases hs : 0 < x.re ∨ x.im ≠ 0
· exact Real.Angle.continuous_coe.continuousAt.comp (continuousAt_arg hs)
· rw [← Function.comp.right_id (((↑) : ℝ → Real.Angle) ∘ arg),
(Function.funext_iff.2 fun _ => (neg_n... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.621_0.CflASCTDE9UCom5 | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
x z : ℂ
h : x ≠ 0
ha : Function.update ((Angle.coe ∘ arg) ∘ Neg.neg) 0 ↑π = fun z => ↑(arg z) + ↑π
hs : x.re ≤ 0 ∧ x.im = 0
⊢ x.re < 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | exact hs.1.lt_of_ne fun h0 => h (ext_iff.2 ⟨h0, hs.2⟩) | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x := by
by_cases hs : 0 < x.re ∨ x.im ≠ 0
· exact Real.Angle.continuous_coe.continuousAt.comp (continuousAt_arg hs)
· rw [← Function.comp.right_id (((↑) : ℝ → Real.Angle) ∘ arg),
(Function.funext_iff.2 fun _ => (neg_n... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.621_0.CflASCTDE9UCom5 | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
inst✝⁴ : NormedField 𝕜
inst✝³ : NormedField 𝕜'
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜' E
r : ℝ
c : ↑(closedBall 0 1)
x : ↑(ball 0 r)
⊢ ‖↑c • ↑x‖ < r | /-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Heather Macbeth
-/
import Mathlib.Analysis.Normed.Field.UnitBall
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.normed_space.ball_action from "lean... | simpa only [norm_smul, one_mul] using
mul_lt_mul' (mem_closedBall_zero_iff.1 c.2) (mem_ball_zero_iff.1 x.2) (norm_nonneg _)
one_pos | instance mulActionClosedBallBall : MulAction (closedBall (0 : 𝕜) 1) (ball (0 : E) r) where
smul c x :=
⟨(c : 𝕜) • ↑x,
mem_ball_zero_iff.2 <| by
| Mathlib.Analysis.NormedSpace.BallAction.29_0.NWEJH2CyESYnSt3 | instance mulActionClosedBallBall : MulAction (closedBall (0 : 𝕜) 1) (ball (0 : E) r) where
smul c x | Mathlib_Analysis_NormedSpace_BallAction |
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
inst✝⁴ : NormedField 𝕜
inst✝³ : NormedField 𝕜'
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜' E
r : ℝ
c : ↑(closedBall 0 1)
x : ↑(closedBall 0 r)
⊢ ‖↑c • ↑x‖ ≤ r | /-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Heather Macbeth
-/
import Mathlib.Analysis.Normed.Field.UnitBall
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.normed_space.ball_action from "lean... | simpa only [norm_smul, one_mul] using
mul_le_mul (mem_closedBall_zero_iff.1 c.2) (mem_closedBall_zero_iff.1 x.2) (norm_nonneg _)
zero_le_one | instance mulActionClosedBallClosedBall : MulAction (closedBall (0 : 𝕜) 1) (closedBall (0 : E) r)
where
smul c x :=
⟨(c : 𝕜) • ↑x,
mem_closedBall_zero_iff.2 <| by
| Mathlib.Analysis.NormedSpace.BallAction.44_0.NWEJH2CyESYnSt3 | instance mulActionClosedBallClosedBall : MulAction (closedBall (0 : 𝕜) 1) (closedBall (0 : E) r)
where
smul c x | Mathlib_Analysis_NormedSpace_BallAction |
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
inst✝⁴ : NormedField 𝕜
inst✝³ : NormedField 𝕜'
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜' E
r : ℝ
c : ↑(sphere 0 1)
x : ↑(sphere 0 r)
⊢ ‖↑c • ↑x‖ = r | /-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Heather Macbeth
-/
import Mathlib.Analysis.Normed.Field.UnitBall
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.normed_space.ball_action from "lean... | rw [norm_smul, mem_sphere_zero_iff_norm.1 c.coe_prop, mem_sphere_zero_iff_norm.1 x.coe_prop,
one_mul] | instance mulActionSphereSphere : MulAction (sphere (0 : 𝕜) 1) (sphere (0 : E) r) where
smul c x :=
⟨(c : 𝕜) • ↑x,
mem_sphere_zero_iff_norm.2 <| by
| Mathlib.Analysis.NormedSpace.BallAction.86_0.NWEJH2CyESYnSt3 | instance mulActionSphereSphere : MulAction (sphere (0 : 𝕜) 1) (sphere (0 : E) r) where
smul c x | Mathlib_Analysis_NormedSpace_BallAction |
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
inst✝⁵ : NormedField 𝕜
inst✝⁴ : NormedField 𝕜'
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜' E
r✝ : ℝ
inst✝ : CharZero 𝕜
r : ℝ
hr : r ≠ 0
x : ↑(sphere 0 r)
h : x = -x
⊢ ↑x = -↑x | /-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Heather Macbeth
-/
import Mathlib.Analysis.Normed.Field.UnitBall
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.normed_space.ball_action from "lean... | conv_lhs => rw [h] | theorem ne_neg_of_mem_sphere {r : ℝ} (hr : r ≠ 0) (x : sphere (0 : E) r) : x ≠ -x := fun h =>
ne_zero_of_mem_sphere hr x ((self_eq_neg 𝕜 _).mp (by | Mathlib.Analysis.NormedSpace.BallAction.203_0.NWEJH2CyESYnSt3 | theorem ne_neg_of_mem_sphere {r : ℝ} (hr : r ≠ 0) (x : sphere (0 : E) r) : x ≠ -x | Mathlib_Analysis_NormedSpace_BallAction |
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
inst✝⁵ : NormedField 𝕜
inst✝⁴ : NormedField 𝕜'
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜' E
r✝ : ℝ
inst✝ : CharZero 𝕜
r : ℝ
hr : r ≠ 0
x : ↑(sphere 0 r)
h : x = -x
| ↑x | /-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Heather Macbeth
-/
import Mathlib.Analysis.Normed.Field.UnitBall
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.normed_space.ball_action from "lean... | rw [h] | theorem ne_neg_of_mem_sphere {r : ℝ} (hr : r ≠ 0) (x : sphere (0 : E) r) : x ≠ -x := fun h =>
ne_zero_of_mem_sphere hr x ((self_eq_neg 𝕜 _).mp (by conv_lhs => | Mathlib.Analysis.NormedSpace.BallAction.203_0.NWEJH2CyESYnSt3 | theorem ne_neg_of_mem_sphere {r : ℝ} (hr : r ≠ 0) (x : sphere (0 : E) r) : x ≠ -x | Mathlib_Analysis_NormedSpace_BallAction |
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
inst✝⁵ : NormedField 𝕜
inst✝⁴ : NormedField 𝕜'
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜' E
r✝ : ℝ
inst✝ : CharZero 𝕜
r : ℝ
hr : r ≠ 0
x : ↑(sphere 0 r)
h : x = -x
| ↑x | /-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Heather Macbeth
-/
import Mathlib.Analysis.Normed.Field.UnitBall
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.normed_space.ball_action from "lean... | rw [h] | theorem ne_neg_of_mem_sphere {r : ℝ} (hr : r ≠ 0) (x : sphere (0 : E) r) : x ≠ -x := fun h =>
ne_zero_of_mem_sphere hr x ((self_eq_neg 𝕜 _).mp (by conv_lhs => | Mathlib.Analysis.NormedSpace.BallAction.203_0.NWEJH2CyESYnSt3 | theorem ne_neg_of_mem_sphere {r : ℝ} (hr : r ≠ 0) (x : sphere (0 : E) r) : x ≠ -x | Mathlib_Analysis_NormedSpace_BallAction |
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
inst✝⁵ : NormedField 𝕜
inst✝⁴ : NormedField 𝕜'
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜' E
r✝ : ℝ
inst✝ : CharZero 𝕜
r : ℝ
hr : r ≠ 0
x : ↑(sphere 0 r)
h : x = -x
| ↑x | /-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Heather Macbeth
-/
import Mathlib.Analysis.Normed.Field.UnitBall
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.normed_space.ball_action from "lean... | rw [h] | theorem ne_neg_of_mem_sphere {r : ℝ} (hr : r ≠ 0) (x : sphere (0 : E) r) : x ≠ -x := fun h =>
ne_zero_of_mem_sphere hr x ((self_eq_neg 𝕜 _).mp (by conv_lhs => | Mathlib.Analysis.NormedSpace.BallAction.203_0.NWEJH2CyESYnSt3 | theorem ne_neg_of_mem_sphere {r : ℝ} (hr : r ≠ 0) (x : sphere (0 : E) r) : x ≠ -x | Mathlib_Analysis_NormedSpace_BallAction |
R : Type u_1
S : Type u_2
M : Type u_3
a✝ b : R
s : S
inst✝³ : SMul R M
inst✝² : SMul R S
inst✝¹ : SMul S M
inst✝ : IsScalarTower R S M
a : R
ab : IsSMulRegular M (a • s)
c d : M
cd : ((fun m => a • m) ∘ fun x => s • x) c = ((fun m => a • m) ∘ fun x => s • x) d
⊢ c = d | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | dsimp only [Function.comp_def] at cd | /-- If an element `b` becomes `M`-regular after multiplying it on the left by an `M`-regular
element, then `b` is `M`-regular. -/
theorem of_smul (a : R) (ab : IsSMulRegular M (a • s)) : IsSMulRegular M s :=
@Function.Injective.of_comp _ _ _ (fun m : M => a • m) _ fun c d cd => by
| Mathlib.Algebra.Regular.SMul.72_0.jyFiiORgtZ4G1XR | /-- If an element `b` becomes `M`-regular after multiplying it on the left by an `M`-regular
element, then `b` is `M`-regular. -/
theorem of_smul (a : R) (ab : IsSMulRegular M (a • s)) : IsSMulRegular M s | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a✝ b : R
s : S
inst✝³ : SMul R M
inst✝² : SMul R S
inst✝¹ : SMul S M
inst✝ : IsScalarTower R S M
a : R
ab : IsSMulRegular M (a • s)
c d : M
cd : a • s • c = a • s • d
⊢ c = d | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | rw [← smul_assoc, ← smul_assoc] at cd | /-- If an element `b` becomes `M`-regular after multiplying it on the left by an `M`-regular
element, then `b` is `M`-regular. -/
theorem of_smul (a : R) (ab : IsSMulRegular M (a • s)) : IsSMulRegular M s :=
@Function.Injective.of_comp _ _ _ (fun m : M => a • m) _ fun c d cd => by
dsimp only [Function.comp_def] at ... | Mathlib.Algebra.Regular.SMul.72_0.jyFiiORgtZ4G1XR | /-- If an element `b` becomes `M`-regular after multiplying it on the left by an `M`-regular
element, then `b` is `M`-regular. -/
theorem of_smul (a : R) (ab : IsSMulRegular M (a • s)) : IsSMulRegular M s | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a✝ b : R
s : S
inst✝³ : SMul R M
inst✝² : SMul R S
inst✝¹ : SMul S M
inst✝ : IsScalarTower R S M
a : R
ab : IsSMulRegular M (a • s)
c d : M
cd : (a • s) • c = (a • s) • d
⊢ c = d | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | exact ab cd | /-- If an element `b` becomes `M`-regular after multiplying it on the left by an `M`-regular
element, then `b` is `M`-regular. -/
theorem of_smul (a : R) (ab : IsSMulRegular M (a • s)) : IsSMulRegular M s :=
@Function.Injective.of_comp _ _ _ (fun m : M => a • m) _ fun c d cd => by
dsimp only [Function.comp_def] at ... | Mathlib.Algebra.Regular.SMul.72_0.jyFiiORgtZ4G1XR | /-- If an element `b` becomes `M`-regular after multiplying it on the left by an `M`-regular
element, then `b` is `M`-regular. -/
theorem of_smul (a : R) (ab : IsSMulRegular M (a • s)) : IsSMulRegular M s | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝⁵ : SMul R M
inst✝⁴ : SMul R S
inst✝³ : SMul S M
inst✝² : IsScalarTower R S M
inst✝¹ : Mul R
inst✝ : IsScalarTower R R M
ab : IsSMulRegular M (a * b)
⊢ IsSMulRegular M b | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | rw [← smul_eq_mul] at ab | theorem of_mul [Mul R] [IsScalarTower R R M] (ab : IsSMulRegular M (a * b)) :
IsSMulRegular M b := by
| Mathlib.Algebra.Regular.SMul.102_0.jyFiiORgtZ4G1XR | theorem of_mul [Mul R] [IsScalarTower R R M] (ab : IsSMulRegular M (a * b)) :
IsSMulRegular M b | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝⁵ : SMul R M
inst✝⁴ : SMul R S
inst✝³ : SMul S M
inst✝² : IsScalarTower R S M
inst✝¹ : Mul R
inst✝ : IsScalarTower R R M
ab : IsSMulRegular M (a • b)
⊢ IsSMulRegular M b | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | exact ab.of_smul _ | theorem of_mul [Mul R] [IsScalarTower R R M] (ab : IsSMulRegular M (a * b)) :
IsSMulRegular M b := by
rw [← smul_eq_mul] at ab
| Mathlib.Algebra.Regular.SMul.102_0.jyFiiORgtZ4G1XR | theorem of_mul [Mul R] [IsScalarTower R R M] (ab : IsSMulRegular M (a * b)) :
IsSMulRegular M b | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝⁵ : SMul R M
inst✝⁴ : SMul R S
inst✝³ : SMul S M
inst✝² : IsScalarTower R S M
inst✝¹ : Mul R
inst✝ : IsScalarTower R R M
⊢ IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) ↔ IsSMulRegular M a ∧ IsSMulRegular M b | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | refine' ⟨_, _⟩ | /-- Two elements `a` and `b` are `M`-regular if and only if both products `a * b` and `b * a`
are `M`-regular. -/
theorem mul_and_mul_iff [Mul R] [IsScalarTower R R M] :
IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) ↔ IsSMulRegular M a ∧ IsSMulRegular M b := by
| Mathlib.Algebra.Regular.SMul.114_0.jyFiiORgtZ4G1XR | /-- Two elements `a` and `b` are `M`-regular if and only if both products `a * b` and `b * a`
are `M`-regular. -/
theorem mul_and_mul_iff [Mul R] [IsScalarTower R R M] :
IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) ↔ IsSMulRegular M a ∧ IsSMulRegular M b | Mathlib_Algebra_Regular_SMul |
case refine'_1
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝⁵ : SMul R M
inst✝⁴ : SMul R S
inst✝³ : SMul S M
inst✝² : IsScalarTower R S M
inst✝¹ : Mul R
inst✝ : IsScalarTower R R M
⊢ IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) → IsSMulRegular M a ∧ IsSMulRegular M b | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | rintro ⟨ab, ba⟩ | /-- Two elements `a` and `b` are `M`-regular if and only if both products `a * b` and `b * a`
are `M`-regular. -/
theorem mul_and_mul_iff [Mul R] [IsScalarTower R R M] :
IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) ↔ IsSMulRegular M a ∧ IsSMulRegular M b := by
refine' ⟨_, _⟩
· | Mathlib.Algebra.Regular.SMul.114_0.jyFiiORgtZ4G1XR | /-- Two elements `a` and `b` are `M`-regular if and only if both products `a * b` and `b * a`
are `M`-regular. -/
theorem mul_and_mul_iff [Mul R] [IsScalarTower R R M] :
IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) ↔ IsSMulRegular M a ∧ IsSMulRegular M b | Mathlib_Algebra_Regular_SMul |
case refine'_1.intro
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝⁵ : SMul R M
inst✝⁴ : SMul R S
inst✝³ : SMul S M
inst✝² : IsScalarTower R S M
inst✝¹ : Mul R
inst✝ : IsScalarTower R R M
ab : IsSMulRegular M (a * b)
ba : IsSMulRegular M (b * a)
⊢ IsSMulRegular M a ∧ IsSMulRegular M b | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | refine' ⟨ba.of_mul, ab.of_mul⟩ | /-- Two elements `a` and `b` are `M`-regular if and only if both products `a * b` and `b * a`
are `M`-regular. -/
theorem mul_and_mul_iff [Mul R] [IsScalarTower R R M] :
IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) ↔ IsSMulRegular M a ∧ IsSMulRegular M b := by
refine' ⟨_, _⟩
· rintro ⟨ab, ba⟩
| Mathlib.Algebra.Regular.SMul.114_0.jyFiiORgtZ4G1XR | /-- Two elements `a` and `b` are `M`-regular if and only if both products `a * b` and `b * a`
are `M`-regular. -/
theorem mul_and_mul_iff [Mul R] [IsScalarTower R R M] :
IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) ↔ IsSMulRegular M a ∧ IsSMulRegular M b | Mathlib_Algebra_Regular_SMul |
case refine'_2
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝⁵ : SMul R M
inst✝⁴ : SMul R S
inst✝³ : SMul S M
inst✝² : IsScalarTower R S M
inst✝¹ : Mul R
inst✝ : IsScalarTower R R M
⊢ IsSMulRegular M a ∧ IsSMulRegular M b → IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | rintro ⟨ha, hb⟩ | /-- Two elements `a` and `b` are `M`-regular if and only if both products `a * b` and `b * a`
are `M`-regular. -/
theorem mul_and_mul_iff [Mul R] [IsScalarTower R R M] :
IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) ↔ IsSMulRegular M a ∧ IsSMulRegular M b := by
refine' ⟨_, _⟩
· rintro ⟨ab, ba⟩
refine' ⟨... | Mathlib.Algebra.Regular.SMul.114_0.jyFiiORgtZ4G1XR | /-- Two elements `a` and `b` are `M`-regular if and only if both products `a * b` and `b * a`
are `M`-regular. -/
theorem mul_and_mul_iff [Mul R] [IsScalarTower R R M] :
IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) ↔ IsSMulRegular M a ∧ IsSMulRegular M b | Mathlib_Algebra_Regular_SMul |
case refine'_2.intro
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝⁵ : SMul R M
inst✝⁴ : SMul R S
inst✝³ : SMul S M
inst✝² : IsScalarTower R S M
inst✝¹ : Mul R
inst✝ : IsScalarTower R R M
ha : IsSMulRegular M a
hb : IsSMulRegular M b
⊢ IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | exact ⟨ha.mul hb, hb.mul ha⟩ | /-- Two elements `a` and `b` are `M`-regular if and only if both products `a * b` and `b * a`
are `M`-regular. -/
theorem mul_and_mul_iff [Mul R] [IsScalarTower R R M] :
IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) ↔ IsSMulRegular M a ∧ IsSMulRegular M b := by
refine' ⟨_, _⟩
· rintro ⟨ab, ba⟩
refine' ⟨... | Mathlib.Algebra.Regular.SMul.114_0.jyFiiORgtZ4G1XR | /-- Two elements `a` and `b` are `M`-regular if and only if both products `a * b` and `b * a`
are `M`-regular. -/
theorem mul_and_mul_iff [Mul R] [IsScalarTower R R M] :
IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) ↔ IsSMulRegular M a ∧ IsSMulRegular M b | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a✝ b✝ : R
s : S
inst✝¹ : Monoid R
inst✝ : MulAction R M
a b : M
ab : (fun x => 1 • x) a = (fun x => 1 • x) b
⊢ a = b | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | dsimp only [Function.comp_def] at ab | /-- One is always `M`-regular. -/
@[simp]
theorem one : IsSMulRegular M (1 : R) := fun a b ab => by
| Mathlib.Algebra.Regular.SMul.133_0.jyFiiORgtZ4G1XR | /-- One is always `M`-regular. -/
@[simp]
theorem one : IsSMulRegular M (1 : R) | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a✝ b✝ : R
s : S
inst✝¹ : Monoid R
inst✝ : MulAction R M
a b : M
ab : 1 • a = 1 • b
⊢ a = b | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | rw [one_smul, one_smul] at ab | /-- One is always `M`-regular. -/
@[simp]
theorem one : IsSMulRegular M (1 : R) := fun a b ab => by
dsimp only [Function.comp_def] at ab
| Mathlib.Algebra.Regular.SMul.133_0.jyFiiORgtZ4G1XR | /-- One is always `M`-regular. -/
@[simp]
theorem one : IsSMulRegular M (1 : R) | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a✝ b✝ : R
s : S
inst✝¹ : Monoid R
inst✝ : MulAction R M
a b : M
ab : a = b
⊢ a = b | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | assumption | /-- One is always `M`-regular. -/
@[simp]
theorem one : IsSMulRegular M (1 : R) := fun a b ab => by
dsimp only [Function.comp_def] at ab
rw [one_smul, one_smul] at ab
| Mathlib.Algebra.Regular.SMul.133_0.jyFiiORgtZ4G1XR | /-- One is always `M`-regular. -/
@[simp]
theorem one : IsSMulRegular M (1 : R) | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝¹ : Monoid R
inst✝ : MulAction R M
h : a * b = 1
⊢ IsSMulRegular M (?m.10627 h * b) | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | rw [h] | /-- An element of `R` admitting a left inverse is `M`-regular. -/
theorem of_mul_eq_one (h : a * b = 1) : IsSMulRegular M b :=
of_mul
(by
| Mathlib.Algebra.Regular.SMul.143_0.jyFiiORgtZ4G1XR | /-- An element of `R` admitting a left inverse is `M`-regular. -/
theorem of_mul_eq_one (h : a * b = 1) : IsSMulRegular M b | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝¹ : Monoid R
inst✝ : MulAction R M
h : a * b = 1
⊢ IsSMulRegular M 1 | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | exact one M | /-- An element of `R` admitting a left inverse is `M`-regular. -/
theorem of_mul_eq_one (h : a * b = 1) : IsSMulRegular M b :=
of_mul
(by
rw [h]
| Mathlib.Algebra.Regular.SMul.143_0.jyFiiORgtZ4G1XR | /-- An element of `R` admitting a left inverse is `M`-regular. -/
theorem of_mul_eq_one (h : a * b = 1) : IsSMulRegular M b | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝¹ : Monoid R
inst✝ : MulAction R M
n : ℕ
ra : IsSMulRegular M a
⊢ IsSMulRegular M (a ^ n) | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | induction' n with n hn | /-- Any power of an `M`-regular element is `M`-regular. -/
theorem pow (n : ℕ) (ra : IsSMulRegular M a) : IsSMulRegular M (a ^ n) := by
| Mathlib.Algebra.Regular.SMul.151_0.jyFiiORgtZ4G1XR | /-- Any power of an `M`-regular element is `M`-regular. -/
theorem pow (n : ℕ) (ra : IsSMulRegular M a) : IsSMulRegular M (a ^ n) | Mathlib_Algebra_Regular_SMul |
case zero
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝¹ : Monoid R
inst✝ : MulAction R M
ra : IsSMulRegular M a
⊢ IsSMulRegular M (a ^ Nat.zero) | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | rw [pow_zero] | /-- Any power of an `M`-regular element is `M`-regular. -/
theorem pow (n : ℕ) (ra : IsSMulRegular M a) : IsSMulRegular M (a ^ n) := by
induction' n with n hn
· | Mathlib.Algebra.Regular.SMul.151_0.jyFiiORgtZ4G1XR | /-- Any power of an `M`-regular element is `M`-regular. -/
theorem pow (n : ℕ) (ra : IsSMulRegular M a) : IsSMulRegular M (a ^ n) | Mathlib_Algebra_Regular_SMul |
case zero
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝¹ : Monoid R
inst✝ : MulAction R M
ra : IsSMulRegular M a
⊢ IsSMulRegular M 1 | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | simp only [one] | /-- Any power of an `M`-regular element is `M`-regular. -/
theorem pow (n : ℕ) (ra : IsSMulRegular M a) : IsSMulRegular M (a ^ n) := by
induction' n with n hn
· rw [pow_zero]; | Mathlib.Algebra.Regular.SMul.151_0.jyFiiORgtZ4G1XR | /-- Any power of an `M`-regular element is `M`-regular. -/
theorem pow (n : ℕ) (ra : IsSMulRegular M a) : IsSMulRegular M (a ^ n) | Mathlib_Algebra_Regular_SMul |
case succ
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝¹ : Monoid R
inst✝ : MulAction R M
ra : IsSMulRegular M a
n : ℕ
hn : IsSMulRegular M (a ^ n)
⊢ IsSMulRegular M (a ^ Nat.succ n) | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | rw [pow_succ] | /-- Any power of an `M`-regular element is `M`-regular. -/
theorem pow (n : ℕ) (ra : IsSMulRegular M a) : IsSMulRegular M (a ^ n) := by
induction' n with n hn
· rw [pow_zero]; simp only [one]
· | Mathlib.Algebra.Regular.SMul.151_0.jyFiiORgtZ4G1XR | /-- Any power of an `M`-regular element is `M`-regular. -/
theorem pow (n : ℕ) (ra : IsSMulRegular M a) : IsSMulRegular M (a ^ n) | Mathlib_Algebra_Regular_SMul |
case succ
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝¹ : Monoid R
inst✝ : MulAction R M
ra : IsSMulRegular M a
n : ℕ
hn : IsSMulRegular M (a ^ n)
⊢ IsSMulRegular M (a * a ^ n) | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | exact (ra.smul_iff (a ^ n)).mpr hn | /-- Any power of an `M`-regular element is `M`-regular. -/
theorem pow (n : ℕ) (ra : IsSMulRegular M a) : IsSMulRegular M (a ^ n) := by
induction' n with n hn
· rw [pow_zero]; simp only [one]
· rw [pow_succ]
| Mathlib.Algebra.Regular.SMul.151_0.jyFiiORgtZ4G1XR | /-- Any power of an `M`-regular element is `M`-regular. -/
theorem pow (n : ℕ) (ra : IsSMulRegular M a) : IsSMulRegular M (a ^ n) | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝¹ : Monoid R
inst✝ : MulAction R M
n : ℕ
n0 : 0 < n
⊢ IsSMulRegular M (a ^ n) ↔ IsSMulRegular M a | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | refine' ⟨_, pow n⟩ | /-- An element `a` is `M`-regular if and only if a positive power of `a` is `M`-regular. -/
theorem pow_iff {n : ℕ} (n0 : 0 < n) : IsSMulRegular M (a ^ n) ↔ IsSMulRegular M a := by
| Mathlib.Algebra.Regular.SMul.159_0.jyFiiORgtZ4G1XR | /-- An element `a` is `M`-regular if and only if a positive power of `a` is `M`-regular. -/
theorem pow_iff {n : ℕ} (n0 : 0 < n) : IsSMulRegular M (a ^ n) ↔ IsSMulRegular M a | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝¹ : Monoid R
inst✝ : MulAction R M
n : ℕ
n0 : 0 < n
⊢ IsSMulRegular M (a ^ n) → IsSMulRegular M a | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | rw [← Nat.succ_pred_eq_of_pos n0, pow_succ', ← smul_eq_mul] | /-- An element `a` is `M`-regular if and only if a positive power of `a` is `M`-regular. -/
theorem pow_iff {n : ℕ} (n0 : 0 < n) : IsSMulRegular M (a ^ n) ↔ IsSMulRegular M a := by
refine' ⟨_, pow n⟩
| Mathlib.Algebra.Regular.SMul.159_0.jyFiiORgtZ4G1XR | /-- An element `a` is `M`-regular if and only if a positive power of `a` is `M`-regular. -/
theorem pow_iff {n : ℕ} (n0 : 0 < n) : IsSMulRegular M (a ^ n) ↔ IsSMulRegular M a | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝¹ : Monoid R
inst✝ : MulAction R M
n : ℕ
n0 : 0 < n
⊢ IsSMulRegular M (a ^ Nat.pred n • a) → IsSMulRegular M a | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | exact of_smul _ | /-- An element `a` is `M`-regular if and only if a positive power of `a` is `M`-regular. -/
theorem pow_iff {n : ℕ} (n0 : 0 < n) : IsSMulRegular M (a ^ n) ↔ IsSMulRegular M a := by
refine' ⟨_, pow n⟩
rw [← Nat.succ_pred_eq_of_pos n0, pow_succ', ← smul_eq_mul]
| Mathlib.Algebra.Regular.SMul.159_0.jyFiiORgtZ4G1XR | /-- An element `a` is `M`-regular if and only if a positive power of `a` is `M`-regular. -/
theorem pow_iff {n : ℕ} (n0 : 0 < n) : IsSMulRegular M (a ^ n) ↔ IsSMulRegular M a | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝⁴ : Monoid S
inst✝³ : SMul R M
inst✝² : SMul R S
inst✝¹ : MulAction S M
inst✝ : IsScalarTower R S M
h : a • s = 1
⊢ IsSMulRegular M (a • s) | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | rw [h] | /-- An element of `S` admitting a left inverse in `R` is `M`-regular. -/
theorem of_smul_eq_one (h : a • s = 1) : IsSMulRegular M s :=
of_smul a
(by
| Mathlib.Algebra.Regular.SMul.172_0.jyFiiORgtZ4G1XR | /-- An element of `S` admitting a left inverse in `R` is `M`-regular. -/
theorem of_smul_eq_one (h : a • s = 1) : IsSMulRegular M s | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝⁴ : Monoid S
inst✝³ : SMul R M
inst✝² : SMul R S
inst✝¹ : MulAction S M
inst✝ : IsScalarTower R S M
h : a • s = 1
⊢ IsSMulRegular M 1 | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | exact one M | /-- An element of `S` admitting a left inverse in `R` is `M`-regular. -/
theorem of_smul_eq_one (h : a • s = 1) : IsSMulRegular M s :=
of_smul a
(by
rw [h]
| Mathlib.Algebra.Regular.SMul.172_0.jyFiiORgtZ4G1XR | /-- An element of `S` admitting a left inverse in `R` is `M`-regular. -/
theorem of_smul_eq_one (h : a • s = 1) : IsSMulRegular M s | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a✝ b✝ : R
s : S
inst✝⁶ : MonoidWithZero R
inst✝⁵ : MonoidWithZero S
inst✝⁴ : Zero M
inst✝³ : MulActionWithZero R M
inst✝² : MulActionWithZero R S
inst✝¹ : MulActionWithZero S M
inst✝ : IsScalarTower R S M
h : IsSMulRegular M 0
a b : M
⊢ (fun x => 0 • x) a = (fun x => 0 • x) b | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | dsimp only [Function.comp_def] | /-- The element `0` is `M`-regular if and only if `M` is trivial. -/
protected theorem subsingleton (h : IsSMulRegular M (0 : R)) : Subsingleton M :=
⟨fun a b => h (by | Mathlib.Algebra.Regular.SMul.187_0.jyFiiORgtZ4G1XR | /-- The element `0` is `M`-regular if and only if `M` is trivial. -/
protected theorem subsingleton (h : IsSMulRegular M (0 : R)) : Subsingleton M | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a✝ b✝ : R
s : S
inst✝⁶ : MonoidWithZero R
inst✝⁵ : MonoidWithZero S
inst✝⁴ : Zero M
inst✝³ : MulActionWithZero R M
inst✝² : MulActionWithZero R S
inst✝¹ : MulActionWithZero S M
inst✝ : IsScalarTower R S M
h : IsSMulRegular M 0
a b : M
⊢ 0 • a = 0 • b | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | repeat' rw [MulActionWithZero.zero_smul] | /-- The element `0` is `M`-regular if and only if `M` is trivial. -/
protected theorem subsingleton (h : IsSMulRegular M (0 : R)) : Subsingleton M :=
⟨fun a b => h (by dsimp only [Function.comp_def]; | Mathlib.Algebra.Regular.SMul.187_0.jyFiiORgtZ4G1XR | /-- The element `0` is `M`-regular if and only if `M` is trivial. -/
protected theorem subsingleton (h : IsSMulRegular M (0 : R)) : Subsingleton M | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a✝ b✝ : R
s : S
inst✝⁶ : MonoidWithZero R
inst✝⁵ : MonoidWithZero S
inst✝⁴ : Zero M
inst✝³ : MulActionWithZero R M
inst✝² : MulActionWithZero R S
inst✝¹ : MulActionWithZero S M
inst✝ : IsScalarTower R S M
h : IsSMulRegular M 0
a b : M
⊢ 0 • a = 0 • b | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | rw [MulActionWithZero.zero_smul] | /-- The element `0` is `M`-regular if and only if `M` is trivial. -/
protected theorem subsingleton (h : IsSMulRegular M (0 : R)) : Subsingleton M :=
⟨fun a b => h (by dsimp only [Function.comp_def]; repeat' | Mathlib.Algebra.Regular.SMul.187_0.jyFiiORgtZ4G1XR | /-- The element `0` is `M`-regular if and only if `M` is trivial. -/
protected theorem subsingleton (h : IsSMulRegular M (0 : R)) : Subsingleton M | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a✝ b✝ : R
s : S
inst✝⁶ : MonoidWithZero R
inst✝⁵ : MonoidWithZero S
inst✝⁴ : Zero M
inst✝³ : MulActionWithZero R M
inst✝² : MulActionWithZero R S
inst✝¹ : MulActionWithZero S M
inst✝ : IsScalarTower R S M
h : IsSMulRegular M 0
a b : M
⊢ 0 = 0 • b | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | rw [MulActionWithZero.zero_smul] | /-- The element `0` is `M`-regular if and only if `M` is trivial. -/
protected theorem subsingleton (h : IsSMulRegular M (0 : R)) : Subsingleton M :=
⟨fun a b => h (by dsimp only [Function.comp_def]; repeat' | Mathlib.Algebra.Regular.SMul.187_0.jyFiiORgtZ4G1XR | /-- The element `0` is `M`-regular if and only if `M` is trivial. -/
protected theorem subsingleton (h : IsSMulRegular M (0 : R)) : Subsingleton M | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝⁶ : MonoidWithZero R
inst✝⁵ : MonoidWithZero S
inst✝⁴ : Zero M
inst✝³ : MulActionWithZero R M
inst✝² : MulActionWithZero R S
inst✝¹ : MulActionWithZero S M
inst✝ : IsScalarTower R S M
⊢ ¬IsSMulRegular M 0 ↔ Nontrivial M | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | rw [nontrivial_iff, not_iff_comm, zero_iff_subsingleton, subsingleton_iff] | /-- The `0` element is not `M`-regular, on a non-trivial module. -/
theorem not_zero_iff : ¬IsSMulRegular M (0 : R) ↔ Nontrivial M := by
| Mathlib.Algebra.Regular.SMul.197_0.jyFiiORgtZ4G1XR | /-- The `0` element is not `M`-regular, on a non-trivial module. -/
theorem not_zero_iff : ¬IsSMulRegular M (0 : R) ↔ Nontrivial M | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝⁶ : MonoidWithZero R
inst✝⁵ : MonoidWithZero S
inst✝⁴ : Zero M
inst✝³ : MulActionWithZero R M
inst✝² : MulActionWithZero R S
inst✝¹ : MulActionWithZero S M
inst✝ : IsScalarTower R S M
⊢ (¬∃ x y, x ≠ y) ↔ ∀ (x y : M), x = y | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | push_neg | /-- The `0` element is not `M`-regular, on a non-trivial module. -/
theorem not_zero_iff : ¬IsSMulRegular M (0 : R) ↔ Nontrivial M := by
rw [nontrivial_iff, not_iff_comm, zero_iff_subsingleton, subsingleton_iff]
| Mathlib.Algebra.Regular.SMul.197_0.jyFiiORgtZ4G1XR | /-- The `0` element is not `M`-regular, on a non-trivial module. -/
theorem not_zero_iff : ¬IsSMulRegular M (0 : R) ↔ Nontrivial M | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝⁶ : MonoidWithZero R
inst✝⁵ : MonoidWithZero S
inst✝⁴ : Zero M
inst✝³ : MulActionWithZero R M
inst✝² : MulActionWithZero R S
inst✝¹ : MulActionWithZero S M
inst✝ : IsScalarTower R S M
⊢ (∀ (x y : M), x = y) ↔ ∀ (x y : M), x = y | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | exact Iff.rfl | /-- The `0` element is not `M`-regular, on a non-trivial module. -/
theorem not_zero_iff : ¬IsSMulRegular M (0 : R) ↔ Nontrivial M := by
rw [nontrivial_iff, not_iff_comm, zero_iff_subsingleton, subsingleton_iff]
push_neg
| Mathlib.Algebra.Regular.SMul.197_0.jyFiiORgtZ4G1XR | /-- The `0` element is not `M`-regular, on a non-trivial module. -/
theorem not_zero_iff : ¬IsSMulRegular M (0 : R) ↔ Nontrivial M | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝² : CommSemigroup R
inst✝¹ : SMul R M
inst✝ : IsScalarTower R R M
⊢ IsSMulRegular M (a * b) ↔ IsSMulRegular M a ∧ IsSMulRegular M b | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | rw [← mul_and_mul_iff] | /-- A product is `M`-regular if and only if the factors are. -/
theorem mul_iff : IsSMulRegular M (a * b) ↔ IsSMulRegular M a ∧ IsSMulRegular M b := by
| Mathlib.Algebra.Regular.SMul.220_0.jyFiiORgtZ4G1XR | /-- A product is `M`-regular if and only if the factors are. -/
theorem mul_iff : IsSMulRegular M (a * b) ↔ IsSMulRegular M a ∧ IsSMulRegular M b | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝² : CommSemigroup R
inst✝¹ : SMul R M
inst✝ : IsScalarTower R R M
⊢ IsSMulRegular M (a * b) ↔ IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | exact ⟨fun ab => ⟨ab, by rwa [mul_comm]⟩, fun rab => rab.1⟩ | /-- A product is `M`-regular if and only if the factors are. -/
theorem mul_iff : IsSMulRegular M (a * b) ↔ IsSMulRegular M a ∧ IsSMulRegular M b := by
rw [← mul_and_mul_iff]
| Mathlib.Algebra.Regular.SMul.220_0.jyFiiORgtZ4G1XR | /-- A product is `M`-regular if and only if the factors are. -/
theorem mul_iff : IsSMulRegular M (a * b) ↔ IsSMulRegular M a ∧ IsSMulRegular M b | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝² : CommSemigroup R
inst✝¹ : SMul R M
inst✝ : IsScalarTower R R M
ab : IsSMulRegular M (a * b)
⊢ IsSMulRegular M (b * a) | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | rwa [mul_comm] | /-- A product is `M`-regular if and only if the factors are. -/
theorem mul_iff : IsSMulRegular M (a * b) ↔ IsSMulRegular M a ∧ IsSMulRegular M b := by
rw [← mul_and_mul_iff]
exact ⟨fun ab => ⟨ab, by | Mathlib.Algebra.Regular.SMul.220_0.jyFiiORgtZ4G1XR | /-- A product is `M`-regular if and only if the factors are. -/
theorem mul_iff : IsSMulRegular M (a * b) ↔ IsSMulRegular M a ∧ IsSMulRegular M b | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
G : Type u_4
inst✝¹ : Group G
inst✝ : MulAction G R
g : G
⊢ IsSMulRegular R g | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | intro x y h | /-- An element of a group acting on a Type is regular. This relies on the availability
of the inverse given by groups, since there is no `LeftCancelSMul` typeclass. -/
theorem isSMulRegular_of_group [MulAction G R] (g : G) : IsSMulRegular R g := by
| Mathlib.Algebra.Regular.SMul.234_0.jyFiiORgtZ4G1XR | /-- An element of a group acting on a Type is regular. This relies on the availability
of the inverse given by groups, since there is no `LeftCancelSMul` typeclass. -/
theorem isSMulRegular_of_group [MulAction G R] (g : G) : IsSMulRegular R g | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
G : Type u_4
inst✝¹ : Group G
inst✝ : MulAction G R
g : G
x y : R
h : (fun x => g • x) x = (fun x => g • x) y
⊢ x = y | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | convert congr_arg (g⁻¹ • ·) h using 1 | /-- An element of a group acting on a Type is regular. This relies on the availability
of the inverse given by groups, since there is no `LeftCancelSMul` typeclass. -/
theorem isSMulRegular_of_group [MulAction G R] (g : G) : IsSMulRegular R g := by
intro x y h
| Mathlib.Algebra.Regular.SMul.234_0.jyFiiORgtZ4G1XR | /-- An element of a group acting on a Type is regular. This relies on the availability
of the inverse given by groups, since there is no `LeftCancelSMul` typeclass. -/
theorem isSMulRegular_of_group [MulAction G R] (g : G) : IsSMulRegular R g | Mathlib_Algebra_Regular_SMul |
case h.e'_2
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
G : Type u_4
inst✝¹ : Group G
inst✝ : MulAction G R
g : G
x y : R
h : (fun x => g • x) x = (fun x => g • x) y
⊢ x = g⁻¹ • (fun x => g • x) x | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | simp [← smul_assoc] | /-- An element of a group acting on a Type is regular. This relies on the availability
of the inverse given by groups, since there is no `LeftCancelSMul` typeclass. -/
theorem isSMulRegular_of_group [MulAction G R] (g : G) : IsSMulRegular R g := by
intro x y h
convert congr_arg (g⁻¹ • ·) h using 1 <;> | Mathlib.Algebra.Regular.SMul.234_0.jyFiiORgtZ4G1XR | /-- An element of a group acting on a Type is regular. This relies on the availability
of the inverse given by groups, since there is no `LeftCancelSMul` typeclass. -/
theorem isSMulRegular_of_group [MulAction G R] (g : G) : IsSMulRegular R g | Mathlib_Algebra_Regular_SMul |
case h.e'_3
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
G : Type u_4
inst✝¹ : Group G
inst✝ : MulAction G R
g : G
x y : R
h : (fun x => g • x) x = (fun x => g • x) y
⊢ y = g⁻¹ • (fun x => g • x) y | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | simp [← smul_assoc] | /-- An element of a group acting on a Type is regular. This relies on the availability
of the inverse given by groups, since there is no `LeftCancelSMul` typeclass. -/
theorem isSMulRegular_of_group [MulAction G R] (g : G) : IsSMulRegular R g := by
intro x y h
convert congr_arg (g⁻¹ • ·) h using 1 <;> | Mathlib.Algebra.Regular.SMul.234_0.jyFiiORgtZ4G1XR | /-- An element of a group acting on a Type is regular. This relies on the availability
of the inverse given by groups, since there is no `LeftCancelSMul` typeclass. -/
theorem isSMulRegular_of_group [MulAction G R] (g : G) : IsSMulRegular R g | Mathlib_Algebra_Regular_SMul |
R : Type u_1
S : Type u_2
M : Type u_3
a b : R
s : S
inst✝¹ : Monoid R
inst✝ : MulAction R M
ua : IsUnit a
⊢ IsSMulRegular M a | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | rcases ua with ⟨a, rfl⟩ | /-- A unit is `M`-regular. -/
theorem IsUnit.isSMulRegular (ua : IsUnit a) : IsSMulRegular M a := by
| Mathlib.Algebra.Regular.SMul.252_0.jyFiiORgtZ4G1XR | /-- A unit is `M`-regular. -/
theorem IsUnit.isSMulRegular (ua : IsUnit a) : IsSMulRegular M a | Mathlib_Algebra_Regular_SMul |
case intro
R : Type u_1
S : Type u_2
M : Type u_3
b : R
s : S
inst✝¹ : Monoid R
inst✝ : MulAction R M
a : Rˣ
⊢ IsSMulRegular M ↑a | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.SMulWithZero
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fd... | exact a.isSMulRegular M | /-- A unit is `M`-regular. -/
theorem IsUnit.isSMulRegular (ua : IsUnit a) : IsSMulRegular M a := by
rcases ua with ⟨a, rfl⟩
| Mathlib.Algebra.Regular.SMul.252_0.jyFiiORgtZ4G1XR | /-- A unit is `M`-regular. -/
theorem IsUnit.isSMulRegular (ua : IsUnit a) : IsSMulRegular M a | Mathlib_Algebra_Regular_SMul |
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝¹ : NonUnitalNonAssocSemiring α
inst✝ : NonUnitalNonAssocSemiring β
f g : α →ₙ+* β
h : (fun f => f.toFun) f = (fun f => f.toFun) g
⊢ f = g | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | cases f | instance : NonUnitalRingHomClass (α →ₙ+* β) α β where
coe f := f.toFun
coe_injective' f g h := by
| Mathlib.Algebra.Ring.Hom.Defs.104_0.KyHvVYrIs9pW9ZQ | instance : NonUnitalRingHomClass (α →ₙ+* β) α β where
coe f | Mathlib_Algebra_Ring_Hom_Defs |
case mk
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝¹ : NonUnitalNonAssocSemiring α
inst✝ : NonUnitalNonAssocSemiring β
g : α →ₙ+* β
toMulHom✝ : α →ₙ* β
map_zero'✝ : MulHom.toFun toMulHom✝ 0 = 0
map_add'✝ : ∀ (x y : α), MulHom.toFun toMulHom✝ (x + y) = MulHom.toFun toMulHom✝ x + MulHom.toFun toMulHom✝ y
h ... | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | cases g | instance : NonUnitalRingHomClass (α →ₙ+* β) α β where
coe f := f.toFun
coe_injective' f g h := by
cases f
| Mathlib.Algebra.Ring.Hom.Defs.104_0.KyHvVYrIs9pW9ZQ | instance : NonUnitalRingHomClass (α →ₙ+* β) α β where
coe f | Mathlib_Algebra_Ring_Hom_Defs |
case mk.mk
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝¹ : NonUnitalNonAssocSemiring α
inst✝ : NonUnitalNonAssocSemiring β
toMulHom✝¹ : α →ₙ* β
map_zero'✝¹ : MulHom.toFun toMulHom✝¹ 0 = 0
map_add'✝¹ : ∀ (x y : α), MulHom.toFun toMulHom✝¹ (x + y) = MulHom.toFun toMulHom✝¹ x + MulHom.toFun toMulHom✝¹ y
toMul... | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | congr | instance : NonUnitalRingHomClass (α →ₙ+* β) α β where
coe f := f.toFun
coe_injective' f g h := by
cases f
cases g
| Mathlib.Algebra.Ring.Hom.Defs.104_0.KyHvVYrIs9pW9ZQ | instance : NonUnitalRingHomClass (α →ₙ+* β) α β where
coe f | Mathlib_Algebra_Ring_Hom_Defs |
case mk.mk.e_toMulHom
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝¹ : NonUnitalNonAssocSemiring α
inst✝ : NonUnitalNonAssocSemiring β
toMulHom✝¹ : α →ₙ* β
map_zero'✝¹ : MulHom.toFun toMulHom✝¹ 0 = 0
map_add'✝¹ : ∀ (x y : α), MulHom.toFun toMulHom✝¹ (x + y) = MulHom.toFun toMulHom✝¹ x + MulHom.toFun toMulHo... | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | apply FunLike.coe_injective' | instance : NonUnitalRingHomClass (α →ₙ+* β) α β where
coe f := f.toFun
coe_injective' f g h := by
cases f
cases g
congr
| Mathlib.Algebra.Ring.Hom.Defs.104_0.KyHvVYrIs9pW9ZQ | instance : NonUnitalRingHomClass (α →ₙ+* β) α β where
coe f | Mathlib_Algebra_Ring_Hom_Defs |
case mk.mk.e_toMulHom.a
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝¹ : NonUnitalNonAssocSemiring α
inst✝ : NonUnitalNonAssocSemiring β
toMulHom✝¹ : α →ₙ* β
map_zero'✝¹ : MulHom.toFun toMulHom✝¹ 0 = 0
map_add'✝¹ : ∀ (x y : α), MulHom.toFun toMulHom✝¹ (x + y) = MulHom.toFun toMulHom✝¹ x + MulHom.toFun toMul... | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | exact h | instance : NonUnitalRingHomClass (α →ₙ+* β) α β where
coe f := f.toFun
coe_injective' f g h := by
cases f
cases g
congr
apply FunLike.coe_injective'
| Mathlib.Algebra.Ring.Hom.Defs.104_0.KyHvVYrIs9pW9ZQ | instance : NonUnitalRingHomClass (α →ₙ+* β) α β where
coe f | Mathlib_Algebra_Ring_Hom_Defs |
F : Type u_1
α✝ : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : NonUnitalNonAssocSemiring α✝
inst✝¹ : NonUnitalNonAssocSemiring β
α : Type u_5
inst✝ : NonUnitalNonAssocSemiring α
⊢ α →ₙ+* α | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | refine' { toFun := id.. } | /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/
protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by
| Mathlib.Algebra.Ring.Hom.Defs.199_0.KyHvVYrIs9pW9ZQ | /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/
protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α | Mathlib_Algebra_Ring_Hom_Defs |
case refine'_1
F : Type u_1
α✝ : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : NonUnitalNonAssocSemiring α✝
inst✝¹ : NonUnitalNonAssocSemiring β
α : Type u_5
inst✝ : NonUnitalNonAssocSemiring α
⊢ ∀ (x y : α), id (x * y) = id x * id y | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | intros | /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/
protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by
refine' { toFun := id.. } <;> | Mathlib.Algebra.Ring.Hom.Defs.199_0.KyHvVYrIs9pW9ZQ | /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/
protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α | Mathlib_Algebra_Ring_Hom_Defs |
case refine'_2
F : Type u_1
α✝ : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : NonUnitalNonAssocSemiring α✝
inst✝¹ : NonUnitalNonAssocSemiring β
α : Type u_5
inst✝ : NonUnitalNonAssocSemiring α
⊢ MulHom.toFun { toFun := id, map_mul' := (_ : ∀ (x y : α), id (x * y) = id x * id y) } 0 = 0 | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | intros | /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/
protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by
refine' { toFun := id.. } <;> | Mathlib.Algebra.Ring.Hom.Defs.199_0.KyHvVYrIs9pW9ZQ | /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/
protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α | Mathlib_Algebra_Ring_Hom_Defs |
case refine'_3
F : Type u_1
α✝ : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : NonUnitalNonAssocSemiring α✝
inst✝¹ : NonUnitalNonAssocSemiring β
α : Type u_5
inst✝ : NonUnitalNonAssocSemiring α
⊢ ∀ (x y : α),
MulHom.toFun { toFun := id, map_mul' := (_ : ∀ (x y : α), id (x * y) = id x * id y) } (x + y) =
MulHom.... | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | intros | /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/
protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by
refine' { toFun := id.. } <;> | Mathlib.Algebra.Ring.Hom.Defs.199_0.KyHvVYrIs9pW9ZQ | /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/
protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α | Mathlib_Algebra_Ring_Hom_Defs |
case refine'_1
F : Type u_1
α✝ : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : NonUnitalNonAssocSemiring α✝
inst✝¹ : NonUnitalNonAssocSemiring β
α : Type u_5
inst✝ : NonUnitalNonAssocSemiring α
x✝ y✝ : α
⊢ id (x✝ * y✝) = id x✝ * id y✝ | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | rfl | /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/
protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by
refine' { toFun := id.. } <;> intros <;> | Mathlib.Algebra.Ring.Hom.Defs.199_0.KyHvVYrIs9pW9ZQ | /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/
protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α | Mathlib_Algebra_Ring_Hom_Defs |
case refine'_2
F : Type u_1
α✝ : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : NonUnitalNonAssocSemiring α✝
inst✝¹ : NonUnitalNonAssocSemiring β
α : Type u_5
inst✝ : NonUnitalNonAssocSemiring α
⊢ MulHom.toFun { toFun := id, map_mul' := (_ : ∀ (x y : α), id (x * y) = id (x * y)) } 0 = 0 | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | rfl | /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/
protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by
refine' { toFun := id.. } <;> intros <;> | Mathlib.Algebra.Ring.Hom.Defs.199_0.KyHvVYrIs9pW9ZQ | /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/
protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α | Mathlib_Algebra_Ring_Hom_Defs |
case refine'_3
F : Type u_1
α✝ : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : NonUnitalNonAssocSemiring α✝
inst✝¹ : NonUnitalNonAssocSemiring β
α : Type u_5
inst✝ : NonUnitalNonAssocSemiring α
x✝ y✝ : α
⊢ MulHom.toFun { toFun := id, map_mul' := (_ : ∀ (x y : α), id (x * y) = id (x * y)) } (x✝ + y✝) =
MulHom.toFun { ... | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | rfl | /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/
protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by
refine' { toFun := id.. } <;> intros <;> | Mathlib.Algebra.Ring.Hom.Defs.199_0.KyHvVYrIs9pW9ZQ | /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/
protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α | Mathlib_Algebra_Ring_Hom_Defs |
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : NonUnitalNonAssocSemiring α
inst✝¹ : NonUnitalNonAssocSemiring β
inst✝ : NonUnitalNonAssocSemiring γ
g✝ : β →ₙ+* γ
f : α →ₙ+* β
g : β →ₙ+* γ
⊢ comp g 0 = 0 | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | ext | @[simp]
theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 := by
| Mathlib.Algebra.Ring.Hom.Defs.272_0.KyHvVYrIs9pW9ZQ | @[simp]
theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 | Mathlib_Algebra_Ring_Hom_Defs |
case a
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : NonUnitalNonAssocSemiring α
inst✝¹ : NonUnitalNonAssocSemiring β
inst✝ : NonUnitalNonAssocSemiring γ
g✝ : β →ₙ+* γ
f : α →ₙ+* β
g : β →ₙ+* γ
x✝ : α
⊢ (comp g 0) x✝ = 0 x✝ | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | simp | @[simp]
theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 := by
ext
| Mathlib.Algebra.Ring.Hom.Defs.272_0.KyHvVYrIs9pW9ZQ | @[simp]
theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 | Mathlib_Algebra_Ring_Hom_Defs |
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : NonUnitalNonAssocSemiring α
inst✝¹ : NonUnitalNonAssocSemiring β
inst✝ : NonUnitalNonAssocSemiring γ
g : β →ₙ+* γ
f✝ f : α →ₙ+* β
⊢ comp 0 f = 0 | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | ext | @[simp]
theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 := by
| Mathlib.Algebra.Ring.Hom.Defs.278_0.KyHvVYrIs9pW9ZQ | @[simp]
theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 | Mathlib_Algebra_Ring_Hom_Defs |
case a
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : NonUnitalNonAssocSemiring α
inst✝¹ : NonUnitalNonAssocSemiring β
inst✝ : NonUnitalNonAssocSemiring γ
g : β →ₙ+* γ
f✝ f : α →ₙ+* β
x✝ : α
⊢ (comp 0 f) x✝ = 0 x✝ | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | rfl | @[simp]
theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 := by
ext
| Mathlib.Algebra.Ring.Hom.Defs.278_0.KyHvVYrIs9pW9ZQ | @[simp]
theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 | Mathlib_Algebra_Ring_Hom_Defs |
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : NonUnitalNonAssocSemiring α
inst✝¹ : NonUnitalNonAssocSemiring β
inst✝ : NonUnitalNonAssocSemiring γ
g✝ : β →ₙ+* γ
f : α →ₙ+* β
g : β →ₙ+* γ
f₁ f₂ : α →ₙ+* β
hg : Injective ⇑g
h : comp g f₁ = comp g f₂
x : α
⊢ g (f₁ x) = g (f₂ x) | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | rw [← comp_apply, h, comp_apply] | @[simp]
theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=
⟨fun h => ext fun x => hg <| by | Mathlib.Algebra.Ring.Hom.Defs.328_0.KyHvVYrIs9pW9ZQ | @[simp]
theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | Mathlib_Algebra_Ring_Hom_Defs |
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : NonAssocSemiring α
inst✝¹ : NonAssocSemiring β
inst✝ : RingHomClass F α β
f : F
a : α
⊢ f (bit1 a) = bit1 (f a) | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | simp [bit1] | set_option linter.deprecated false in
/-- Ring homomorphisms preserve `bit1`. -/
@[simp] lemma map_bit1 [NonAssocSemiring α] [NonAssocSemiring β] [RingHomClass F α β]
(f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) := by | Mathlib.Algebra.Ring.Hom.Defs.380_0.KyHvVYrIs9pW9ZQ | set_option linter.deprecated false in
/-- Ring homomorphisms preserve `bit1`. -/
@[simp] lemma map_bit1 [NonAssocSemiring α] [NonAssocSemiring β] [RingHomClass F α β]
(f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) | Mathlib_Algebra_Ring_Hom_Defs |
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
x✝¹ : NonAssocSemiring α
x✝ : NonAssocSemiring β
f g : α →+* β
h : (fun f => f.toFun) f = (fun f => f.toFun) g
⊢ f = g | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | cases f | instance instRingHomClass : RingHomClass (α →+* β) α β where
coe f := f.toFun
coe_injective' f g h := by
| Mathlib.Algebra.Ring.Hom.Defs.416_0.KyHvVYrIs9pW9ZQ | instance instRingHomClass : RingHomClass (α →+* β) α β where
coe f | Mathlib_Algebra_Ring_Hom_Defs |
case mk
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
x✝¹ : NonAssocSemiring α
x✝ : NonAssocSemiring β
g : α →+* β
toMonoidHom✝ : α →* β
map_zero'✝ : OneHom.toFun (↑toMonoidHom✝) 0 = 0
map_add'✝ :
∀ (x y : α), OneHom.toFun (↑toMonoidHom✝) (x + y) = OneHom.toFun (↑toMonoidHom✝) x + OneHom.toFun (↑toMonoidHom✝) y... | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | cases g | instance instRingHomClass : RingHomClass (α →+* β) α β where
coe f := f.toFun
coe_injective' f g h := by
cases f
| Mathlib.Algebra.Ring.Hom.Defs.416_0.KyHvVYrIs9pW9ZQ | instance instRingHomClass : RingHomClass (α →+* β) α β where
coe f | Mathlib_Algebra_Ring_Hom_Defs |
case mk.mk
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
x✝¹ : NonAssocSemiring α
x✝ : NonAssocSemiring β
toMonoidHom✝¹ : α →* β
map_zero'✝¹ : OneHom.toFun (↑toMonoidHom✝¹) 0 = 0
map_add'✝¹ :
∀ (x y : α), OneHom.toFun (↑toMonoidHom✝¹) (x + y) = OneHom.toFun (↑toMonoidHom✝¹) x + OneHom.toFun (↑toMonoidHom✝¹) y
t... | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | congr | instance instRingHomClass : RingHomClass (α →+* β) α β where
coe f := f.toFun
coe_injective' f g h := by
cases f
cases g
| Mathlib.Algebra.Ring.Hom.Defs.416_0.KyHvVYrIs9pW9ZQ | instance instRingHomClass : RingHomClass (α →+* β) α β where
coe f | Mathlib_Algebra_Ring_Hom_Defs |
case mk.mk.e_toMonoidHom
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
x✝¹ : NonAssocSemiring α
x✝ : NonAssocSemiring β
toMonoidHom✝¹ : α →* β
map_zero'✝¹ : OneHom.toFun (↑toMonoidHom✝¹) 0 = 0
map_add'✝¹ :
∀ (x y : α), OneHom.toFun (↑toMonoidHom✝¹) (x + y) = OneHom.toFun (↑toMonoidHom✝¹) x + OneHom.toFun (↑toMo... | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | apply FunLike.coe_injective' | instance instRingHomClass : RingHomClass (α →+* β) α β where
coe f := f.toFun
coe_injective' f g h := by
cases f
cases g
congr
| Mathlib.Algebra.Ring.Hom.Defs.416_0.KyHvVYrIs9pW9ZQ | instance instRingHomClass : RingHomClass (α →+* β) α β where
coe f | Mathlib_Algebra_Ring_Hom_Defs |
case mk.mk.e_toMonoidHom.a
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
x✝¹ : NonAssocSemiring α
x✝ : NonAssocSemiring β
toMonoidHom✝¹ : α →* β
map_zero'✝¹ : OneHom.toFun (↑toMonoidHom✝¹) 0 = 0
map_add'✝¹ :
∀ (x y : α), OneHom.toFun (↑toMonoidHom✝¹) (x + y) = OneHom.toFun (↑toMonoidHom✝¹) x + OneHom.toFun (↑to... | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | exact h | instance instRingHomClass : RingHomClass (α →+* β) α β where
coe f := f.toFun
coe_injective' f g h := by
cases f
cases g
congr
apply FunLike.coe_injective'
| Mathlib.Algebra.Ring.Hom.Defs.416_0.KyHvVYrIs9pW9ZQ | instance instRingHomClass : RingHomClass (α →+* β) α β where
coe f | Mathlib_Algebra_Ring_Hom_Defs |
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
x✝¹ : NonAssocSemiring α
x✝ : NonAssocSemiring β
f : α →+* β
⊢ ⇑(toMonoidWithZeroHom f) = ⇑f | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | rfl | theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f := by
| Mathlib.Algebra.Ring.Hom.Defs.474_0.KyHvVYrIs9pW9ZQ | theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f | Mathlib_Algebra_Ring_Hom_Defs |
F✝ : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
x✝¹ : NonAssocSemiring α
x✝ : NonAssocSemiring β
f✝ : α →+* β
x y : α
F : Type u_5
inst✝¹ : RingHomClass F α β
f : F
p : Prop
inst✝ : Decidable p
⊢ f (if p then 0 else 1) = if p then 0 else 1 | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | split_ifs with h | @[simp]
theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] :
f (ite p 0 1) = ite p 0 1 := by
| Mathlib.Algebra.Ring.Hom.Defs.573_0.KyHvVYrIs9pW9ZQ | @[simp]
theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] :
f (ite p 0 1) = ite p 0 1 | Mathlib_Algebra_Ring_Hom_Defs |
case pos
F✝ : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
x✝¹ : NonAssocSemiring α
x✝ : NonAssocSemiring β
f✝ : α →+* β
x y : α
F : Type u_5
inst✝¹ : RingHomClass F α β
f : F
p : Prop
inst✝ : Decidable p
h : p
⊢ f 0 = 0 | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | simp [h] | @[simp]
theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] :
f (ite p 0 1) = ite p 0 1 := by
split_ifs with h <;> | Mathlib.Algebra.Ring.Hom.Defs.573_0.KyHvVYrIs9pW9ZQ | @[simp]
theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] :
f (ite p 0 1) = ite p 0 1 | Mathlib_Algebra_Ring_Hom_Defs |
case neg
F✝ : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
x✝¹ : NonAssocSemiring α
x✝ : NonAssocSemiring β
f✝ : α →+* β
x y : α
F : Type u_5
inst✝¹ : RingHomClass F α β
f : F
p : Prop
inst✝ : Decidable p
h : ¬p
⊢ f 1 = 1 | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | simp [h] | @[simp]
theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] :
f (ite p 0 1) = ite p 0 1 := by
split_ifs with h <;> | Mathlib.Algebra.Ring.Hom.Defs.573_0.KyHvVYrIs9pW9ZQ | @[simp]
theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] :
f (ite p 0 1) = ite p 0 1 | Mathlib_Algebra_Ring_Hom_Defs |
F✝ : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
x✝¹ : NonAssocSemiring α
x✝ : NonAssocSemiring β
f✝ : α →+* β
x y : α
F : Type u_5
inst✝¹ : RingHomClass F α β
f : F
p : Prop
inst✝ : Decidable p
⊢ f (if p then 1 else 0) = if p then 1 else 0 | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | split_ifs with h | @[simp]
theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] :
f (ite p 1 0) = ite p 1 0 := by
| Mathlib.Algebra.Ring.Hom.Defs.579_0.KyHvVYrIs9pW9ZQ | @[simp]
theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] :
f (ite p 1 0) = ite p 1 0 | Mathlib_Algebra_Ring_Hom_Defs |
case pos
F✝ : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
x✝¹ : NonAssocSemiring α
x✝ : NonAssocSemiring β
f✝ : α →+* β
x y : α
F : Type u_5
inst✝¹ : RingHomClass F α β
f : F
p : Prop
inst✝ : Decidable p
h : p
⊢ f 1 = 1 | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | simp [h] | @[simp]
theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] :
f (ite p 1 0) = ite p 1 0 := by
split_ifs with h <;> | Mathlib.Algebra.Ring.Hom.Defs.579_0.KyHvVYrIs9pW9ZQ | @[simp]
theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] :
f (ite p 1 0) = ite p 1 0 | Mathlib_Algebra_Ring_Hom_Defs |
case neg
F✝ : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
x✝¹ : NonAssocSemiring α
x✝ : NonAssocSemiring β
f✝ : α →+* β
x y : α
F : Type u_5
inst✝¹ : RingHomClass F α β
f : F
p : Prop
inst✝ : Decidable p
h : ¬p
⊢ f 0 = 0 | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | simp [h] | @[simp]
theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] :
f (ite p 1 0) = ite p 1 0 := by
split_ifs with h <;> | Mathlib.Algebra.Ring.Hom.Defs.579_0.KyHvVYrIs9pW9ZQ | @[simp]
theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] :
f (ite p 1 0) = ite p 1 0 | Mathlib_Algebra_Ring_Hom_Defs |
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
x✝¹ : NonAssocSemiring α
x✝ : NonAssocSemiring β
f : α →+* β
x y : α
⊢ 0 = 1 ↔ f 1 = 0 | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | rw [map_one, eq_comm] | /-- `f : α →+* β` has a trivial codomain iff `f 1 = 0`. -/
theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 := by | Mathlib.Algebra.Ring.Hom.Defs.585_0.KyHvVYrIs9pW9ZQ | /-- `f : α →+* β` has a trivial codomain iff `f 1 = 0`. -/
theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 | Mathlib_Algebra_Ring_Hom_Defs |
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
x✝² : NonAssocSemiring α
x✝¹ : NonAssocSemiring β
f : α →+* β
x✝ y : α
h : f 1 = 0
x : α
⊢ f x = 0 | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | rw [← mul_one x, map_mul, h, mul_zero] | /-- `f : α →+* β` has a trivial codomain iff it has a trivial range. -/
theorem codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 :=
f.codomain_trivial_iff_map_one_eq_zero.trans
⟨fun h x => by | Mathlib.Algebra.Ring.Hom.Defs.589_0.KyHvVYrIs9pW9ZQ | /-- `f : α →+* β` has a trivial codomain iff it has a trivial range. -/
theorem codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 | Mathlib_Algebra_Ring_Hom_Defs |
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
x✝¹ : NonAssocSemiring α
x✝ : NonAssocSemiring β
f : α →+* β
x y : α
inst✝ : Nontrivial β
h : 1 = 0
⊢ f 1 = 0 | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
import Mathlib.Data.Pi.Algebra
#align_import algebra.hom.ring from "leanprove... | rw [h, map_zero] | /-- If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. -/
theorem domain_nontrivial [Nontrivial β] : Nontrivial α :=
⟨⟨1, 0, mt (fun h => show f 1 = 0 by | Mathlib.Algebra.Ring.Hom.Defs.600_0.KyHvVYrIs9pW9ZQ | /-- If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. -/
theorem domain_nontrivial [Nontrivial β] : Nontrivial α | Mathlib_Algebra_Ring_Hom_Defs |
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