File: Mathlib/Analysis/Normed/Algebra/TrivSqZeroExt.lean

full_name: _aux_Mathlib_Analysis_Normed_Algebra_TrivSqZeroExt___macroRules_termTsze_1
code: local notation "tsze" => TrivSqZeroExt
goal: Lean.Macro

full_name: _aux_Mathlib_Analysis_Normed_Algebra_TrivSqZeroExt___unexpand_TrivSqZeroExt_1
code: local notation "tsze" => TrivSqZeroExt
goal: Lean.PrettyPrinter.Unexpander

full_name: TrivSqZeroExt.fst_expSeries
code: @[simp] theorem fst_expSeries (x : tsze R M) (n : ℕ) :
    fst (expSeries 𝕜 (tsze R M) n fun _ => x) = expSeries 𝕜 R n fun _ => x.fst 
goal: ((NormedSpace.expSeries 𝕜 (tsze R M) n) fun x_1 => x).fst = (NormedSpace.expSeries 𝕜 R n) fun x_1 => x.fst

full_name: TrivSqZeroExt.snd_expSeries_of_smul_comm
code: theorem snd_expSeries_of_smul_comm
    (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) (n : ℕ) :
    snd (expSeries 𝕜 (tsze R M) (n + 1) fun _ => x) = (expSeries 𝕜 R n fun _ => x.fst) • x.snd 
goal: ((NormedSpace.expSeries 𝕜 (tsze R M) (n + 1)) fun x_1 => x).snd =
  ((NormedSpace.expSeries 𝕜 R n) fun x_1 => x.fst) • x.snd

full_name: TrivSqZeroExt.hasSum_snd_expSeries_of_smul_comm
code: /-- If `exp R x.fst` converges to `e` then `(exp R x).snd` converges to `e • x.snd`. -/
theorem hasSum_snd_expSeries_of_smul_comm (x : tsze R M)
    (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) {e : R}
    (h : HasSum (fun n => expSeries 𝕜 R n fun _ => x.fst) e) :
    HasSum (fun n => snd (expSeries 𝕜 (tsze R M) n fun _ => x)) (e • x.snd) 
goal: HasSum (fun n => ((NormedSpace.expSeries 𝕜 (tsze R M) n) fun x_1 => x).snd) (e • x.snd)

full_name: TrivSqZeroExt.hasSum_expSeries_of_smul_comm
code: /-- If `exp R x.fst` converges to `e` then `exp R x` converges to `inl e + inr (e • x.snd)`. -/
theorem hasSum_expSeries_of_smul_comm
    (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd)
    {e : R} (h : HasSum (fun n => expSeries 𝕜 R n fun _ => x.fst) e) :
    HasSum (fun n => expSeries 𝕜 (tsze R M) n fun _ => x) (inl e + inr (e • x.snd)) 
goal: HasSum (fun n => (NormedSpace.expSeries 𝕜 (tsze R M) n) fun x_1 => x)
  (TrivSqZeroExt.inl e + TrivSqZeroExt.inr (e • x.snd))

full_name: TrivSqZeroExt.exp_def_of_smul_comm
code: theorem exp_def_of_smul_comm (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) :
    exp 𝕜 x = inl (exp 𝕜 x.fst) + inr (exp 𝕜 x.fst • x.snd) 
goal: NormedSpace.exp 𝕜 x = TrivSqZeroExt.inl (NormedSpace.exp 𝕜 x.fst) + TrivSqZeroExt.inr (NormedSpace.exp 𝕜 x.fst • x.snd)

full_name: TrivSqZeroExt.exp_inl
code: @[simp]
theorem exp_inl (x : R) : exp 𝕜 (inl x : tsze R M) = inl (exp 𝕜 x) 
goal: NormedSpace.exp 𝕜 (TrivSqZeroExt.inl x) = TrivSqZeroExt.inl (NormedSpace.exp 𝕜 x)

full_name: TrivSqZeroExt.exp_inr
code: @[simp]
theorem exp_inr (m : M) : exp 𝕜 (inr m : tsze R M) = 1 + inr m 
goal: NormedSpace.exp 𝕜 (TrivSqZeroExt.inr m) = 1 + TrivSqZeroExt.inr m

full_name: TrivSqZeroExt.exp_def
code: theorem exp_def (x : tsze R M) : exp 𝕜 x = inl (exp 𝕜 x.fst) + inr (exp 𝕜 x.fst • x.snd) 
goal: NormedSpace.exp 𝕜 x = TrivSqZeroExt.inl (NormedSpace.exp 𝕜 x.fst) + TrivSqZeroExt.inr (NormedSpace.exp 𝕜 x.fst • x.snd)

full_name: TrivSqZeroExt.fst_exp
code: @[simp]
theorem fst_exp (x : tsze R M) : fst (exp 𝕜 x) = exp 𝕜 x.fst 
goal: (NormedSpace.exp 𝕜 x).fst = NormedSpace.exp 𝕜 x.fst

full_name: TrivSqZeroExt.snd_exp
code: @[simp]
theorem snd_exp (x : tsze R M) : snd (exp 𝕜 x) = exp 𝕜 x.fst • x.snd 
goal: (NormedSpace.exp 𝕜 x).snd = NormedSpace.exp 𝕜 x.fst • x.snd

full_name: TrivSqZeroExt.eq_smul_exp_of_invertible
code: /-- Polar form of trivial-square-zero extension. -/
theorem eq_smul_exp_of_invertible (x : tsze R M) [Invertible x.fst] :
    x = x.fst • exp 𝕜 (⅟ x.fst • inr x.snd) 
goal: x = x.fst • NormedSpace.exp 𝕜 (⅟x.fst • TrivSqZeroExt.inr x.snd)

full_name: TrivSqZeroExt.eq_smul_exp_of_ne_zero
code: /-- More convenient version of `TrivSqZeroExt.eq_smul_exp_of_invertible` for when `R` is a
field. -/
theorem eq_smul_exp_of_ne_zero (x : tsze R M) (hx : x.fst ≠ 0) :
    x = x.fst • exp 𝕜 (x.fst⁻¹ • inr x.snd) 
goal: x = x.fst • NormedSpace.exp 𝕜 (x.fst⁻¹ • TrivSqZeroExt.inr x.snd)

full_name: TrivSqZeroExt.instL1SeminormedAddCommGroup
code: instance instL1SeminormedAddCommGroup : SeminormedAddCommGroup (tsze R M) 
goal: SeminormedAddCommGroup (tsze R M)

full_name: TrivSqZeroExt.norm_def
code: theorem norm_def (x : tsze R M) : ‖x‖ = ‖fst x‖ + ‖snd x‖ 
goal: ‖x‖ = ‖x.fst‖ + ‖x.snd‖

full_name: TrivSqZeroExt.nnnorm_def
code: theorem nnnorm_def (x : tsze R M) : ‖x‖₊ = ‖fst x‖₊ + ‖snd x‖₊ 
goal: ‖x‖₊ = ‖x.fst‖₊ + ‖x.snd‖₊

full_name: TrivSqZeroExt.norm_inl
code: @[simp] theorem norm_inl (r : R) : ‖(inl r : tsze R M)‖ = ‖r‖ 
goal: ‖TrivSqZeroExt.inl r‖ = ‖r‖

full_name: TrivSqZeroExt.norm_inr
code: @[simp] theorem norm_inr (m : M) : ‖(inr m : tsze R M)‖ = ‖m‖ 
goal: ‖TrivSqZeroExt.inr m‖ = ‖m‖

full_name: TrivSqZeroExt.nnnorm_inl
code: @[simp] theorem nnnorm_inl (r : R) : ‖(inl r : tsze R M)‖₊ = ‖r‖₊ 
goal: ‖TrivSqZeroExt.inl r‖₊ = ‖r‖₊

full_name: TrivSqZeroExt.nnnorm_inr
code: @[simp] theorem nnnorm_inr (m : M) : ‖(inr m : tsze R M)‖₊ = ‖m‖₊ 
goal: ‖TrivSqZeroExt.inr m‖₊ = ‖m‖₊

full_name: TrivSqZeroExt.instL1SeminormedRing
code: instance instL1SeminormedRing : SeminormedRing (tsze R M) 
goal: SeminormedRing (tsze R M)

full_name: TrivSqZeroExt.instL1BoundedSMul
code: instance instL1BoundedSMul : BoundedSMul S (tsze R M) 
goal: BoundedSMul S (tsze R M)

full_name: TrivSqZeroExt.instNormOneClass
code: instance [NormOneClass R] : NormOneClass (tsze R M) 
goal: NormOneClass (tsze R M)

full_name: TrivSqZeroExt.instL1SeminormedCommRing
code: instance instL1SeminormedCommRing : SeminormedCommRing (tsze R M) 
goal: SeminormedCommRing (tsze R M)

full_name: TrivSqZeroExt.instL1NormedAddCommGroup
code: instance instL1NormedAddCommGroup : NormedAddCommGroup (tsze R M) 
goal: NormedAddCommGroup (tsze R M)

full_name: TrivSqZeroExt.instL1NormedRing
code: instance instL1NormedRing : NormedRing (tsze R M) 
goal: NormedRing (tsze R M)

full_name: TrivSqZeroExt.instL1NormedCommRing
code: instance instL1NormedCommRing : NormedCommRing (tsze R M) 
goal: NormedCommRing (tsze R M)

full_name: TrivSqZeroExt.instL1NormedSpace
code: instance instL1NormedSpace : NormedSpace 𝕜 (tsze R M) 
goal: NormedSpace 𝕜 (tsze R M)

full_name: TrivSqZeroExt.instL1NormedAlgebra
code: instance instL1NormedAlgebra : NormedAlgebra 𝕜 (tsze R M) 
goal: NormedAlgebra 𝕜 (tsze R M)