File: Mathlib/Analysis/Normed/Ring/SeminormFromConst.lean

full_name: seminormFromConst_seq
code: /-- For a ring seminorm `f` on `R` and `c ∈ R`, the sequence given by `(f (x * c^n))/((f c)^n)`. -/
def seminormFromConst_seq (x : R) : ℕ → ℝ 
goal: ℕ → ℝ

full_name: seminormFromConst_seq_def
code: lemma seminormFromConst_seq_def (x : R) :
    seminormFromConst_seq c f x = fun n ↦ f (x * c ^ n) / f c ^ n 
goal: seminormFromConst_seq c f x = fun n => f (x * c ^ n) / f c ^ n

full_name: seminormFromConst_seq_nonneg
code: /-- The terms in the sequence `seminormFromConst_seq c f x` are nonnegative. -/
theorem seminormFromConst_seq_nonneg (x : R) : 0 ≤ seminormFromConst_seq c f x 
goal: 0 ≤ seminormFromConst_seq c f x

full_name: seminormFromConst_bddBelow
code: /-- The image of `seminormFromConst_seq c f x` is bounded below by zero. -/
theorem seminormFromConst_bddBelow (x : R) :
    BddBelow (Set.range (seminormFromConst_seq c f x)) 
goal: BddBelow (Set.range (seminormFromConst_seq c f x))

full_name: seminormFromConst_seq_zero
code: /-- `seminormFromConst_seq c f 0` is the constant sequence zero. -/
theorem seminormFromConst_seq_zero (hf : f 0 = 0) : seminormFromConst_seq c f 0 = 0 
goal: seminormFromConst_seq c f 0 = 0

full_name: seminormFromConst_seq_one
code: /-- If `1 ≤ n`, then `seminormFromConst_seq c f 1 n = 1`. -/
theorem seminormFromConst_seq_one (n : ℕ) (hn : 1 ≤ n) : seminormFromConst_seq c f 1 n = 1 
goal: seminormFromConst_seq c f 1 n = 1

full_name: seminormFromConst_seq_antitone
code: /-- `seminormFromConst_seq c f x` is antitone. -/
theorem seminormFromConst_seq_antitone (x : R) : Antitone (seminormFromConst_seq c f x) 
goal: Antitone (seminormFromConst_seq c f x)

full_name: seminormFromConst'
code: /-- The real-valued function sending `x ∈ R` to the limit of `(f (x * c^n))/((f c)^n)`. -/
def seminormFromConst' (x : R) : ℝ 
goal: ℝ

full_name: seminormFromConst_isLimit
code: /-- We prove that `seminormFromConst' hf1 hc hpm x` is the limit of the sequence
  `seminormFromConst_seq c f x` as `n` tends to infinity. -/
theorem seminormFromConst_isLimit (x : R) :
    Tendsto (seminormFromConst_seq c f x) atTop (𝓝 (seminormFromConst' hf1 hc hpm x)) 
goal: Filter.Tendsto (seminormFromConst_seq c f x) Filter.atTop (𝓝 (seminormFromConst' hf1 hc hpm x))

full_name: seminormFromConst_one
code: /-- `seminormFromConst' hf1 hc hpm 1 = 1`. -/
theorem seminormFromConst_one : seminormFromConst' hf1 hc hpm 1 = 1 
goal: seminormFromConst' hf1 hc hpm 1 = 1

full_name: seminormFromConst
code: /-- The function `seminormFromConst` is a `RingSeminorm` on `R`. -/
def seminormFromConst : RingSeminorm R 
goal: RingSeminorm R

full_name: seminormFromConst_def
code: theorem seminormFromConst_def (x : R) :
    seminormFromConst hf1 hc hpm x = seminormFromConst' hf1 hc hpm x 
goal: (seminormFromConst hf1 hc hpm) x = seminormFromConst' hf1 hc hpm x

full_name: seminormFromConst_one_le
code: /-- `seminormFromConst' hf1 hc hpm 1 ≤ 1`. -/
theorem seminormFromConst_one_le : seminormFromConst' hf1 hc hpm 1 ≤ 1 
goal: seminormFromConst' hf1 hc hpm 1 ≤ 1

full_name: seminormFromConst_isNonarchimedean
code: /-- The function `seminormFromConst' hf1 hc hpm` is nonarchimedean. -/
theorem seminormFromConst_isNonarchimedean (hna : IsNonarchimedean f) :
    IsNonarchimedean (seminormFromConst' hf1 hc hpm) 
goal: IsNonarchimedean (seminormFromConst' hf1 hc hpm)

full_name: seminormFromConst_isPowMul
code: /-- The function `seminormFromConst' hf1 hc hpm` is power-multiplicative. -/
theorem seminormFromConst_isPowMul : IsPowMul (seminormFromConst' hf1 hc hpm) 
goal: IsPowMul (seminormFromConst' hf1 hc hpm)

full_name: seminormFromConst_le_seminorm
code: /-- The function `seminormFromConst' hf1 hc hpm` is bounded above by `x`. -/
theorem seminormFromConst_le_seminorm (x : R) : seminormFromConst' hf1 hc hpm x ≤ f x 
goal: seminormFromConst' hf1 hc hpm x ≤ f x

full_name: seminormFromConst_apply_of_isMul
code: /-- If `x : R` is multiplicative for `f`, then `seminormFromConst' hf1 hc hpm x = f x`. -/
theorem seminormFromConst_apply_of_isMul {x : R} (hx : ∀ y : R, f (x * y) = f x * f y) :
    seminormFromConst' hf1 hc hpm x = f x 
goal: seminormFromConst' hf1 hc hpm x = f x

full_name: seminormFromConst_isMul_of_isMul
code: /-- If `x : R` is multiplicative for `f`, then it is multiplicative for
  `seminormFromConst' hf1 hc hpm`. -/
theorem seminormFromConst_isMul_of_isMul {x : R} (hx : ∀ y : R, f (x * y) = f x * f y) (y : R) :
    seminormFromConst' hf1 hc hpm (x * y) =
      seminormFromConst' hf1 hc hpm x * seminormFromConst' hf1 hc hpm y 
goal: seminormFromConst' hf1 hc hpm (x * y) = seminormFromConst' hf1 hc hpm x * seminormFromConst' hf1 hc hpm y

full_name: seminormFromConst_apply_c
code: /-- `seminormFromConst' hf1 hc hpm c = f c`. -/
theorem seminormFromConst_apply_c : seminormFromConst' hf1 hc hpm c = f c 
goal: seminormFromConst' hf1 hc hpm c = f c

full_name: seminormFromConst_const_mul
code: /-- For every `x : R`, `seminormFromConst' hf1 hc hpm (c * x)` equals the product
  `seminormFromConst' hf1 hc hpm c * SeminormFromConst' hf1 hc hpm x`. -/
theorem seminormFromConst_const_mul (x : R) :
    seminormFromConst' hf1 hc hpm (c * x) =
      seminormFromConst' hf1 hc hpm c * seminormFromConst' hf1 hc hpm x 
goal: seminormFromConst' hf1 hc hpm (c * x) = seminormFromConst' hf1 hc hpm c * seminormFromConst' hf1 hc hpm x

full_name: normFromConst
code: /-- If `K` is a field, the function `seminormFromConst` is a `RingNorm` on `K`. -/
def normFromConst {k : K} {g : RingSeminorm K} (hg1 : g 1 ≤ 1) (hg_k : g k ≠ 0)
    (hg_pm : IsPowMul g) : RingNorm K 
goal: RingNorm K

full_name: seminormFromConstRingNormOfField_def
code: theorem seminormFromConstRingNormOfField_def {k : K} {g : RingSeminorm K} (hg1 : g 1 ≤ 1)
    (hg_k : g k ≠ 0) (hg_pm : IsPowMul g) (x : K) :
    normFromConst hg1 hg_k hg_pm x = seminormFromConst' hg1 hg_k hg_pm x 
goal: (normFromConst hg1 hg_k hg_pm) x = seminormFromConst' hg1 hg_k hg_pm x