Theorem: StieltjesFunction.eq_of_measure_of_tendsto_atBot
File: Mathlib/MeasureTheory/Measure/Stieltjes.lean

tactic: ext x
state_before: f g : StieltjesFunction
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
⊢ f = g
state_after: case h
f g : StieltjesFunction
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : ℝ
⊢ ↑f x = ↑g x

tactic: have hf := measure_Iic f hfl x
state_before: case h
f g : StieltjesFunction
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : ℝ
⊢ ↑f x = ↑g x
state_after: case h
f g : StieltjesFunction
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : ℝ
hf : f.measure (Iic x) = ofReal (↑f x - l)
⊢ ↑f x = ↑g x

tactic: rw [hfg, measure_Iic g hgl x, ENNReal.ofReal_eq_ofReal_iff, eq_comm] at hf
state_before: case h
f g : StieltjesFunction
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : ℝ
hf : f.measure (Iic x) = ofReal (↑f x - l)
⊢ ↑f x = ↑g x
state_after: case h
f g : StieltjesFunction
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : ℝ
hf : ↑f x - l = ↑g x - l
⊢ ↑f x = ↑g x

case h.hp
f g : StieltjesFunction
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : ℝ
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ 0 ≤ ↑g x - l

case h.hq
f g : StieltjesFunction
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : ℝ
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ 0 ≤ ↑f x - l

tactic: simpa using hf
state_before: case h
f g : StieltjesFunction
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : ℝ
hf : ↑f x - l = ↑g x - l
⊢ ↑f x = ↑g x
state_after: no goals

tactic: rw [sub_nonneg]
state_before: case h.hp
f g : StieltjesFunction
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : ℝ
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ 0 ≤ ↑g x - l
state_after: case h.hp
f g : StieltjesFunction
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : ℝ
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ l ≤ ↑g x

tactic: exact Monotone.le_of_tendsto g.mono hgl x
state_before: case h.hp
f g : StieltjesFunction
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : ℝ
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ l ≤ ↑g x
state_after: no goals

tactic: rw [sub_nonneg]
state_before: case h.hq
f g : StieltjesFunction
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : ℝ
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ 0 ≤ ↑f x - l
state_after: case h.hq
f g : StieltjesFunction
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : ℝ
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ l ≤ ↑f x

tactic: exact Monotone.le_of_tendsto f.mono hfl x
state_before: case h.hq
f g : StieltjesFunction
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : ℝ
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ l ≤ ↑f x
state_after: no goals