Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Has Deriv At of has Deriv At of Real comp

HasDerivAt.of_hasDerivAt_ofReal_comp

analytic number theory prime numbers asymptotics auxiliary

Plain-language statement

Let $f:\mathbb R\to\mathbb R$ . If the same function, viewed as taking values in $\mathbb C$ , has derivative $u\in\mathbb C$ at a real point $z$ , then $u$ is real and equals the ordinary real derivative of $f$ at $z$ .

Exact Lean statement

lemma HasDerivAt.of_hasDerivAt_ofReal_comp {z : ℝ} {f : ℝ → ℝ} {u : ℂ}
    (hf : HasDerivAt (fun y ↦ (f y : ℂ)) u z) :
    ∃ u' : ℝ, u = u' ∧ HasDerivAt f u' z

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma HasDerivAt.of_hasDerivAt_ofReal_comp {z : ℝ} {f : ℝ → ℝ} {u : ℂ}    (hf : HasDerivAt (fun y ↦ (f y : ℂ)) u z) :    ∃ u' : ℝ, u = u' ∧ HasDerivAt f u' z := by  lift u to ℝ  · have H := (imCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt.deriv    simp only [Function.comp_def, imCLM_apply, ofReal_im, deriv_const] at H    rwa [eq_comm, comp_apply, imCLM_apply, toSpanSingleton_apply_one] at H  refine ⟨u, rfl, ?_⟩  convert! (reCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt  rw [comp_apply, toSpanSingleton_apply_one, reCLM_apply, ofReal_re]

Project: Prime Number Theorem and More
License: Apache-2.0
Commit: a93551347dce
Source: PrimeNumberTheoremAnd/Auxiliary.lean:48-57

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Source project: Prime Number Theorem and More

Person-level attribution pending.

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Plain-language statement

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Source project: Prime Number Theorem and More

Person-level attribution pending.

View proof record