Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Complex nhds has Basis square

Complex.nhds_hasBasis_square

analytic number theory prime numbers asymptotics rectangle

Plain-language statement

At every point $p\in\mathbb C$ , the axis-parallel closed squares centered at $p$ with positive half-width form a neighborhood basis. Equivalently, a set is a neighborhood of $p$ exactly when it contains one of these sufficiently small squares.

Exact Lean statement

theorem Complex.nhds_hasBasis_square (p : ℂ) : (𝓝 p).HasBasis (0 < ·) (Square p ·)

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

theorem Complex.nhds_hasBasis_square (p : ℂ) : (𝓝 p).HasBasis (0 < ·) (Square p ·) := by  suffices      (𝓝 p.re ×ˢ 𝓝 p.im).HasBasis (0 < ·)        (equivRealProdCLM.symm.toHomeomorph ⁻¹' Square p ·)    by simpa only [← nhds_prod_eq, Homeomorph.map_nhds_eq, Homeomorph.image_preimage]      using! this.map equivRealProdCLM.symm.toHomeomorph  apply ((nhds_basis_Icc_pos p.re).prod_same_index_mono (nhds_basis_Icc_pos p.im) ?_ ?_).congr  · intro; rfl  · intros    rw [← uIcc_of_lt (by linarith), ← uIcc_of_lt (by linarith)]    simpa [Square, Rectangle] using by ring_nf  all_goals exact (antitone_const_tsub.Icc (monotone_id.const_add _)).monotoneOn _

Project: Prime Number Theorem and More
License: Apache-2.0
Commit: a93551347dce
Source: PrimeNumberTheoremAnd/Rectangle.lean:253-264

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