Fmod GReal differentiable On
FmodGReal_differentiableOn
Plain-language statement
FmodGReal is differentiable on (0, ∞).
Exact Lean statement
theorem FmodGReal_differentiableOn : DifferentiableOn ℝ FmodGReal (Set.Ioi 0)
Formal artifact
Lean source
theorem FmodGReal_differentiableOn : DifferentiableOn ℝ FmodGReal (Set.Ioi 0) := by intro t ht simp only [Set.mem_Ioi] at ht have hF_re_diff := (hasDerivAt_resToImagAxis_re F_holo ht).differentiableAt have hG_re_diff := (hasDerivAt_resToImagAxis_re G_holo ht).differentiableAt have hG_ne : (G.resToImagAxis t).re ≠ 0 := ne_of_gt (G_imag_axis_pos.2 t ht) apply (hF_re_diff.div hG_re_diff hG_ne).differentiableWithinAt.congr_of_eventuallyEq_of_mem · filter_upwards [self_mem_nhdsWithin] with s hs simp only [Set.mem_Ioi] at hs simp [FmodGReal, FReal, GReal, hs, ResToImagAxis] · simp [ht]- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/FG.lean:967-978
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Related declarations
Anti Der Pos
antiDerPos
Plain-language statement
If is a modular form where is positive for sufficiently large (i.e. constant term is positive) and the derivative is positive, then is also positive.
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
Anti Serre Der Pos
antiSerreDerPos
Plain-language statement
Let be a holomorphic function where is real for all . Assume that Serre derivative is positive on the imaginary axis. If is positive for sufficiently large , then is positive for all .
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
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closedBall_center_subset_upperHalfPlane
Plain-language statement
Closed ball centered at z with radius z.im/2 is contained in the upper half plane.
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.