Theorems · Theorem · ordinary differential equations

ODE.FunSpace.mem_closedBall

∀ {E : Type u_1} [inst : NormedAddCommGroup E] {tmin tmax : ℝ} {t₀ : ↑(Set.Icc tmin tmax)} {x₀ : E} {a r L : NNReal}
  {α : ODE.FunSpace t₀ x₀ r L},
  ↑L * max (tmax - ↑t₀) (↑t₀ - tmin) ≤ ↑a - ↑r → ∀ {t : ↑(Set.Icc tmin tmax)}, α.toFun t ∈ Metric.closedBall x₀ ↑a

The image of a function in FunSpace is contained within a closed ball.

Defined in: Mathlib.Analysis.ODE.PicardLindelof
Cited by: 3 results in Mathlib
Foundations: Depth 154 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NormedAddCommGroup

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Cites24

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Setstatement · cited by 53,352
Realstatement and proof · cited by 25,697
NormedAddCommGroupstatement and proof · cited by 15,752
Set.Elemstatement and proof · cited by 7,166
NNRealstatement and proof · cited by 4,310
le_reflproof · cited by 2,061
absproof · cited by 1,814
Set.Iccstatement and proof · cited by 1,702
NNReal.toRealstatement and proof · cited by 1,260
Metric.closedBallstatement · cited by 704
add_le_addproof · cited by 666
mul_le_mul_of_nonneg_leftproof · cited by 361

Cited by3

Results whose statement or proof uses this declaration.

ODE.FunSpace.compProj_mem_closedBallproof · cited by 2
ODE.FunSpace.dist_comp_iterate_next_leproof · cited by 1
ODE.FunSpace.continuousOn_comp_compProjproof · cited by 1