Theorems · Theorem · functional analysis
LowerSemicontinuousOn.isClosed_re_epigraph
∀ {𝕜 : Type u_1} {E : Type u_2} {s : Set E} {φ : E → ℝ} [inst : RCLike 𝕜] [inst_1 : TopologicalSpace E],
IsClosed s → LowerSemicontinuousOn φ s → IsClosed {p | p.1 ∈ s ∧ φ p.1 ≤ RCLike.re p.2}- Defined in
- Mathlib.Analysis.Convex.Approximation
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 168 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- RCLikeTopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites25
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement and proof · cited by 53,352
- Realstatement and proof · cited by 25,697
- TopologicalSpacestatement and proof · cited by 24,529
- Set.ofPredstatement and proof · cited by 6,101
- Set.preimageproof · cited by 4,946
- AddMonoidHomstatement · cited by 3,230
- RCLikestatement and proof · cited by 2,829
- IsClosedstatement and proof · cited by 1,639
- ERealproof · cited by 793
- Continuous.compproof · cited by 371
- AddMonoid.toZerostatement · cited by 325
Cited by1
Results whose statement or proof uses this declaration.
- ConvexOn.exists_affine_le_of_ltproof · cited by 4