Theorems · Theorem · order theory
MonotoneOn.dual
∀ {α : Type u} {β : Type v} [inst : Preorder α] [inst_1 : Preorder β] {f : α → β} {s : Set α},
MonotoneOn f s → MonotoneOn (⇑OrderDual.toDual ∘ f ∘ ⇑OrderDual.ofDual) sAlias of the reverse direction of monotoneOn_dual_iff.
- Defined in
- Mathlib.Order.Monotone.Basic
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 16 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Setstatement and proof · cited by 53,352
- Equivstatement · cited by 8,337
- Preorderstatement and proof · cited by 7,952
- OrderDualstatement · cited by 927
- OrderDual.toDualstatement · cited by 481
- OrderDual.ofDualstatement · cited by 400
- MonotoneOnstatement · cited by 311
- monotoneOn_dual_iffproof · cited by 1
Cited by10
Results whose statement or proof uses this declaration.
- MonotoneOn.convex_geproof · cited by 3
- continuousWithinAt_left_of_monotoneOn_of_closure_image_mem_nhdsWithinproof · cited by 2
- IsGLB.isGLB_of_tendstoproof · cited by 2
- MonotoneOn.map_sInf_of_continuousWithinAtproof · cited by 1
- continuousWithinAt_left_of_monotoneOn_of_exists_betweenproof · cited by 1
- MonotoneOn.tendsto_nhdsGTproof · cited by 1
- MonotoneOn.convex_gtproof · cited by 1
- MonotoneOn.countable_not_continuousWithinAt_Iioproof · cited by 1
- IsGLB.mem_lowerBounds_of_tendstoproof · cited by 1
- MonotoneOn.map_csInf_of_continuousWithinAtproof · cited by 1