Theorems · Theorem · functional analysis
preimage_add_ball
∀ {E : Type u_5} [inst : SeminormedAddCommGroup E] (a b : E) (r : ℝ),
(fun x => b + x) ⁻¹' Metric.ball a r = Metric.ball (a - b) r- Defined in
- Mathlib.Analysis.Normed.Group.Basic
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SeminormedAddCommGroup
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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
- Norm.normproof · cited by 5,413
- Set.preimagestatement · cited by 4,946
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- Set.extproof · cited by 2,266
- add_commproof · cited by 1,535
- sub_eq_add_negproof · cited by 1,023
- add_assocproof · cited by 746
- Metric.ballstatement and proof · cited by 735
- neg_add_revproof · cited by 236
- Set.mem_preimageproof · cited by 190
Cited by3
Results whose statement or proof uses this declaration.
- singleton_add_ballproof · cited by 4
- MeasureTheory.Measure.addHaar_ball_centerproof · cited by 2
- singleton_add_ball_zeroproof · cited by 0