Theorems · Theorem · functional analysis
IsometryEquiv.map_midpoint
∀ {E : Type u_1} {PE : Type u_2} {F : Type u_3} {PF : Type u_4} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
[inst_2 : MetricSpace PE] [inst_3 : NormedAddTorsor E PE] [inst_4 : NormedAddCommGroup F] [inst_5 : NormedSpace ℝ F]
[inst_6 : MetricSpace PF] [inst_7 : NormedAddTorsor F PF] (f : PE ≃ᵢ PF) (x y : PE),
f (midpoint ℝ x y) = midpoint ℝ (f x) (f y)A bijective isometry sends midpoints to midpoints.
- Defined in
- Mathlib.Analysis.Normed.Affine.MazurUlam
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 162 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites24
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
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- MetricSpacestatement and proof · cited by 1,684
- NormedAddTorsorstatement and proof · cited by 1,325
- IsometryEquivstatement and proof · cited by 177
- midpointstatement and proof · cited by 123
- AffineIsometryEquivproof · cited by 118
- IsometryEquiv.symmproof · cited by 75
- AffineIsometryEquiv.pointReflectionproof · cited by 14
- AffineIsometryEquiv.toIsometryEquivproof · cited by 8
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