Theorems · Theorem · functional analysis
sameRay_iff_of_norm_eq
∀ {F : Type u_2} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace ℝ F] {x y : F}, ‖x‖ = ‖y‖ → (SameRay ℝ x y ↔ x = y)Two vectors of the same norm are on the same ray if and only if they are equal.
- Defined in
- Mathlib.Analysis.Normed.Module.Ray
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 159 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- Norm.normstatement and proof · cited by 5,413
- eq_or_neproof · cited by 1,117
- norm_zeroproof · cited by 366
- SameRaystatement and proof · cited by 149
- norm_eq_zeroproof · cited by 43
- SameRay.rflproof · cited by 11
- SameRay.zero_rightproof · cited by 7
- norm_injOn_ray_rightproof · cited by 1
Cited by3
Results whose statement or proof uses this declaration.
- SameRay.eq_of_norm_eqproof · cited by 2
- not_sameRay_iff_of_norm_eqproof · cited by 1
- StrictConvexSpace.of_norm_addproof · cited by 0