Theorems · Theorem · functional analysis
real_inner_add_sub_eq_zero_iff
∀ {F : Type u_3} [inst : SeminormedAddCommGroup F] [inst_1 : InnerProductSpace ℝ F] (x y : F),
inner ℝ (x + y) (x - y) = 0 ↔ ‖x‖ = ‖y‖The sum and difference of two vectors are orthogonal if and only if they have the same norm.
- Defined in
- Mathlib.Analysis.InnerProductSpace.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 163 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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
- Norm.normstatement and proof · cited by 5,413
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikeproof · cited by 2,829
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- Inner.innerstatement and proof · cited by 1,089
- norm_nonnegproof · cited by 725
- real_inner_commproof · cited by 57
- inner_add_leftproof · cited by 28
- inner_sub_rightproof · cited by 19
- inner_self_eq_norm_mul_normproof · cited by 9
- mul_self_inj_of_nonnegproof · cited by 9
Cited by2
Results whose statement or proof uses this declaration.
- AffineSubspace.mem_perpBisector_iff_dist_eqproof · cited by 1
- Submodule.reflection_subproof · cited by 1