Theorems · Theorem · functional analysis
norm_add_le_of_le
∀ {E : Type u_5} [inst : SeminormedAddGroup E] {a₁ a₂ : E} {r₁ r₂ : ℝ}, ‖a₁‖ ≤ r₁ → ‖a₂‖ ≤ r₂ → ‖a₁ + a₂‖ ≤ r₁ + r₂Triangle inequality for the norm.
- Defined in
- Mathlib.Analysis.Normed.Group.Basic
- Cited by
- 13 results in Mathlib
- Foundations
- Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SeminormedAddGroup
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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
- LE.le.transproof · cited by 3,151
- add_le_addproof · cited by 666
- SeminormedAddGroupstatement and proof · cited by 331
- norm_add_leproof · cited by 68
Cited by13
Results whose statement or proof uses this declaration.
- norm_add₃_leproof · cited by 9
- Asymptotics.IsBigOWith.addproof · cited by 6
- Bornology.IsBounded.addproof · cited by 2
- IsBoundedLinearMap.addproof · cited by 2
- Finset.norm_prod_one_add_sub_one_leproof · cited by 1
- RightDerivMeasurableAux.D_subset_differentiable_setproof · cited by 1
- Asymptotics.IsBigOWith.add_addproof · cited by 1
- Complex.norm_log_one_add_leproof · cited by 1
- FDerivMeasurableAux.D_subset_differentiable_setproof · cited by 1
- exists_forall_closed_ball_dist_add_le_two_subproof · cited by 1
- ContinuousMultilinearMap.opNorm_add_leproof · cited by 0
- norm_add₄_leproof · cited by 0