Nuprl Lemma : real-vec-norm-diff-bound
∀[n:ℕ]. ∀[x,y:ℝ^n]. (|||x|| - ||y||| ≤ d(x;y))
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y),
real-vec-norm: ||x||,
real-vec: ℝ^n,
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
nat: ℕ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cand: A c∧ B,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
real-vec-triangle-inequality,
int-to-real_wf,
int_seg_wf,
le_witness_for_triv,
real-vec_wf,
istype-nat,
real-vec-dist_wf,
real-vec-norm_wf,
radd_wf,
rleq_functionality,
real-vec-dist-from-zero,
radd_functionality,
req_weakening,
rabs-difference-bound-rleq,
rleq-implies-rleq,
rsub_wf,
rleq_wf,
squash_wf,
true_wf,
real_wf,
radd_comm_eq,
subtype_rel_self,
iff_weakening_equal,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
req-iff-rsub-is-0,
real-vec-dist-symmetry,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
setElimination,
rename,
productElimination,
hypothesis,
universeIsType,
natural_numberEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
because_Cache,
applyEquality,
independent_functionElimination,
independent_pairFormation,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
approximateComputation,
int_eqEquality,
voidElimination
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbR{}\^{}n]. (|||x|| - ||y||| \mleq{} d(x;y))
Date html generated:
2019_10_30-AM-08_42_49
Last ObjectModification:
2019_07_08-AM-11_35_44
Theory : reals
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