Nuprl Lemma : rv-norm-triangle-inequality2
∀[rv:InnerProductSpace]. ∀[x,y,z:Point]. (||x - z|| ≤ (||x - y|| + ||y - z||))
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
rv-sub: x - y,
inner-product-space: InnerProductSpace,
ss-point: Point,
rleq: x ≤ y,
radd: a + b,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
rv-sub: x - y,
uiff: uiff(P;Q),
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
guard: {T},
prop: ℙ,
subtype_rel: A ⊆r B,
false: False,
implies: P ⇒ Q,
not: ¬A,
and: P ∧ Q,
le: A ≤ B,
all: ∀x:A. B[x],
rnonneg: rnonneg(x),
rleq: x ≤ y,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rv-0-add,
rv-add-minus,
rv-add_functionality,
ss-eq_inversion,
ss-eq_transitivity,
uiff_transitivity,
rv-add-assoc,
ss-eq_functionality,
ss-eq_weakening,
rv-0_wf,
ss-eq_wf,
rv-add-cancel-right,
rv-minus_wf,
rv-norm_functionality,
req_weakening,
rleq_functionality,
rv-norm-triangle-inequality,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rv-add_wf,
separation-space_wf,
real-vector-space_wf,
inner-product-space_wf,
subtype_rel_transitivity,
real-vector-space_subtype1,
ss-point_wf,
nat_plus_wf,
rv-ip_wf,
rmul_wf,
req_wf,
int-to-real_wf,
rleq_wf,
real_wf,
inner-product-space_subtype,
rv-sub_wf,
rv-norm_wf,
radd_wf,
rsub_wf,
less_than'_wf
Rules used in proof :
independent_functionElimination,
voidElimination,
isect_memberEquality,
independent_isectElimination,
instantiate,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
minusEquality,
natural_numberEquality,
productEquality,
setEquality,
rename,
setElimination,
hypothesis,
applyEquality,
isectElimination,
extract_by_obid,
because_Cache,
independent_pairEquality,
productElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
lambdaEquality,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[x,y,z:Point]. (||x - z|| \mleq{} (||x - y|| + ||y - z||))
Date html generated:
2016_11_08-AM-09_17_31
Last ObjectModification:
2016_11_01-PM-05_50_51
Theory : inner!product!spaces
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