Nuprl Lemma : ip-between-rleq
∀[rv:InnerProductSpace]. ∀[a,b,c:Point]. {(||a - b|| ≤ ||a - c||) ∧ (||b - c|| ≤ ||a - c||)} supposing a_b_c
Proof
Definitions occuring in Statement :
ip-between: a_b_c,
rv-norm: ||x||,
rv-sub: x - y,
inner-product-space: InnerProductSpace,
rleq: x ≤ y,
ss-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
guard: {T},
and: P ∧ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
prop: ℙ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
ip-dist-between,
less_than'_wf,
rsub_wf,
rv-norm_wf,
rv-sub_wf,
inner-product-space_subtype,
real_wf,
rleq_wf,
int-to-real_wf,
req_wf,
rmul_wf,
rv-ip_wf,
nat_plus_wf,
ip-between_wf,
ss-point_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
radd_wf,
rleq_functionality,
req_weakening,
radd-preserves-rleq,
rminus_wf,
rv-norm-nonneg,
uiff_transitivity,
req_transitivity,
radd_functionality,
rminus-as-rmul,
req_inversion,
rmul-identity1,
rmul-distrib2,
radd-assoc,
rmul_functionality,
radd-int,
rmul-zero-both,
radd-zero-both,
radd-ac,
radd_comm,
radd-rminus-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
sqequalRule,
productElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
because_Cache,
applyEquality,
setElimination,
rename,
setEquality,
productEquality,
natural_numberEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
instantiate,
voidElimination,
independent_pairFormation,
addLevel,
addEquality,
independent_functionElimination
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[a,b,c:Point].
\{(||a - b|| \mleq{} ||a - c||) \mwedge{} (||b - c|| \mleq{} ||a - c||)\} supposing a\_b\_c
Date html generated:
2017_10_05-AM-00_01_42
Last ObjectModification:
2017_03_13-PM-06_12_14
Theory : inner!product!spaces
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