Nuprl Lemma : rv-sep-iff-norm
∀rv:InnerProductSpace. ∀x,y:Point.  (x # y 
⇐⇒ r0 < ||x - y||)
Proof
Definitions occuring in Statement : 
rv-norm: ||x||
, 
rv-sub: x - y
, 
inner-product-space: InnerProductSpace
, 
rless: x < y
, 
int-to-real: r(n)
, 
ss-sep: x # y
, 
ss-point: Point
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
natural_number: $n
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
, 
subtype_rel: A ⊆r B
, 
rev_implies: P 
⇐ Q
, 
implies: P 
⇒ Q
, 
prop: ℙ
, 
guard: {T}
, 
uimplies: b supposing a
Lemmas referenced : 
rv-norm-positive-iff-ext, 
rv-sub_wf, 
inner-product-space_subtype, 
rless_wf, 
int-to-real_wf, 
rv-norm_wf, 
real_wf, 
rleq_wf, 
req_wf, 
rmul_wf, 
rv-ip_wf, 
iff_wf, 
ss-sep_wf, 
rv-0_wf, 
ss-point_wf, 
real-vector-space_subtype1, 
subtype_rel_transitivity, 
inner-product-space_wf, 
real-vector-space_wf, 
separation-space_wf, 
rv-sep-iff
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
cut, 
addLevel, 
sqequalHypSubstitution, 
productElimination, 
thin, 
independent_pairFormation, 
impliesFunctionality, 
hypothesis, 
introduction, 
extract_by_obid, 
dependent_functionElimination, 
hypothesisEquality, 
isectElimination, 
applyEquality, 
sqequalRule, 
independent_functionElimination, 
because_Cache, 
natural_numberEquality, 
lambdaEquality, 
setElimination, 
rename, 
setEquality, 
productEquality, 
instantiate, 
independent_isectElimination
Latex:
\mforall{}rv:InnerProductSpace.  \mforall{}x,y:Point.    (x  \#  y  \mLeftarrow{}{}\mRightarrow{}  r0  <  ||x  -  y||)
Date html generated:
2017_10_04-PM-11_51_36
Last ObjectModification:
2017_03_12-PM-09_48_27
Theory : inner!product!spaces
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