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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