Nuprl Lemma : rv-norm-positive
∀rv:InnerProductSpace. ∀x:Point.  (x # 0 
⇒ (r0 < ||x||))
Proof
Definitions occuring in Statement : 
rv-norm: ||x||
, 
inner-product-space: InnerProductSpace
, 
rv-0: 0
, 
ss-sep: x # y
, 
ss-point: Point
, 
rless: x < y
, 
int-to-real: r(n)
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
natural_number: $n
Definitions unfolded in proof : 
uimplies: b supposing a
, 
guard: {T}
, 
subtype_rel: A ⊆r B
, 
prop: ℙ
, 
uall: ∀[x:A]. B[x]
, 
rv-norm: ||x||
, 
member: t ∈ T
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
Lemmas referenced : 
ss-point_wf, 
rv-0_wf, 
separation-space_wf, 
real-vector-space_wf, 
inner-product-space_wf, 
subtype_rel_transitivity, 
inner-product-space_subtype, 
real-vector-space_subtype1, 
ss-sep_wf, 
int-to-real_wf, 
rless_wf, 
rsqrt-positive, 
rv-ip-positive
Rules used in proof : 
independent_isectElimination, 
instantiate, 
applyEquality, 
natural_numberEquality, 
isectElimination, 
because_Cache, 
dependent_set_memberEquality, 
sqequalRule, 
hypothesis, 
independent_functionElimination, 
hypothesisEquality, 
thin, 
dependent_functionElimination, 
sqequalHypSubstitution, 
extract_by_obid, 
introduction, 
cut, 
lambdaFormation, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution
Latex:
\mforall{}rv:InnerProductSpace.  \mforall{}x:Point.    (x  \#  0  {}\mRightarrow{}  (r0  <  ||x||))
Date html generated:
2016_11_08-AM-09_16_16
Last ObjectModification:
2016_11_02-PM-03_23_11
Theory : inner!product!spaces
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