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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