Nuprl Lemma : rv-norm-positive-iff-ext
∀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],
iff: P ⇐⇒ Q,
natural_number: $n
Definitions unfolded in proof :
rnexp-rless,
rless-iff4,
iff_weakening_equal,
regular-less-iff,
rless-iff-large-diff,
rless_functionality,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
squash: ↓T,
or: P ∨ Q,
guard: {T},
prop: ℙ,
has-value: (a)↓,
implies: P ⇒ Q,
all: ∀x:A. B[x],
and: P ∧ Q,
strict4: strict4(F),
uimplies: b supposing a,
top: Top,
so_apply: x[s1;s2;s3;s4],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
uall: ∀[x:A]. B[x],
rv-ip-positive,
rv-ip: x ⋅ y,
sq_stable__rless,
rsqrt-positive,
rv-norm-positive,
rv-norm-positive-iff,
member: t ∈ T
Lemmas referenced :
top_wf,
is-exception_wf,
base_wf,
has-value_wf_base,
lifting-strict-spread,
rv-norm-positive-iff,
rnexp-rless,
rless-iff4,
iff_weakening_equal,
regular-less-iff,
rless-iff-large-diff,
rless_functionality,
rv-ip-positive,
sq_stable__rless,
rsqrt-positive,
rv-norm-positive
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
dependent_functionElimination,
inlFormation,
exceptionSqequal,
imageElimination,
imageMemberEquality,
inrFormation,
applyExceptionCases,
hypothesisEquality,
closedConclusion,
baseApply,
callbyvalueApply,
lambdaFormation,
independent_pairFormation,
independent_isectElimination,
voidEquality,
voidElimination,
isect_memberEquality,
baseClosed,
isectElimination,
sqequalHypSubstitution,
thin,
sqequalRule,
hypothesis,
extract_by_obid,
instantiate,
cut,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
introduction
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}x:Point. (x \# 0 \mLeftarrow{}{}\mRightarrow{} r0 < ||x||)
Date html generated:
2016_11_08-AM-09_16_29
Last ObjectModification:
2016_11_02-PM-04_14_30
Theory : inner!product!spaces
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