Nuprl Lemma : rv-norm-positive-iff
∀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 :
not: ¬A,
false: False,
le: A ≤ B,
nat: ℕ,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
nat_plus: ℕ+,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
rv-norm-squared,
rv-ip-positive,
iff_weakening_equal,
true_wf,
squash_wf,
req_weakening,
rnexp-int,
rless_functionality,
exp-zero,
exp_wf2,
le_wf,
false_wf,
rnexp_wf,
less_than_wf,
rleq_weakening_equal,
rnexp-rless,
ss-point_wf,
rv-ip_wf,
rmul_wf,
req_wf,
rleq_wf,
real_wf,
rv-norm_wf,
int-to-real_wf,
rless_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,
rv-norm-positive
Rules used in proof :
universeEquality,
intEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
productElimination,
baseClosed,
imageMemberEquality,
dependent_set_memberEquality,
because_Cache,
productEquality,
setEquality,
rename,
setElimination,
lambdaEquality,
natural_numberEquality,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
isectElimination,
hypothesis,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}x:Point. (x \# 0 \mLeftarrow{}{}\mRightarrow{} r0 < ||x||)
Date html generated:
2016_11_08-AM-09_16_27
Last ObjectModification:
2016_11_02-PM-03_05_33
Theory : inner!product!spaces
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