Nuprl Lemma : rv-unit-property
∀rv:InnerProductSpace. ∀x:Point. (x # 0 ⇒ (∃t:ℝ. (t ≠ r0 ∧ rv-unit(rv;x) ≡ t*x)))
Proof
Definitions occuring in Statement :
rv-unit: rv-unit(rv;x),
inner-product-space: InnerProductSpace,
rv-mul: a*x,
rv-0: 0,
ss-eq: x ≡ y,
ss-sep: x # y,
ss-point: Point,
rneq: x ≠ y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
rv-unit: rv-unit(rv;x),
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
rmul-rdiv-cancel2,
req_weakening,
rmul-zero-both,
rless_functionality,
rv-ip_wf,
req_wf,
rleq_wf,
real_wf,
rmul_wf,
rless-int,
rmul_preserves_rless,
ss-point_wf,
rv-0_wf,
ss-sep_wf,
ss-eq_wf,
rneq_wf,
rv-mul_wf,
separation-space_wf,
real-vector-space_wf,
inner-product-space_wf,
subtype_rel_transitivity,
inner-product-space_subtype,
real-vector-space_subtype1,
ss-eq_weakening,
rless_wf,
rv-norm_wf,
int-to-real_wf,
rdiv_wf,
rv-norm-positive
Rules used in proof :
addLevel,
setEquality,
rename,
setElimination,
lambdaEquality,
baseClosed,
imageMemberEquality,
productElimination,
productEquality,
instantiate,
independent_pairFormation,
inrFormation,
independent_isectElimination,
sqequalRule,
because_Cache,
applyEquality,
natural_numberEquality,
isectElimination,
dependent_pairFormation,
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{} (\mexists{}t:\mBbbR{}. (t \mneq{} r0 \mwedge{} rv-unit(rv;x) \mequiv{} t*x)))
Date html generated:
2016_11_08-AM-09_17_04
Last ObjectModification:
2016_10_31-PM-05_03_03
Theory : inner!product!spaces
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