Nuprl Lemma : rv-unit-squared
∀[rv:InnerProductSpace]. ∀[x:Point]. rv-unit(rv;x)^2 = r1 supposing x # 0
Proof
Definitions occuring in Statement :
rv-unit: rv-unit(rv;x),
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rv-0: 0,
ss-sep: x # y,
ss-point: Point,
req: x = y,
int-to-real: r(n),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
guard: {T},
squash: ↓T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rv-0_wf,
ss-sep_wf,
separation-space_wf,
real-vector-space_wf,
inner-product-space_wf,
subtype_rel_transitivity,
inner-product-space_subtype,
real-vector-space_subtype1,
req_witness,
equal_wf,
sq_stable__req,
int-to-real_wf,
rv-ip_wf,
req_wf,
ss-point_wf,
set_wf,
rv-unit_wf
Rules used in proof :
isect_memberEquality,
instantiate,
setEquality,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
rename,
setElimination,
lambdaFormation,
natural_numberEquality,
lambdaEquality,
sqequalRule,
because_Cache,
applyEquality,
hypothesis,
independent_isectElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[x:Point]. rv-unit(rv;x)\^{}2 = r1 supposing x \# 0
Date html generated:
2016_11_08-AM-09_17_01
Last ObjectModification:
2016_10_31-PM-05_04_48
Theory : inner!product!spaces
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