Nuprl Lemma : rv-norm_wf
∀[rv:InnerProductSpace]. ∀[x:Point]. (||x|| ∈ {r:ℝ| (r0 ≤ r) ∧ ((r * r) = x^2)} )
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
ss-point: Point,
rleq: x ≤ y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
rv-norm: ||x||,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
separation-space_wf,
real-vector-space_wf,
inner-product-space_wf,
subtype_rel_transitivity,
inner-product-space_subtype,
real-vector-space_subtype1,
ss-point_wf,
int-to-real_wf,
rleq_wf,
rv-ip_wf,
rv-ip-nonneg,
rsqrt_wf
Rules used in proof :
because_Cache,
isect_memberEquality,
independent_isectElimination,
instantiate,
applyEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
natural_numberEquality,
dependent_set_memberEquality,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[x:Point]. (||x|| \mmember{} \{r:\mBbbR{}| (r0 \mleq{} r) \mwedge{} ((r * r) = x\^{}2)\} )
Date html generated:
2016_11_08-AM-09_16_05
Last ObjectModification:
2016_10_31-PM-04_37_09
Theory : inner!product!spaces
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