Nuprl Lemma : rv-norm-nonneg
∀[rv:InnerProductSpace]. ∀[x:Point]. (r0 ≤ ||x||)
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
inner-product-space: InnerProductSpace,
ss-point: Point,
rleq: x ≤ y,
int-to-real: r(n),
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
real: ℝ,
subtype_rel: A ⊆r B,
false: False,
not: ¬A,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
squash: ↓T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_apply: x[s],
prop: ℙ,
and: P ∧ Q,
so_lambda: λ2x.t[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,
nat_plus_wf,
rsub_wf,
less_than'_wf,
equal_wf,
sq_stable__rleq,
rv-ip_wf,
rmul_wf,
req_wf,
int-to-real_wf,
rleq_wf,
real_wf,
set_wf,
rv-norm_wf
Rules used in proof :
voidElimination,
isect_memberEquality,
independent_isectElimination,
instantiate,
axiomEquality,
minusEquality,
setEquality,
applyEquality,
because_Cache,
independent_pairEquality,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
imageElimination,
baseClosed,
imageMemberEquality,
productElimination,
independent_functionElimination,
rename,
setElimination,
lambdaFormation,
natural_numberEquality,
productEquality,
lambdaEquality,
sqequalRule,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[x:Point]. (r0 \mleq{} ||x||)
Date html generated:
2016_11_08-AM-09_16_13
Last ObjectModification:
2016_10_31-PM-04_38_16
Theory : inner!product!spaces
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