Nuprl Lemma : rv-ip-rleq
∀[rv:InnerProductSpace]. ∀[a,b:Point]. (a ⋅ b ≤ (||a|| * ||b||))
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rleq: x ≤ y,
rmul: a * b,
ss-point: Point,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
real: ℝ
Lemmas referenced :
rabs-bounds,
rv-ip_wf,
rv-Cauchy-Schwarz',
rleq_transitivity,
rabs_wf,
rmul_wf,
rv-norm_wf,
real_wf,
rleq_wf,
int-to-real_wf,
req_wf,
less_than'_wf,
rsub_wf,
nat_plus_wf,
ss-point_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_pairFormation,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
natural_numberEquality,
sqequalRule,
independent_isectElimination,
dependent_functionElimination,
independent_pairEquality,
because_Cache,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
isect_memberEquality,
voidElimination
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[a,b:Point]. (a \mcdot{} b \mleq{} (||a|| * ||b||))
Date html generated:
2017_10_04-PM-11_52_15
Last ObjectModification:
2017_07_28-AM-08_54_01
Theory : inner!product!spaces
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