Nuprl Lemma : rv-Cauchy-Schwarz'
∀[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,
rabs: |x|,
rmul: a * b,
ss-point: Point,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
and: P ∧ Q,
prop: ℙ,
sq_stable: SqStable(P),
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a,
squash: ↓T,
guard: {T},
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y
Lemmas referenced :
rv-Cauchy-Schwarz,
sq_stable__rleq,
rabs_wf,
rv-ip_wf,
rmul_wf,
rv-norm_wf,
real_wf,
rleq_wf,
int-to-real_wf,
req_wf,
square-rleq-implies,
rmul-nonneg-case1,
rv-norm-nonneg,
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,
rnexp_wf,
false_wf,
le_wf,
rleq_functionality,
req_weakening,
rmul_functionality,
req_inversion,
rv-norm-squared,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_transitivity,
rleq_weakening,
rnexp-rmul,
rnexp2-nonneg,
rabs-rnexp,
rabs-of-nonneg
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
natural_numberEquality,
sqequalRule,
independent_functionElimination,
dependent_functionElimination,
because_Cache,
independent_isectElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination,
instantiate,
dependent_set_memberEquality,
lambdaFormation,
productElimination
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[a,b:Point]. (|a \mcdot{} b| \mleq{} (||a|| * ||b||))
Date html generated:
2017_10_04-PM-11_52_12
Last ObjectModification:
2017_03_10-PM-02_28_59
Theory : inner!product!spaces
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