Nuprl Lemma : rv-Cauchy-Schwarz
∀[rv:InnerProductSpace]. ∀[a,b:Point]. (a ⋅ b^2 ≤ (a^2 * b^2))
Proof
Definitions occuring in Statement :
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
ss-point: Point,
rleq: x ≤ y,
rnexp: x^k1,
rmul: a * b,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
ss-eq: x ≡ y,
rev_uimplies: rev_uimplies(P;Q),
rsub: x - y,
uiff: uiff(P;Q),
rneq: x ≠ y,
or: P ∨ Q,
guard: {T},
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
rnonneg: rnonneg(x),
rleq: x ≤ y,
uimplies: b supposing a,
stable: Stable{P},
prop: ℙ,
implies: P ⇒ Q,
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rmul-int,
rv-ip0,
ss-eq_weakening,
rv-ip_functionality,
rleq_weakening_equal,
rnexp2,
radd-rminus-assoc,
radd-int,
rmul-identity1,
rminus-as-rmul,
rmul-distrib2,
radd-zero-both,
radd-preserves-rleq,
radd_comm,
radd-ac,
radd-assoc,
rmul-ac,
rminus_functionality,
rmul-assoc,
req_inversion,
rmul-zero-both,
rmul_over_rminus,
rmul-distrib,
rmul-rdiv-cancel2,
uiff_transitivity,
rminus_wf,
rmul_preserves_rleq2,
rsub_functionality,
rv-ip-mul2,
rmul_functionality,
rv-ip-mul,
req_transitivity,
radd_functionality,
rv-ip-sub-squared,
req_weakening,
rleq_functionality,
radd_wf,
int-to-real_wf,
rless_wf,
rdiv_wf,
rv-mul_wf,
rv-sub_wf,
rv-ip-nonneg,
rv-ip-positive,
minimal-not-not-excluded-middle,
minimal-double-negation-hyp-elim,
rleq_wf,
not_wf,
rv-0_wf,
ss-sep_wf,
or_wf,
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,
rmul_wf,
rv-ip_wf,
le_wf,
false_wf,
rnexp_wf,
stable__rleq
Rules used in proof :
multiplyEquality,
addEquality,
inrFormation,
unionElimination,
independent_functionElimination,
functionEquality,
voidElimination,
isect_memberEquality,
instantiate,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
minusEquality,
applyEquality,
because_Cache,
independent_pairEquality,
productElimination,
dependent_functionElimination,
lambdaEquality,
independent_isectElimination,
hypothesisEquality,
hypothesis,
lambdaFormation,
independent_pairFormation,
sqequalRule,
natural_numberEquality,
dependent_set_memberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[a,b:Point]. (a \mcdot{} b\^{}2 \mleq{} (a\^{}2 * b\^{}2))
Date html generated:
2016_11_08-AM-09_17_18
Last ObjectModification:
2016_10_31-PM-03_43_11
Theory : inner!product!spaces
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