Nuprl Lemma : rv-Cauchy-Schwarz-equality'
∀rv:InnerProductSpace. ∀a,b:Point. ((|a ⋅ b| = (||a|| * ||b||)) ⇒ b # 0 ⇒ (∃t:ℝ. a ≡ t*b))
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rv-mul: a*x,
rv-0: 0,
rabs: |x|,
req: x = y,
rmul: a * b,
real: ℝ,
ss-eq: x ≡ y,
ss-sep: x # y,
ss-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
and: P ∧ Q,
guard: {T},
uimplies: b supposing a,
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)
Lemmas referenced :
rv-Cauchy-Schwarz-equality,
req_wf,
rabs_wf,
rv-ip_wf,
rmul_wf,
rv-norm_wf,
real_wf,
rleq_wf,
int-to-real_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,
rnexp_wf,
false_wf,
le_wf,
uiff_transitivity,
req_functionality,
req_weakening,
rmul_functionality,
req_inversion,
rv-norm-squared,
rnexp-rmul,
rnexp_functionality,
rabs-rnexp,
rnexp2-nonneg,
rabs-of-nonneg
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
isectElimination,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
natural_numberEquality,
sqequalRule,
instantiate,
independent_isectElimination,
dependent_set_memberEquality,
independent_pairFormation,
because_Cache,
productElimination
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b:Point. ((|a \mcdot{} b| = (||a|| * ||b||)) {}\mRightarrow{} b \# 0 {}\mRightarrow{} (\mexists{}t:\mBbbR{}. a \mequiv{} t*b))
Date html generated:
2017_10_04-PM-11_52_32
Last ObjectModification:
2017_03_10-PM-02_34_47
Theory : inner!product!spaces
Home
Index