Nuprl Lemma : rv-Cauchy-Schwarz-equality
∀rv:InnerProductSpace. ∀a,b:Point(rv). ((a ⋅ b^2 = (a^2 * b^2)) ⇒ b # 0 ⇒ (∃t:ℝ. a ≡ t*b))
Proof
Definitions occuring in Statement :
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rv-mul: a*x,
rv-0: 0,
rnexp: x^k1,
req: x = y,
rmul: a * b,
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
false: False,
rneq: x ≠ y,
or: P ∨ Q,
uiff: uiff(P;Q),
rv-norm: ||x||,
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
top: Top,
exists: ∃x:A. B[x]
Lemmas referenced :
rv-ip-positive,
Error :ss-sep_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
rv-0_wf,
req_wf,
rnexp_wf,
istype-void,
istype-le,
rv-ip_wf,
rmul_wf,
Error :ss-point_wf,
rv-norm-is-zero,
rv-sub_wf,
rv-mul_wf,
rdiv_wf,
rless_wf,
int-to-real_wf,
radd_wf,
rsub_wf,
rmul_preserves_req,
rminus_wf,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
itermMinus_wf,
req_functionality,
rv-ip-sub-squared,
req_weakening,
radd_functionality,
req_transitivity,
rv-ip-mul,
rmul_functionality,
rv-ip-mul2,
rsub_functionality,
rmul-rinv3,
rmul-rinv,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_minus_lemma,
radd-preserves-req,
req_inversion,
rnexp2,
rsqrt_wf,
rv-ip-nonneg,
rleq_wf,
rleq_weakening_equal,
rsqrt0,
rsqrt_functionality,
Error :ss-eq_wf,
rv-sub-is-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
hypothesis,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
voidElimination,
inhabitedIsType,
because_Cache,
inrFormation_alt,
closedConclusion,
minusEquality,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
dependent_pairFormation_alt
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b:Point(rv). ((a \mcdot{} b\^{}2 = (a\^{}2 * b\^{}2)) {}\mRightarrow{} b \# 0 {}\mRightarrow{} (\mexists{}t:\mBbbR{}. a \mequiv{} t*b))
Date html generated:
2020_05_20-PM-01_11_53
Last ObjectModification:
2019_12_09-PM-11_41_12
Theory : inner!product!spaces
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