Nuprl Lemma : rv-norm-equal-iff
∀[rv:InnerProductSpace]. ∀[x,y:Point]. uiff(||x|| = ||y||;x^2 = y^2)
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
req: x = y,
ss-point: Point,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T}
Lemmas referenced :
iff_weakening_uiff,
req_wf,
rv-norm_wf,
real_wf,
rleq_wf,
int-to-real_wf,
rmul_wf,
rv-ip_wf,
rnexp_wf,
false_wf,
le_wf,
rv-norm-eq-iff,
rv-norm-nonneg,
req_witness,
uiff_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,
req_functionality,
req_weakening,
rv-norm-squared
Rules used in proof :
cut,
addLevel,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
productElimination,
thin,
independent_pairFormation,
isect_memberFormation,
introduction,
independent_isectElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
natural_numberEquality,
sqequalRule,
because_Cache,
dependent_set_memberEquality,
lambdaFormation,
independent_functionElimination,
cumulativity,
universeEquality,
instantiate,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[x,y:Point]. uiff(||x|| = ||y||;x\^{}2 = y\^{}2)
Date html generated:
2017_10_04-PM-11_51_33
Last ObjectModification:
2017_03_13-AM-10_44_03
Theory : inner!product!spaces
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