Nuprl Lemma : rv-norm-eq-iff
∀[rv:InnerProductSpace]. ∀[x:Point]. ∀[r:ℝ]. uiff(||x|| = r;x^2 = r^2) supposing r0 ≤ r
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rleq: x ≤ y,
rnexp: x^k1,
req: x = y,
int-to-real: r(n),
real: ℝ,
ss-point: Point,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
req_witness,
rv-ip_wf,
rnexp_wf,
false_wf,
le_wf,
req_wf,
rv-norm_wf,
real_wf,
rleq_wf,
int-to-real_wf,
rmul_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_weakening,
req_functionality,
req_transitivity,
req_inversion,
rv-norm-squared,
rnexp_functionality,
square-req-iff,
rv-norm-nonneg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
lambdaFormation,
independent_functionElimination,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
because_Cache,
productElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
instantiate,
dependent_functionElimination
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[x:Point]. \mforall{}[r:\mBbbR{}]. uiff(||x|| = r;x\^{}2 = r\^{}2) supposing r0 \mleq{} r
Date html generated:
2017_10_04-PM-11_51_30
Last ObjectModification:
2017_03_13-AM-10_42_26
Theory : inner!product!spaces
Home
Index