Nuprl Lemma : rv-norm_functionality
∀[rv:InnerProductSpace]. ∀[x1,x2:Point]. ||x1|| = ||x2|| supposing x1 ≡ x2
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
inner-product-space: InnerProductSpace,
ss-eq: x ≡ y,
ss-point: Point,
req: x = y,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
guard: {T},
implies: P ⇒ Q,
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
rv-norm: ||x||,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rv-ip_functionality,
rsqrt_functionality,
req_functionality,
req_weakening,
rv-ip-nonneg,
rsqrt_wf,
ss-point_wf,
separation-space_wf,
real-vector-space_wf,
inner-product-space_wf,
subtype_rel_transitivity,
inner-product-space_subtype,
real-vector-space_subtype1,
ss-eq_wf,
rv-ip_wf,
rmul_wf,
req_wf,
int-to-real_wf,
rleq_wf,
real_wf,
rv-norm_wf,
req_witness
Rules used in proof :
productElimination,
dependent_set_memberEquality,
equalitySymmetry,
equalityTransitivity,
because_Cache,
isect_memberEquality,
independent_isectElimination,
instantiate,
independent_functionElimination,
sqequalRule,
natural_numberEquality,
productEquality,
setEquality,
rename,
setElimination,
lambdaEquality,
applyEquality,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[x1,x2:Point]. ||x1|| = ||x2|| supposing x1 \mequiv{} x2
Date html generated:
2016_11_08-AM-09_16_07
Last ObjectModification:
2016_10_31-PM-04_44_40
Theory : inner!product!spaces
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