Nuprl Lemma : rv-norm-is-zero
∀[rv:InnerProductSpace]. ∀[x:Point(rv)]. uiff(||x|| = r0;x ≡ 0)
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
inner-product-space: InnerProductSpace,
rv-0: 0,
req: x = y,
int-to-real: r(n),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ss-eq: Error :ss-eq,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
all: ∀x:A. B[x],
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_wf,
rv-norm_wf,
int-to-real_wf,
req_witness,
Error :ss-eq_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,
Error :ss-point_wf,
Error :ss-sep_wf,
rless_irreflexivity,
rleq_weakening,
rv-ip_wf,
rmul_wf,
rleq_wf,
real_wf,
rless_transitivity1,
rv-norm-positive,
rv-norm0,
req_functionality,
rv-norm_functionality,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
functionIsTypeImplies,
inhabitedIsType,
universeIsType,
extract_by_obid,
isectElimination,
hypothesis,
applyEquality,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
independent_functionElimination,
instantiate,
independent_isectElimination,
productElimination,
independent_pairEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
voidElimination,
productEquality,
setEquality,
lambdaEquality,
lambdaFormation
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[x:Point(rv)]. uiff(||x|| = r0;x \mequiv{} 0)
Date html generated:
2020_05_20-PM-01_11_30
Last ObjectModification:
2019_12_09-PM-11_48_28
Theory : inner!product!spaces
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