Nuprl Lemma : rv-ip-nonneg
∀[rv:InnerProductSpace]. ∀[x:Point(rv)]. (r0 ≤ x^2)
Proof
Definitions occuring in Statement :
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rleq: x ≤ y,
int-to-real: r(n),
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
false: False,
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
prop: ℙ,
guard: {T},
subtype_rel: A ⊆r B,
and: P ∧ Q,
le: A ≤ B,
all: ∀x:A. B[x],
rnonneg: rnonneg(x),
rleq: x ≤ y,
uimplies: b supposing a,
stable: Stable{P},
member: t ∈ T,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
ss-eq: Error :ss-eq,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
istype-void,
rleq_wf,
not_wf,
rv-0_wf,
Error :ss-sep_wf,
false_wf,
Error :separation-space_wf,
real-vector-space_wf,
inner-product-space_wf,
subtype_rel_transitivity,
inner-product-space_subtype,
real-vector-space_subtype1,
Error :ss-point_wf,
le_witness_for_triv,
rv-ip_wf,
int-to-real_wf,
stable__rleq,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
rleq_weakening_rless,
rv-ip-positive,
rleq_weakening_equal,
uiff_transitivity,
rleq_functionality,
req_weakening,
rv-ip_functionality,
rv-ip0
Rules used in proof :
unionIsType,
functionIsType,
voidElimination,
independent_functionElimination,
lambdaFormation_alt,
functionEquality,
because_Cache,
unionEquality,
isectIsTypeImplies,
isect_memberEquality_alt,
instantiate,
applyEquality,
universeIsType,
inhabitedIsType,
functionIsTypeImplies,
equalitySymmetry,
equalityTransitivity,
productElimination,
dependent_functionElimination,
lambdaEquality_alt,
sqequalRule,
independent_isectElimination,
hypothesisEquality,
hypothesis,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
unionElimination
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[x:Point(rv)]. (r0 \mleq{} x\^{}2)
Date html generated:
2020_05_20-PM-01_11_07
Last ObjectModification:
2019_12_28-AM-10_59_03
Theory : inner!product!spaces
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