Nuprl Lemma : rv-ip-positive
∀rv:InnerProductSpace. ∀x:Point. (x # 0 ⇐⇒ r0 < x^2)
Proof
Definitions occuring in Statement :
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rv-0: 0,
ss-sep: x # y,
ss-point: Point,
rless: x < y,
int-to-real: r(n),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n
Definitions unfolded in proof :
rv-ip: x ⋅ y,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
and: P ∧ Q,
guard: {T},
uall: ∀[x:A]. B[x],
btrue: tt,
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
subtype_rel: A ⊆r B,
record-select: r.x,
record+: record+,
inner-product-space: InnerProductSpace,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
inner-product-space_wf,
exists_wf,
int-to-real_wf,
rless_wf,
rv-0_wf,
ss-sep_wf,
iff_wf,
rmul_wf,
rv-mul_wf,
radd_wf,
rv-add_wf,
req_wf,
ss-eq_wf,
all_wf,
real_wf,
real-vector-space_subtype1,
ss-point_wf,
subtype_rel_self
Rules used in proof :
rename,
setElimination,
natural_numberEquality,
functionExtensionality,
lambdaEquality,
productEquality,
because_Cache,
equalitySymmetry,
equalityTransitivity,
functionEquality,
setEquality,
isectElimination,
extract_by_obid,
tokenEquality,
applyEquality,
hypothesis,
thin,
dependentIntersectionEqElimination,
sqequalRule,
dependentIntersectionElimination,
sqequalHypSubstitution,
introduction,
hypothesisEquality,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}x:Point. (x \# 0 \mLeftarrow{}{}\mRightarrow{} r0 < x\^{}2)
Date html generated:
2016_11_08-AM-09_15_01
Last ObjectModification:
2016_11_02-PM-03_13_47
Theory : inner!product!spaces
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