Nuprl Lemma : rsqrt-1-plus-ip-positive
∀rv:InnerProductSpace. ∀h:Point. (r0 < rsqrt(r1 + h^2))
Proof
Definitions occuring in Statement :
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rsqrt: rsqrt(x),
rless: x < y,
radd: a + b,
int-to-real: r(n),
ss-point: Point,
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
rge: x ≥ y
Lemmas referenced :
rsqrt-positive,
radd_wf,
int-to-real_wf,
rv-ip_wf,
rless_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,
trivial-rless-radd,
rless-int,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
rv-ip-nonneg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
dependent_set_memberEquality,
isectElimination,
natural_numberEquality,
hypothesis,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
productElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}h:Point. (r0 < rsqrt(r1 + h\^{}2))
Date html generated:
2017_10_04-PM-11_51_07
Last ObjectModification:
2017_06_21-PM-00_45_56
Theory : inner!product!spaces
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