Nuprl Lemma : ip-circle-circle-lemma2
∀rv:InnerProductSpace. ∀r1,r2:{r:ℝ| r0 ≤ r} . ∀b:Point.
((r0 < ||b||)
⇒ ((r1^2 - r2^2) + ||b||^2^2 ≤ (r(4) * ||b||^2 * r1^2))
⇒ (∃u,v:Point
(((||u|| = r1) ∧ (||u - b|| = r2))
∧ ((||v|| = r1) ∧ (||v - b|| = r2))
∧ (((r1^2 - r2^2) + ||b||^2^2 < (r(4) * ||b||^2 * r1^2)) ⇒ u # v))))
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
rv-sub: x - y,
inner-product-space: InnerProductSpace,
rleq: x ≤ y,
rless: x < y,
rnexp: x^k1,
rsub: x - y,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
ss-sep: x # y,
ss-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
rneq: x ≠ y,
or: P ∨ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
less_than: a < b,
squash: ↓T,
true: True,
exp: i^n,
primrec: primrec(n;b;c),
subtract: n - m,
rsub: x - y,
let: let,
cand: A c∧ B,
rv-sub: x - y,
rv-minus: -x,
sq_stable: SqStable(P),
nat_plus: ℕ+
Lemmas referenced :
ip-circle-circle-lemma1,
rv-perp-same-norm,
rv-norm-positive-iff,
rleq_wf,
rnexp_wf,
false_wf,
le_wf,
radd_wf,
rsub_wf,
rv-norm_wf,
real_wf,
int-to-real_wf,
req_wf,
rmul_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,
set_wf,
radd-preserves-rleq,
rdiv_wf,
rless-int,
equal_wf,
rminus_wf,
exp_wf2,
req-int,
rless_transitivity1,
rleq_weakening,
rmul_preserves_rleq,
squash_wf,
true_wf,
nat_wf,
iff_weakening_equal,
uiff_transitivity,
rleq_functionality,
radd_comm,
radd_functionality,
req_weakening,
radd-rminus-assoc,
radd-zero-both,
req_functionality,
rnexp-int,
rless_functionality,
req_transitivity,
req_inversion,
rnexp-rdiv,
rdiv_functionality,
rmul-rdiv-cancel2,
rmul-assoc,
rmul_functionality,
rmul_comm,
rmul-ac,
rnexp-positive,
rsqrt_wf,
all_wf,
or_wf,
ss-eq_wf,
rv-mul_wf,
rv-add_wf,
rv-sub_wf,
ss-eq_weakening,
ss-sep_wf,
exists_wf,
rmul_preserves_rless,
rmul-zero-both,
rv-minus_wf,
rv-0_wf,
ss-eq_functionality,
rv-add_functionality,
rv-mul-linear,
rv-mul-mul,
rv-add-assoc,
ss-eq_transitivity,
rv-add-swap,
rv-add-comm,
rv-mul-add-alt,
rv-mul-add,
rv-mul_functionality,
rminus-as-rmul,
rmul-minus,
rmul_over_rminus,
rminus_functionality,
rmul-one-both,
rminus-rminus,
rmul-distrib2,
rmul-identity1,
radd-int,
rv-mul0,
rv-0-add,
rv-sep-iff-norm,
rv-norm_functionality,
rabs_wf,
rmul-is-positive,
rleq-int,
rleq_weakening_rless,
sq_stable__rleq,
rv-norm-mul,
rabs-of-nonneg,
rabs-rmul,
rmul-int,
square-rless-implies,
less_than_wf,
rnexp0,
rnexp2,
sq_stable__and,
sq_stable__req,
req_witness,
radd-preserves-rless
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
because_Cache,
productElimination,
rename,
isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
setElimination,
applyEquality,
lambdaEquality,
setEquality,
productEquality,
instantiate,
independent_isectElimination,
inrFormation,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
imageElimination,
universeEquality,
functionEquality,
inlFormation,
dependent_pairFormation,
addLevel,
minusEquality,
addEquality,
multiplyEquality,
promote_hyp,
isect_memberEquality,
levelHypothesis
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}r1,r2:\{r:\mBbbR{}| r0 \mleq{} r\} . \mforall{}b:Point.
((r0 < ||b||)
{}\mRightarrow{} ((r1\^{}2 - r2\^{}2) + ||b||\^{}2\^{}2 \mleq{} (r(4) * ||b||\^{}2 * r1\^{}2))
{}\mRightarrow{} (\mexists{}u,v:Point
(((||u|| = r1) \mwedge{} (||u - b|| = r2))
\mwedge{} ((||v|| = r1) \mwedge{} (||v - b|| = r2))
\mwedge{} (((r1\^{}2 - r2\^{}2) + ||b||\^{}2\^{}2 < (r(4) * ||b||\^{}2 * r1\^{}2)) {}\mRightarrow{} u \# v))))
Date html generated:
2017_10_05-AM-00_09_57
Last ObjectModification:
2017_03_14-PM-02_46_41
Theory : inner!product!spaces
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