Nuprl Lemma : ip-line-circle-lemma
∀rv:InnerProductSpace. ∀r:ℝ. ∀p,q:Point.
(p # q
⇒ (||p|| ≤ r)
⇒ let v = q - p in
(r0 ≤ (((r(2) * p ⋅ v) * r(2) * p ⋅ v) - r(4) * ||v||^2 * (||p||^2 - r^2)))
∧ (||p + quadratic1(||v||^2;r(2) * p ⋅ v;||p||^2 - r^2)*v|| = r)
∧ (||p + quadratic2(||v||^2;r(2) * p ⋅ v;||p||^2 - r^2)*v|| = r))
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
rv-sub: x - y,
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rv-mul: a*x,
rv-add: x + y,
quadratic2: quadratic2(a;b;c),
quadratic1: quadratic1(a;b;c),
rleq: x ≤ y,
rnexp: x^k1,
rsub: x - y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
ss-sep: x # y,
ss-point: Point,
let: let,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
and: P ∧ Q,
guard: {T},
uimplies: b supposing a,
rsub: x - y,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
so_apply: x[s],
so_lambda: λ2x.t[x],
rge: x ≥ y,
rnonneg: rnonneg(x),
rleq: x ≤ y,
iff: P ⇐⇒ Q,
let: let,
rev_implies: P ⇐ Q,
real: ℝ,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
or: P ∨ Q,
rneq: x ≠ y
Lemmas referenced :
rleq_wf,
rv-norm_wf,
real_wf,
int-to-real_wf,
req_wf,
rmul_wf,
rv-ip_wf,
ss-sep_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
ss-point_wf,
radd-zero-both,
req_weakening,
radd-rminus-both,
radd_functionality,
radd_comm,
radd-ac,
rleq_functionality,
uiff_transitivity,
rv-norm-nonneg,
rnexp-rleq,
rminus_wf,
le_wf,
false_wf,
radd_wf,
rnexp_wf,
rsub_wf,
radd-preserves-rleq,
equal_wf,
set_wf,
rv-sub_wf,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rmul_comm,
rmul-zero-both,
nat_plus_wf,
less_than'_wf,
rnexp2-nonneg,
rmul_preserves_rleq2,
ss-sep-symmetry,
rv-sep-iff-norm,
rless_wf,
rleq-int,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
req-iff-rsub-is-0,
itermVar_wf,
itermAdd_wf,
real_term_value_var_lemma,
real_term_value_add_lemma,
rsub_functionality,
rnexp2,
rnexp-positive,
rleq_transitivity,
req-implies-req,
iff_wf,
rv-add_wf,
rv-mul_wf,
rv-norm-eq-iff,
req_functionality,
req_transitivity,
rv-ip-add-squared,
rv-ip-mul2,
rmul_functionality,
rv-ip-mul,
rmul-assoc,
req_inversion,
rv-norm-squared,
quadratic2_wf,
quadratic1_wf,
quadratic-formula1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
natural_numberEquality,
sqequalRule,
instantiate,
independent_isectElimination,
because_Cache,
independent_functionElimination,
dependent_functionElimination,
independent_pairFormation,
dependent_set_memberEquality,
productElimination,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
minusEquality,
voidElimination,
independent_pairEquality,
isect_memberFormation,
computeAll,
intEquality,
isect_memberEquality,
voidEquality,
int_eqEquality,
addLevel,
impliesFunctionality,
promote_hyp,
allFunctionality,
inrFormation,
inlFormation
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}r:\mBbbR{}. \mforall{}p,q:Point.
(p \# q
{}\mRightarrow{} (||p|| \mleq{} r)
{}\mRightarrow{} let v = q - p in
(r0 \mleq{} (((r(2) * p \mcdot{} v) * r(2) * p \mcdot{} v) - r(4) * ||v||\^{}2 * (||p||\^{}2 - r\^{}2)))
\mwedge{} (||p + quadratic1(||v||\^{}2;r(2) * p \mcdot{} v;||p||\^{}2 - r\^{}2)*v|| = r)
\mwedge{} (||p + quadratic2(||v||\^{}2;r(2) * p \mcdot{} v;||p||\^{}2 - r\^{}2)*v|| = r))
Date html generated:
2017_10_05-AM-00_05_27
Last ObjectModification:
2017_07_28-AM-08_54_56
Theory : inner!product!spaces
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