Nuprl Lemma : rv-line-circle-lemma
∀n:ℕ. ∀r:ℝ. ∀p,q:ℝ^n.
(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 :
real-vec-sep: a ≠ b,
real-vec-norm: ||x||,
dot-product: x⋅y,
real-vec-mul: a*X,
real-vec-sub: X - Y,
real-vec-add: X + Y,
real-vec: ℝ^n,
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,
nat: ℕ,
let: let,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
real-vec-sep: a ≠ b,
real-vec-dist: d(x;y),
let: let,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
rev_uimplies: rev_uimplies(P;Q),
top: Top,
req_int_terms: t1 ≡ t2,
itermConstant: "const",
uiff: uiff(P;Q),
real: ℝ,
rnonneg: rnonneg(x),
rleq: x ≤ y,
guard: {T},
rge: x ≥ y,
or: P ∨ Q,
rneq: x ≠ y
Lemmas referenced :
nat_wf,
real_wf,
real-vec_wf,
real-vec-sep_wf,
real-vec-norm_wf,
rleq_wf,
rv-line-circle-lemma0,
real-vec-dist-symmetry,
req_weakening,
rless_functionality,
real-vec-dist_wf,
int-to-real_wf,
equal_wf,
real-vec-sub_wf,
rmul_wf,
dot-product_wf,
le_wf,
false_wf,
rnexp_wf,
rsub_wf,
rless_wf,
rmul_functionality,
radd_functionality,
real_term_value_add_lemma,
real_term_value_var_lemma,
itermAdd_wf,
itermVar_wf,
req_transitivity,
req-iff-rsub-is-0,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_term_value_const_lemma,
itermConstant_wf,
itermMultiply_wf,
itermSubtract_wf,
real_term_polynomial,
rnexp2,
rsub_functionality,
rleq_functionality,
radd_wf,
nat_plus_wf,
less_than'_wf,
rleq-int,
rmul_preserves_rleq2,
rnexp-positive,
real-vec-norm-squared,
dot-product-linearity2,
dot-product-linearity1,
req_functionality,
req-implies-req,
dot-product-comm,
iff_wf,
req_wf,
real-vec-mul_wf,
real-vec-add_wf,
real-vec-norm-eq-iff,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
real-vec-norm-nonneg,
req_inversion,
quadratic2_wf,
quadratic1_wf,
quadratic-formula1
Rules used in proof :
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
dependent_functionElimination,
independent_functionElimination,
productElimination,
independent_isectElimination,
because_Cache,
sqequalRule,
setEquality,
rename,
setElimination,
lambdaEquality,
applyEquality,
natural_numberEquality,
equalitySymmetry,
equalityTransitivity,
independent_pairFormation,
dependent_set_memberEquality,
int_eqEquality,
voidEquality,
isect_memberEquality,
intEquality,
computeAll,
axiomEquality,
minusEquality,
voidElimination,
independent_pairEquality,
isect_memberFormation,
andLevelFunctionality,
productEquality,
impliesFunctionality,
addLevel,
promote_hyp,
allFunctionality,
inrFormation,
inlFormation
Latex:
\mforall{}n:\mBbbN{}. \mforall{}r:\mBbbR{}. \mforall{}p,q:\mBbbR{}\^{}n.
(p \mneq{} 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:
2018_05_22-PM-02_29_40
Last ObjectModification:
2018_03_23-PM-04_32_53
Theory : reals
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