Nuprl Lemma : rv-line-circle-0-ext
∀n:ℕ. ∀a,b,p,q:ℝ^n.
(p ≠ q
⇒ (d(a;p) ≤ d(a;b))
⇒ (d(a;b) ≤ d(a;q))
⇒ (∃u:∃u:ℝ^n [(ab=au ∧ (¬(q ≠ u ∧ u ≠ p ∧ (¬q-u-p))))]
(∃v:ℝ^n [(ab=av
∧ (¬(q ≠ p ∧ p ≠ v ∧ (¬q-p-v)))
∧ ((d(a;p) < d(a;b)) ⇒ (q-p-v ∧ ((d(a;b) < d(a;q)) ⇒ q-u-p)))
∧ ((d(a;p) = d(a;b))
⇒ ((u ≠ v ⇒ ((req-vec(n;u;p) ∧ (r0 < p - a⋅q - p)) ∨ (req-vec(n;v;p) ∧ (p - a⋅q - p < r0))))
∧ (req-vec(n;u;v) ⇒ ((p - a⋅q - p = r0) ∧ req-vec(n;u;p))))))])))
Proof
Definitions occuring in Statement :
rv-between: a-b-c,
real-vec-sep: a ≠ b,
rv-congruent: ab=cd,
real-vec-dist: d(x;y),
dot-product: x⋅y,
real-vec-sub: X - Y,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
rleq: x ≤ y,
rless: x < y,
req: x = y,
int-to-real: r(n),
nat: ℕ,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ
Lemmas referenced :
rvlinecircle0_wf,
rleq_wf,
real-vec-dist_wf,
real_wf,
int-to-real_wf,
real-vec-sep_wf,
real-vec_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
sqequalRule,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
applyEquality,
because_Cache,
lambdaEquality,
setElimination,
rename,
setEquality,
natural_numberEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,p,q:\mBbbR{}\^{}n.
(p \mneq{} q
{}\mRightarrow{} (d(a;p) \mleq{} d(a;b))
{}\mRightarrow{} (d(a;b) \mleq{} d(a;q))
{}\mRightarrow{} (\mexists{}u:\mexists{}u:\mBbbR{}\^{}n [(ab=au \mwedge{} (\mneg{}(q \mneq{} u \mwedge{} u \mneq{} p \mwedge{} (\mneg{}q-u-p))))]
(\mexists{}v:\mBbbR{}\^{}n [(ab=av
\mwedge{} (\mneg{}(q \mneq{} p \mwedge{} p \mneq{} v \mwedge{} (\mneg{}q-p-v)))
\mwedge{} ((d(a;p) < d(a;b)) {}\mRightarrow{} (q-p-v \mwedge{} ((d(a;b) < d(a;q)) {}\mRightarrow{} q-u-p)))
\mwedge{} ((d(a;p) = d(a;b))
{}\mRightarrow{} ((u \mneq{} v
{}\mRightarrow{} ((req-vec(n;u;p) \mwedge{} (r0 < p - a\mcdot{}q - p))
\mvee{} (req-vec(n;v;p) \mwedge{} (p - a\mcdot{}q - p < r0))))
\mwedge{} (req-vec(n;u;v) {}\mRightarrow{} ((p - a\mcdot{}q - p = r0) \mwedge{} req-vec(n;u;p))))))])))
Date html generated:
2018_05_22-PM-02_34_25
Last ObjectModification:
2018_03_27-PM-05_14_24
Theory : reals
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