Nuprl Lemma : rv-line-circle-1
∀n:ℕ. ∀a,b,p,q:ℝ^n.
(a ≠ b
⇒ 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:{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)))))
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),
real-vec: ℝ^n,
rleq: x ≤ y,
rless: x < y,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
sq_exists: ∃x:A [B[x]],
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
rv-line-circle-0,
rv-congruent_wf,
not_wf,
real-vec-sep_wf,
rv-between_wf,
rless_wf,
real-vec-dist_wf,
exists_wf,
real-vec_wf,
rleq_wf,
real_wf,
int-to-real_wf,
nat_wf,
sq_stable__all,
sq_stable__and,
sq_stable__rv-between
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
productElimination,
dependent_pairFormation,
sqequalRule,
setElimination,
rename,
dependent_set_memberEquality,
independent_pairFormation,
productEquality,
isectElimination,
functionEquality,
applyEquality,
because_Cache,
setEquality,
lambdaEquality,
natural_numberEquality,
isect_memberEquality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,p,q:\mBbbR{}\^{}n.
(a \mneq{} b
{}\mRightarrow{} p \mneq{} q
{}\mRightarrow{} (d(a;p) \mleq{} d(a;b))
{}\mRightarrow{} (d(a;b) \mleq{} d(a;q))
{}\mRightarrow{} (\mexists{}u:\{u:\mBbbR{}\^{}n| ab=au \mwedge{} (\mneg{}(q \mneq{} u \mwedge{} u \mneq{} p \mwedge{} (\mneg{}q-u-p)))\}
\mexists{}v:\{v:\mBbbR{}\^{}n| ab=av \mwedge{} (\mneg{}(q \mneq{} p \mwedge{} p \mneq{} v \mwedge{} (\mneg{}q-p-v)))\}
((d(a;p) < d(a;b)) {}\mRightarrow{} (q-p-v \mwedge{} ((d(a;b) < d(a;q)) {}\mRightarrow{} q-u-p)))))
Date html generated:
2018_05_22-PM-02_34_42
Last ObjectModification:
2018_03_27-AM-11_34_05
Theory : reals
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