Nuprl Lemma : r2-compass-compass-simple
∀a,b:ℝ^2. ∀c:{c:ℝ^2| d(a;a) < d(a;c)} . ∀d:{d:ℝ^2|
↓∃p,q:ℝ^2
(((¬((d(a;b) < d(a;p)) ∨ (d(a;p) < d(a;b)))) ∧ (d(c;p) < d(c;d)))
∧ (¬((d(c;d) < d(c;q)) ∨ (d(c;q) < d(c;d))))
∧ (d(a;q) < d(a;b)))} .
(∃u:ℝ^2 [((¬((d(a;b) < d(a;u)) ∨ (d(a;u) < d(a;b)))) ∧ (¬((d(c;d) < d(c;u)) ∨ (d(c;u) < d(c;d)))) ∧ r2-left(u;a;c))])
Proof
Definitions occuring in Statement :
r2-left: r2-left(p;q;r),
real-vec-dist: d(x;y),
real-vec: ℝ^n,
rless: x < y,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
or: 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,
uall: ∀[x:A]. B[x],
prop: ℙ,
exists: ∃x:A. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
false: False,
or: P ∨ Q,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
squash: ↓T,
cand: A c∧ B,
sq_exists: ∃x:A [B[x]]
Latex:
\mforall{}a,b:\mBbbR{}\^{}2. \mforall{}c:\{c:\mBbbR{}\^{}2| d(a;a) < d(a;c)\} . \mforall{}d:\{d:\mBbbR{}\^{}2|
\mdownarrow{}\mexists{}p,q:\mBbbR{}\^{}2
(((\mneg{}((d(a;b) < d(a;p)) \mvee{} (d(a;p) < d(a;b))))
\mwedge{} (d(c;p) < d(c;d)))
\mwedge{} (\mneg{}((d(c;d) < d(c;q)) \mvee{} (d(c;q) < d(c;d))))
\mwedge{} (d(a;q) < d(a;b)))\} .
(\mexists{}u:\mBbbR{}\^{}2 [((\mneg{}((d(a;b) < d(a;u)) \mvee{} (d(a;u) < d(a;b))))
\mwedge{} (\mneg{}((d(c;d) < d(c;u)) \mvee{} (d(c;u) < d(c;d))))
\mwedge{} r2-left(u;a;c))])
Date html generated:
2020_05_20-PM-01_04_52
Last ObjectModification:
2019_12_11-PM-03_26_00
Theory : reals
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