Nuprl Lemma : r2-plane-separation
∀a,b:ℝ^2. ∀u:{u:ℝ^2| r2-left(u;a;b)} . ∀v:{v:ℝ^2| r2-left(v;b;a)} .
(∃x:ℝ^2 [((¬(r2-left(a;b;x) ∨ r2-left(a;x;b)))
∧ ((¬(d(u;v) < d(u;x))) ∧ (¬(d(u;v) < d(x;v))))
∧ (¬(r2-left(u;x;v) ∨ r2-left(u;v;x))))])
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]],
not: ¬A,
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,
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
false: False,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Latex:
\mforall{}a,b:\mBbbR{}\^{}2. \mforall{}u:\{u:\mBbbR{}\^{}2| r2-left(u;a;b)\} . \mforall{}v:\{v:\mBbbR{}\^{}2| r2-left(v;b;a)\} .
(\mexists{}x:\mBbbR{}\^{}2 [((\mneg{}(r2-left(a;b;x) \mvee{} r2-left(a;x;b)))
\mwedge{} ((\mneg{}(d(u;v) < d(u;x))) \mwedge{} (\mneg{}(d(u;v) < d(x;v))))
\mwedge{} (\mneg{}(r2-left(u;x;v) \mvee{} r2-left(u;v;x))))])
Date html generated:
2020_05_20-PM-01_03_55
Last ObjectModification:
2019_12_11-PM-00_22_07
Theory : reals
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