Nuprl Lemma : r2-straightedge-compass-simple
∀c,d,a:ℝ^2. ∀b:{b:ℝ^2|
(d(b;b) < d(b;a))
∧ ((¬(d(c;d) < d(c;b))) ∧ (¬(d(c;d) < d(b;d))))
∧ (¬(r2-left(c;b;d) ∨ r2-left(c;d;b)))} .
(∃u:ℝ^2 [((¬((d(c;u) < d(c;d)) ∨ (d(c;d) < d(c;u))))
∧ (((¬(d(a;u) < d(a;b))) ∧ (¬(d(a;u) < d(b;u)))) ∧ (¬(r2-left(a;b;u) ∨ r2-left(a;u;b))))
∧ ((d(b;b) < d(b;d)) ⇒ (d(b;b) < d(b;u))))])
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,
implies: P ⇒ Q,
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,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
prop: ℙ,
or: P ∨ Q,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_exists: ∃x:A [B[x]]
Latex:
\mforall{}c,d,a:\mBbbR{}\^{}2. \mforall{}b:\{b:\mBbbR{}\^{}2|
(d(b;b) < d(b;a))
\mwedge{} ((\mneg{}(d(c;d) < d(c;b))) \mwedge{} (\mneg{}(d(c;d) < d(b;d))))
\mwedge{} (\mneg{}(r2-left(c;b;d) \mvee{} r2-left(c;d;b)))\} .
(\mexists{}u:\mBbbR{}\^{}2 [((\mneg{}((d(c;u) < d(c;d)) \mvee{} (d(c;d) < d(c;u))))
\mwedge{} (((\mneg{}(d(a;u) < d(a;b))) \mwedge{} (\mneg{}(d(a;u) < d(b;u)))) \mwedge{} (\mneg{}(r2-left(a;b;u) \mvee{} r2-left(a;u;b))))
\mwedge{} ((d(b;b) < d(b;d)) {}\mRightarrow{} (d(b;b) < d(b;u))))])
Date html generated:
2020_05_20-PM-01_04_26
Last ObjectModification:
2019_12_11-PM-01_04_37
Theory : reals
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