Nuprl Lemma : geo-not-colinear
∀e:BasicGeometry. ∀[a,b,c:Point]. (¬Colinear(a;b;c) ⇐⇒ ¬(B(abc) ∨ B(bca) ∨ B(cab)))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-between: B(abc),
geo-point: Point,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
or: P ∨ Q
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
uimplies: b supposing a,
guard: {T},
or: P ∨ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
geo-colinear: Colinear(a;b;c),
false: False,
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
basic-geometry: BasicGeometry,
subtract: n - m,
cons: [a / b],
select: L[n],
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
cand: A c∧ B,
basic-geometry-: BasicGeometry-,
euclidean-plane: EuclideanPlane
Latex:
\mforall{}e:BasicGeometry. \mforall{}[a,b,c:Point]. (\mneg{}Colinear(a;b;c) \mLeftarrow{}{}\mRightarrow{} \mneg{}(B(abc) \mvee{} B(bca) \mvee{} B(cab)))
Date html generated:
2020_05_20-AM-09_49_03
Last ObjectModification:
2019_12_20-PM-07_38_01
Theory : euclidean!plane!geometry
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