Nuprl Lemma : geo-colinear-iff
∀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,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
not: ¬A,
false: False,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
rev_implies: P ⇐ Q,
geo-colinear: Colinear(a;b;c),
cand: A c∧ B,
or: P ∨ Q,
basic-geometry-: BasicGeometry-,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
select: L[n],
cons: [a / b],
subtract: n - m
Latex:
\mforall{}e:BasicGeometry-. \mforall{}a,b,c:Point. (Colinear(a;b;c) \mLeftarrow{}{}\mRightarrow{} \mneg{}((\mneg{}B(abc)) \mwedge{} (\mneg{}B(bca)) \mwedge{} (\mneg{}B(cab))))
Date html generated:
2020_05_20-AM-09_48_35
Last ObjectModification:
2019_12_06-PM-05_37_55
Theory : euclidean!plane!geometry
Home
Index