Nuprl Lemma : geo-intersect-lines
∀e:EuclideanPlane. ∀p,l:Line.
(p \/ l
⇐⇒ ∃a,b:Point
(Colinear(a;fst(p);fst(snd(p)))
∧ Colinear(b;fst(p);fst(snd(p)))
∧ a leftof fst(l)fst(snd(l))
∧ b leftof fst(snd(l))fst(l)))
Proof
Definitions occuring in Statement :
geo-intersect: L \/ M,
geo-line: Line,
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-left: a leftof bc,
geo-point: Point,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
guard: {T},
uimplies: b supposing a,
geo-line: Line,
pi1: fst(t),
pi2: snd(t),
uiff: uiff(P;Q),
oriented-plane: OrientedPlane,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
true: True,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
append: as @ bs,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
geo-lsep: a # bc,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
sq_stable: SqStable(P),
geo-intersect: L \/ M
Latex:
\mforall{}e:EuclideanPlane. \mforall{}p,l:Line.
(p \mbackslash{}/ l
\mLeftarrow{}{}\mRightarrow{} \mexists{}a,b:Point
(Colinear(a;fst(p);fst(snd(p)))
\mwedge{} Colinear(b;fst(p);fst(snd(p)))
\mwedge{} a leftof fst(l)fst(snd(l))
\mwedge{} b leftof fst(snd(l))fst(l)))
Date html generated:
2020_05_20-AM-10_45_55
Last ObjectModification:
2020_01_13-PM-05_50_24
Theory : euclidean!plane!geometry
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