Nuprl Lemma : geo-parallel-iff-not-intersect
∀e:EuclideanPlane. ∀a,b,c,d:Point. (geo-parallel-points(e;a;b;c;d) ⇐⇒ (a # b ∧ c # d) ∧ (¬ab \/ cd))
Proof
Definitions occuring in Statement :
geo-parallel-points: geo-parallel-points(e;a;b;c;d),
geo-intersect-points: ab \/ cd,
euclidean-plane: EuclideanPlane,
geo-sep: a # b,
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,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
prop: ℙ,
rev_implies: P ⇐ Q,
geo-parallel-points: geo-parallel-points(e;a;b;c;d),
cand: A c∧ B,
geo-intersect-points: ab \/ cd,
exists: ∃x:A. B[x]
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point.
(geo-parallel-points(e;a;b;c;d) \mLeftarrow{}{}\mRightarrow{} (a \# b \mwedge{} c \# d) \mwedge{} (\mneg{}ab \mbackslash{}/ cd))
Date html generated:
2020_05_20-AM-10_04_57
Last ObjectModification:
2020_01_13-PM-10_21_19
Theory : euclidean!plane!geometry
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