Nuprl Lemma : parallelogram-construction
∀e:EuclideanPlane. ∀a,b,c,x,y:Point.
(a # bc
⇒ a-x-b
⇒ a-y-c
⇒ (∃t:Point
(geo-parallel-points(e;b;x;y;t)
∧ (¬¬((a leftof bc ⇒ (t leftof ac ∧ t leftof bc)) ∧ (a leftof cb ⇒ (t leftof ca ∧ t leftof cb)))))))
Proof
Definitions occuring in Statement :
geo-parallel-points: geo-parallel-points(e;a;b;c;d),
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-lsep: a # bc,
geo-left: a leftof bc,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
prop: ℙ,
not: ¬A,
uall: ∀[x:A]. B[x],
false: False,
guard: {T},
uimplies: b supposing a
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y:Point.
(a \# bc
{}\mRightarrow{} a-x-b
{}\mRightarrow{} a-y-c
{}\mRightarrow{} (\mexists{}t:Point
(geo-parallel-points(e;b;x;y;t)
\mwedge{} (\mneg{}\mneg{}((a leftof bc {}\mRightarrow{} (t leftof ac \mwedge{} t leftof bc))
\mwedge{} (a leftof cb {}\mRightarrow{} (t leftof ca \mwedge{} t leftof cb)))))))
Date html generated:
2020_05_20-AM-10_44_35
Last ObjectModification:
2020_01_13-PM-05_38_27
Theory : euclidean!plane!geometry
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