Nuprl Lemma : parallelogram-construction2
∀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)
∧ bx ≅ yt
∧ xt ≅ by
∧ xby ≅a xty
∧ (¬¬((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),
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-congruent: ab ≅ cd,
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],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
sq_exists: ∃x:A [B[x]],
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
squash: ↓T,
basic-geometry: BasicGeometry,
exists: ∃x:A. B[x],
cand: A c∧ B,
geo-midpoint: a=m=b,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
basic-geometry-: BasicGeometry-,
uiff: uiff(P;Q),
geo-tri: Triangle(a;b;c),
geo-cong-angle: abc ≅a xyz,
geo-strict-between: a-b-c,
geo-lsep: a # bc,
oriented-plane: OrientedPlane,
stable: Stable{P},
geo-eq: a ≡ b,
iff: P ⇐⇒ Q,
geo-out: out(p ab),
geo-parallel-points: geo-parallel-points(e;a;b;c;d),
so_apply: x[s1;s2;s3],
so_lambda: so_lambda3,
append: as @ bs,
ge: i ≥ j ,
true: True,
less_than: a < b,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
l_member: (x ∈ l)
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{} bx \mcong{} yt
\mwedge{} xt \mcong{} by
\mwedge{} xby \mcong{}\msuba{} xty
\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_19
Last ObjectModification:
2020_01_27-PM-09_56_20
Theory : euclidean!plane!geometry
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