Nuprl Lemma : Euclid-Prop27
∀e:EuclideanPlane. ∀a,b,c,d,x,y:Point.
(((Colinear(x;a;b) ∧ Colinear(y;c;d)) ∧ (a # b ∧ c # d) ∧ a leftof yx ∧ c leftof xy ∧ axy ≅a cyx)
⇒ geo-parallel-points(e;a;b;c;d))
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-colinear: Colinear(a;b;c),
geo-left: a leftof bc,
geo-sep: a # b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
geo-parallel-points: geo-parallel-points(e;a;b;c;d),
not: ¬A,
false: False,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
basic-geometry: BasicGeometry,
sq_stable: SqStable(P),
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
top: Top,
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),
basic-geometry-: BasicGeometry-,
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-eq: a ≡ b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
geo-colinear: Colinear(a;b;c),
geo-cong-angle: abc ≅a xyz,
geo-out: out(p ab)
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d,x,y:Point.
(((Colinear(x;a;b) \mwedge{} Colinear(y;c;d)) \mwedge{} (a \# b \mwedge{} c \# d) \mwedge{} a leftof yx \mwedge{} c leftof xy \mwedge{} axy \mcong{}\msuba{} cyx)
{}\mRightarrow{} geo-parallel-points(e;a;b;c;d))
Date html generated:
2020_05_20-AM-10_43_12
Last ObjectModification:
2019_12_03-AM-09_47_55
Theory : euclidean!plane!geometry
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