Nuprl Lemma : Euclid-Prop28_2
∀e:EuclideanPlane. ∀a,b,c,d,x,y,p:Point.
(((Colinear(x;a;b) ∧ Colinear(y;c;d)) ∧ (a leftof yx ∧ a-x-b) ∧ (c leftof xy ∧ c-y-d) ∧ p-x-y ∧ Ryxa ∧ Rxyc)
⇒ geo-parallel-points(e;a;b;c;d))
Proof
Definitions occuring in Statement :
geo-parallel-points: geo-parallel-points(e;a;b;c;d),
euclidean-plane: EuclideanPlane,
right-angle: Rabc,
geo-colinear: Colinear(a;b;c),
geo-strict-between: a-b-c,
geo-left: a leftof bc,
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,
member: t ∈ T,
guard: {T},
iff: P ⇐⇒ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
prop: ℙ,
cand: A c∧ B,
basic-geometry: BasicGeometry
Lemmas referenced :
adjacent-right-angles-supplementary,
left-implies-sep,
geo-colinear_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-left_wf,
geo-strict-between_wf,
right-angle_wf,
geo-point_wf,
Euclid-Prop27,
geo-sep-sym,
geo-right-angles-congruent,
geo-cong-angle-symmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
sqequalRule,
productIsType,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
inhabitedIsType,
independent_pairFormation
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d,x,y,p:Point.
(((Colinear(x;a;b) \mwedge{} Colinear(y;c;d))
\mwedge{} (a leftof yx \mwedge{} a-x-b)
\mwedge{} (c leftof xy \mwedge{} c-y-d)
\mwedge{} p-x-y
\mwedge{} Ryxa
\mwedge{} Rxyc)
{}\mRightarrow{} geo-parallel-points(e;a;b;c;d))
Date html generated:
2019_10_16-PM-02_38_19
Last ObjectModification:
2019_06_20-PM-05_43_18
Theory : euclidean!plane!geometry
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