Nuprl Lemma : geo-colinear-preserves-parallel2
∀e:EuclideanPlane. ∀a,b,c,d,x,y:Point.
(geo-parallel-points(e;a;b;c;d)
⇒ Colinear(a;b;x)
⇒ Colinear(c;d;y)
⇒ c ≠ y
⇒ a ≠ x
⇒ geo-parallel-points(e;a;x;c;y))
Proof
Definitions occuring in Statement :
geo-parallel-points: geo-parallel-points(e;a;b;c;d),
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: 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
Lemmas referenced :
geo-colinear-preserves-parallel,
geo-parallel-points-symmetry,
geo-sep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-colinear_wf,
geo-parallel-points_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
inhabitedIsType
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d,x,y:Point.
(geo-parallel-points(e;a;b;c;d)
{}\mRightarrow{} Colinear(a;b;x)
{}\mRightarrow{} Colinear(c;d;y)
{}\mRightarrow{} c \mneq{} y
{}\mRightarrow{} a \mneq{} x
{}\mRightarrow{} geo-parallel-points(e;a;x;c;y))
Date html generated:
2019_10_16-PM-01_47_28
Last ObjectModification:
2019_08_23-PM-10_43_18
Theory : euclidean!plane!geometry
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