Nuprl Lemma : geo-parallel-not-intersect
∀e:EuclideanPlane. ∀a,b,c,d:Point. (geo-parallel-points(e;a;b;c;d) ⇒ (¬ab \/ cd))
Proof
Definitions occuring in Statement :
geo-parallel-points: geo-parallel-points(e;a;b;c;d),
geo-intersect-points: ab \/ cd,
euclidean-plane: EuclideanPlane,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
prop: ℙ,
geo-intersect-points: ab \/ cd,
and: P ∧ Q,
geo-parallel-points: geo-parallel-points(e;a;b;c;d)
Lemmas referenced :
geo-intersect-points_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-parallel-points_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
isectElimination,
independent_isectElimination,
sqequalRule,
independent_functionElimination,
voidElimination,
because_Cache,
inhabitedIsType,
productElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point. (geo-parallel-points(e;a;b;c;d) {}\mRightarrow{} (\mneg{}ab \mbackslash{}/ cd))
Date html generated:
2019_10_16-PM-01_46_15
Last ObjectModification:
2019_08_19-PM-03_38_11
Theory : euclidean!plane!geometry
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