Nuprl Lemma : geo-intersect-points-symmetry
∀e:EuclideanPlane. ∀a,b,c,d:Point. (ab \/ cd ⇒ cd \/ ab)
Proof
Definitions occuring in Statement :
geo-intersect-points: ab \/ cd,
euclidean-plane: EuclideanPlane,
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,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
prop: ℙ,
cand: A c∧ B,
basic-geometry-: BasicGeometry-,
oriented-plane: OrientedPlane
Lemmas referenced :
geo-intersect-points-iff,
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-point_wf,
geo-strict-between-sym,
geo-strict-between_wf,
geo-left_wf,
geo-colinear_wf,
between-preserves-left-5,
geo-sep-sym,
geo-strict-between-sep3,
geo-strict-between-implies-between,
left-symmetry,
between-preserves-left-1,
between-preserves-left-2,
geo-strict-between-sep2,
left-between-implies-right1,
between-preserves-left-3,
geo-between-symmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
independent_pairFormation,
universeIsType,
applyEquality,
instantiate,
isectElimination,
independent_isectElimination,
sqequalRule,
inhabitedIsType,
because_Cache,
dependent_pairFormation_alt,
productIsType
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point. (ab \mbackslash{}/ cd {}\mRightarrow{} cd \mbackslash{}/ ab)
Date html generated:
2019_10_16-PM-01_45_36
Last ObjectModification:
2019_08_19-PM-01_44_57
Theory : euclidean!plane!geometry
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