Nuprl Lemma : geo-between-implies-out3
∀e:BasicGeometry. ∀p,a,b,c:Point. (p-b-a ⇒ p-c-a ⇒ out(p bc))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
basic-geometry: BasicGeometry,
geo-strict-between: a-b-c,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-strict-between: a-b-c,
and: P ∧ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
uall: ∀[x:A]. B[x],
geo-out: out(p ab),
guard: {T},
uimplies: b supposing a,
prop: ℙ
Lemmas referenced :
geo-between-out,
geo-strict-between-sep1,
subtype_rel_self,
basic-geometry-_wf,
geo-out_transitivity,
geo-out_inversion,
geo-strict-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
applyEquality,
sqequalRule,
instantiate,
isectElimination,
universeIsType,
independent_isectElimination,
because_Cache,
inhabitedIsType
Latex:
\mforall{}e:BasicGeometry. \mforall{}p,a,b,c:Point. (p-b-a {}\mRightarrow{} p-c-a {}\mRightarrow{} out(p bc))
Date html generated:
2019_10_16-PM-01_23_42
Last ObjectModification:
2019_01_03-PM-09_43_08
Theory : euclidean!plane!geometry
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