Nuprl Lemma : geo-out-strict_transitivity
∀e:BasicGeometry. ∀a,b,c,d:Point.
(b ≠ d ⇒ geo-out-strict(e;a;b;c) ⇒ geo-out-strict(e;a;c;d) ⇒ geo-out-strict(e;a;b;d))
Proof
Definitions occuring in Statement :
geo-out-strict: geo-out-strict(e;p;a;b),
basic-geometry: BasicGeometry,
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,
geo-out-strict: geo-out-strict(e;p;a;b),
not: ¬A,
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],
false: False,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
geo-strict-between: a-b-c,
cand: A c∧ B
Lemmas referenced :
geo-strict-between-sym,
subtype_rel_self,
basic-geometry-_wf,
geo-strict-between-trans,
geo-strict-between-trans2,
geo-strict-between-trans3,
geo-strict-between_wf,
istype-void,
geo-out-strict_wf,
geo-sep_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,
geo-between-same-side,
geo-strict-between-implies-between,
geo-strict-between-sep2,
geo-between_wf,
geo-strict-between-sep1,
geo-sep-sym,
geo-between-middle,
geo-between-symmetry,
geo-between-inner-trans
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
independent_functionElimination,
independent_pairFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
instantiate,
isectElimination,
because_Cache,
voidElimination,
universeIsType,
productIsType,
functionIsType,
independent_isectElimination,
inhabitedIsType
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,d:Point.
(b \mneq{} d {}\mRightarrow{} geo-out-strict(e;a;b;c) {}\mRightarrow{} geo-out-strict(e;a;c;d) {}\mRightarrow{} geo-out-strict(e;a;b;d))
Date html generated:
2019_10_16-PM-01_23_00
Last ObjectModification:
2019_08_27-AM-08_00_41
Theory : euclidean!plane!geometry
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