Nuprl Lemma : geo-between-out-implies-out
∀e:EuclideanPlane. ∀a,b,c,c':Point. (out(a cc') ⇒ (a-b-c ∨ a-c-b) ⇒ out(a c'b))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
basic-geometry: BasicGeometry,
geo-out: out(p ab),
and: P ∧ Q,
basic-geometry-: BasicGeometry-
Lemmas referenced :
geo-strict-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-out_wf,
geo-point_wf,
geo-between-out,
geo-strict-between-sep2,
geo-strict-between-implies-between,
geo-out_transitivity,
geo-out_inversion,
geo-strict-between-sep1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
unionElimination,
thin,
sqequalRule,
unionIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
because_Cache,
inhabitedIsType,
dependent_functionElimination,
independent_functionElimination,
productElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,c':Point. (out(a cc') {}\mRightarrow{} (a-b-c \mvee{} a-c-b) {}\mRightarrow{} out(a c'b))
Date html generated:
2019_10_16-PM-01_23_52
Last ObjectModification:
2018_11_08-AM-09_36_45
Theory : euclidean!plane!geometry
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