Nuprl Lemma : Euclid-prop14
∀g:EuclideanPlane. ∀a,b,x,y:Point. (a ≠ b ⇒ x leftof ab ⇒ y leftof ba ⇒ Rxba ⇒ Ryba ⇒ Colinear(x;b;y))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
right-angle: Rabc,
geo-colinear: Colinear(a;b;c),
geo-left: a leftof bc,
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,
member: t ∈ T,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ
Lemmas referenced :
adjacent-right-angles,
geo-sep-sym,
right-angle_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-left_wf,
geo-sep_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
hypothesisEquality,
independent_functionElimination,
because_Cache,
hypothesis,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
inhabitedIsType
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,x,y:Point.
(a \mneq{} b {}\mRightarrow{} x leftof ab {}\mRightarrow{} y leftof ba {}\mRightarrow{} Rxba {}\mRightarrow{} Ryba {}\mRightarrow{} Colinear(x;b;y))
Date html generated:
2019_10_16-PM-01_55_51
Last ObjectModification:
2018_10_15-PM-00_32_06
Theory : euclidean!plane!geometry
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