Nuprl Lemma : geo-parallel-points-implies
∀e:EuclideanPlane. ∀a,b,x,y:Point.
(geo-parallel-points(e;a;b;x;y)
⇒ (a ≠ b ∧ x ≠ y)
⇒ (∀x1,y1:{z:Point| Colinear(z;a;b)} . (x1 leftof xy ⇒ (¬y1 leftof yx))))
Proof
Definitions occuring in Statement :
geo-parallel-points: geo-parallel-points(e;a;b;c;d),
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-left: a leftof bc,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
geo-parallel-points: geo-parallel-points(e;a;b;c;d),
exists: ∃x:A. B[x],
cand: A c∧ B
Lemmas referenced :
geo-left_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-colinear_wf,
geo-sep_wf,
geo-parallel-points_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
sqequalHypSubstitution,
productElimination,
hypothesis,
independent_functionElimination,
voidElimination,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
setElimination,
rename,
because_Cache,
inhabitedIsType,
setIsType,
productIsType,
dependent_functionElimination,
dependent_pairFormation_alt,
independent_pairFormation
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,x,y:Point.
(geo-parallel-points(e;a;b;x;y)
{}\mRightarrow{} (a \mneq{} b \mwedge{} x \mneq{} y)
{}\mRightarrow{} (\mforall{}x1,y1:\{z:Point| Colinear(z;a;b)\} . (x1 leftof xy {}\mRightarrow{} (\mneg{}y1 leftof yx))))
Date html generated:
2019_10_16-PM-01_46_30
Last ObjectModification:
2019_08_19-PM-03_02_51
Theory : euclidean!plane!geometry
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