Nuprl Lemma : not-proj-point-sep-is-equiv
∀e:EuclideanParPlane
((∀l,m:Line. (l \/ m ⇒ (∀n:Line. (l \/ n ∨ m \/ n)))) ⇒ EquivRel(Point + Line;p,q.¬proj-point-sep(e;p;q)))
Proof
Definitions occuring in Statement :
proj-point-sep: proj-point-sep(eu;p;q),
euclidean-parallel-plane: EuclideanParPlane,
geo-intersect: L \/ M,
geo-line: Line,
geo-point: Point,
equiv_rel: EquivRel(T;x,y.E[x; y]),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
union: left + right
Definitions unfolded in proof :
or: P ∨ Q,
so_apply: x[s],
euclidean-parallel-plane: EuclideanParPlane,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
trans: Trans(T;x,y.E[x; y]),
prop: ℙ,
false: False,
not: ¬A,
sym: Sym(T;x,y.E[x; y]),
cand: A c∧ B,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
refl: Refl(T;x,y.E[x; y]),
and: P ∧ Q,
equiv_rel: EquivRel(T;x,y.E[x; y]),
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
proj-point-sep-cotrans,
proj-point-sep-symmetry,
or_wf,
geoline-subtype1,
geo-intersect_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
euclidean-parallel-plane_wf,
subtype_rel_transitivity,
euclidean-planes-subtype,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
all_wf,
not_wf,
proj-point-sep_wf,
geo-line_wf,
geo-point_wf,
proj-point-sep-irrefl
Rules used in proof :
unionElimination,
rename,
setElimination,
functionEquality,
lambdaEquality,
independent_isectElimination,
instantiate,
voidElimination,
independent_functionElimination,
sqequalRule,
because_Cache,
applyEquality,
isectElimination,
unionEquality,
hypothesis,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:EuclideanParPlane
((\mforall{}l,m:Line. (l \mbackslash{}/ m {}\mRightarrow{} (\mforall{}n:Line. (l \mbackslash{}/ n \mvee{} m \mbackslash{}/ n))))
{}\mRightarrow{} EquivRel(Point + Line;p,q.\mneg{}proj-point-sep(e;p;q)))
Date html generated:
2018_05_22-PM-01_14_14
Last ObjectModification:
2018_05_21-PM-02_23_51
Theory : euclidean!plane!geometry
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