Nuprl Lemma : proj-point-sep-cotrans
∀e:EuclideanParPlane
((∀l,m:Line. (l \/ m ⇒ (∀n:Line. (l \/ n ∨ m \/ n))))
⇒ (∀x,y:Point + Line.
(proj-point-sep(e;x;y) ⇒ (∀z:Point + Line. (proj-point-sep(e;x;z) ∨ proj-point-sep(e;y;z))))))
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,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
union: left + right
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
true: True,
or: P ∨ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
euclidean-parallel-plane: EuclideanParPlane,
member: t ∈ T,
proj-point-sep: proj-point-sep(eu;p;q),
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
or_wf,
all_wf,
proj-point-sep_wf,
geo-line_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,
geo-point_wf,
geoline-subtype1,
geo-intersect_wf,
true_wf,
geo-sep_wf,
geo-sep-or
Rules used in proof :
independent_functionElimination,
functionEquality,
lambdaEquality,
independent_isectElimination,
instantiate,
unionEquality,
inrFormation,
natural_numberEquality,
inlFormation,
because_Cache,
applyEquality,
isectElimination,
dependent_set_memberEquality,
hypothesis,
hypothesisEquality,
rename,
setElimination,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
thin,
unionElimination,
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{} (\mforall{}x,y:Point + Line.
(proj-point-sep(e;x;y)
{}\mRightarrow{} (\mforall{}z:Point + Line. (proj-point-sep(e;x;z) \mvee{} proj-point-sep(e;y;z))))))
Date html generated:
2018_05_22-PM-01_14_03
Last ObjectModification:
2018_05_21-PM-02_18_25
Theory : euclidean!plane!geometry
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