Nuprl Lemma : proj-point-sep-iff-pp-sep
∀e:EuclideanParPlane
((∀l,m:Line. (l \/ m ⇒ (∀n:Line. (l \/ n ∨ m \/ n))))
⇒ (∀p,q:Point + Line. (proj-point-sep(e;p;q) ⇐⇒ ∃n:Line?. ((¬pp-sep(e;p;n)) ∧ pp-sep(e;q;n)))))
Proof
Definitions occuring in Statement :
proj-point-sep: proj-point-sep(eu;p;q),
pp-sep: pp-sep(eu;p;l),
euclidean-parallel-plane: EuclideanParPlane,
geo-intersect: L \/ M,
geo-line: Line,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
unit: Unit,
union: left + right
Definitions unfolded in proof :
euclidean-parallel-plane: EuclideanParPlane,
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
cand: A c∧ B,
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
or_wf,
geoline-subtype1,
geo-intersect_wf,
all_wf,
geo-point_wf,
pp-sep_wf,
not_wf,
unit_wf2,
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-line_wf,
exists_wf,
proj-point-sep_defB,
proj-point-sep_wf,
proj-point-sep_defA
Rules used in proof :
rename,
setElimination,
functionEquality,
because_Cache,
productEquality,
lambdaEquality,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
unionEquality,
isectElimination,
hypothesis,
independent_functionElimination,
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{} (\mforall{}p,q:Point + Line. (proj-point-sep(e;p;q) \mLeftarrow{}{}\mRightarrow{} \mexists{}n:Line?. ((\mneg{}pp-sep(e;p;n)) \mwedge{} pp-sep(e;q;n)))))
Date html generated:
2018_05_22-PM-01_16_32
Last ObjectModification:
2018_05_21-PM-03_29_25
Theory : euclidean!plane!geometry
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