Nuprl Lemma : proj-point-sep_defB
∀e:EuclideanParPlane. ∀p,q:Point + Line.
(((∃n:Line?. ((¬pp-sep(e;p;n)) ∧ pp-sep(e;q;n))) ∧ (∀l,m:Line. (l \/ m ⇒ (∀n:Line. (l \/ n ∨ m \/ n)))))
⇒ proj-point-sep(e;p;q))
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],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
unit: Unit,
union: left + right
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
pp-sep: pp-sep(eu;p;l),
proj-point-sep: proj-point-sep(eu;p;q),
true: True,
member: t ∈ T,
not: ¬A,
false: False,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
euclidean-parallel-plane: EuclideanParPlane,
or: P ∨ Q,
geo-line: Line,
geo-plsep: p # l,
pi1: fst(t),
pi2: snd(t),
iff: P ⇐⇒ Q
Lemmas referenced :
exists_wf,
geo-line_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
euclidean-planes-subtype,
subtype_rel_transitivity,
euclidean-parallel-plane_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
unit_wf2,
not_wf,
pp-sep_wf,
all_wf,
geo-intersect_wf,
geoline-subtype1,
or_wf,
geo-point_wf,
lsep-iff-all-sep,
not-lsep-iff-colinear,
geo-sep-sym,
geo-intersect-symmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
unionElimination,
sqequalRule,
natural_numberEquality,
independent_functionElimination,
voidElimination,
productEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
unionEquality,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
lambdaEquality,
because_Cache,
functionEquality,
setElimination,
rename
Latex:
\mforall{}e:EuclideanParPlane. \mforall{}p,q:Point + Line.
(((\mexists{}n:Line?. ((\mneg{}pp-sep(e;p;n)) \mwedge{} pp-sep(e;q;n)))
\mwedge{} (\mforall{}l,m:Line. (l \mbackslash{}/ m {}\mRightarrow{} (\mforall{}n:Line. (l \mbackslash{}/ n \mvee{} m \mbackslash{}/ n)))))
{}\mRightarrow{} proj-point-sep(e;p;q))
Date html generated:
2018_05_22-PM-01_16_21
Last ObjectModification:
2018_05_19-PM-11_16_40
Theory : euclidean!plane!geometry
Home
Index