Nuprl Lemma : projective-points-exist
∀e:EuclideanParPlane. ∀l,m:Line?. (¬¬(∃p:Point + Line. ((¬pp-sep(e;p;l)) ∧ (¬pp-sep(e;p;m)))))
Proof
Definitions occuring in Statement :
pp-sep: pp-sep(eu;p;l),
euclidean-parallel-plane: EuclideanParPlane,
geo-line: Line,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
unit: Unit,
union: left + right
Definitions unfolded in proof :
geo-line: Line,
euclidean-plane: EuclideanPlane,
geo-Aparallel: l || m,
pp-sep: pp-sep(eu;p;l),
cand: A c∧ B,
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
stable: Stable{P},
euclidean-parallel-plane: EuclideanParPlane,
so_apply: x[s],
and: P ∧ Q,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
false: False,
implies: P ⇒ Q,
not: ¬A,
all: ∀x:A. B[x]
Lemmas referenced :
geo-sep_wf,
geo-X_wf,
geo-O_wf,
geo-sep-O-X,
geo-intersect-irreflexive,
geo-incident-not-plsep,
geo-intersect-lines-iff,
minimal-not-not-excluded-middle,
minimal-double-negation-hyp-elim,
geoline-subtype1,
geo-intersect_wf,
or_wf,
false_wf,
stable__not,
unit_wf2,
pp-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,
exists_wf,
not_wf
Rules used in proof :
dependent_pairEquality,
inrEquality,
independent_pairFormation,
dependent_pairFormation,
productElimination,
functionEquality,
rename,
setElimination,
inlEquality,
productEquality,
lambdaEquality,
because_Cache,
dependent_functionElimination,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesisEquality,
unionEquality,
isectElimination,
extract_by_obid,
introduction,
voidElimination,
independent_functionElimination,
sqequalHypSubstitution,
hypothesis,
unionElimination,
thin,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:EuclideanParPlane. \mforall{}l,m:Line?. (\mneg{}\mneg{}(\mexists{}p:Point + Line. ((\mneg{}pp-sep(e;p;l)) \mwedge{} (\mneg{}pp-sep(e;p;m)))))
Date html generated:
2018_05_22-PM-01_15_11
Last ObjectModification:
2018_05_21-PM-06_21_36
Theory : euclidean!plane!geometry
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