Nuprl Lemma : projective-lines-exist
∀e:EuclideanParPlane. ∀p1,p2:Point + Line.
(proj-point-sep(e;p1;p2) ⇒ (∃l:Line?. ((¬pp-sep(e;p1;l)) ∧ (¬pp-sep(e;p2;l)))))
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-line: Line,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
unit: Unit,
union: left + right
Definitions unfolded in proof :
false: False,
not: ¬A,
geo-Aparallel: l || m,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
euclidean-parallel-plane: EuclideanParPlane,
pi2: snd(t),
pi1: fst(t),
geo-plsep: p # l,
pp-sep: pp-sep(eu;p;l),
cand: A c∧ B,
and: P ∧ Q,
geo-line: Line,
exists: ∃x:A. B[x],
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
proj-point-sep: proj-point-sep(eu;p;q),
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
false_wf,
it_wf,
geoline-subtype1,
geo-Aparallel_sym,
geo-incident-not-plsep,
Euclid-parallel-exists,
not-lsep-iff-colinear,
basic-geometry_wf,
euclidean-plane-subtype-basic,
geo-colinear-same,
pp-sep_wf,
not_wf,
unit_wf2,
geo-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,
proj-point-sep_wf
Rules used in proof :
voidElimination,
inrEquality,
independent_functionElimination,
productElimination,
setElimination,
independent_pairFormation,
productEquality,
dependent_pairEquality,
inlEquality,
dependent_pairFormation,
rename,
because_Cache,
dependent_functionElimination,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
unionEquality,
hypothesis,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
cut,
sqequalHypSubstitution,
thin,
unionElimination,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:EuclideanParPlane. \mforall{}p1,p2:Point + Line.
(proj-point-sep(e;p1;p2) {}\mRightarrow{} (\mexists{}l:Line?. ((\mneg{}pp-sep(e;p1;l)) \mwedge{} (\mneg{}pp-sep(e;p2;l)))))
Date html generated:
2018_05_22-PM-01_14_26
Last ObjectModification:
2018_05_21-PM-02_39_11
Theory : euclidean!plane!geometry
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