Nuprl Lemma : proj-point-sep_defA
∀e:EuclideanParPlane. ∀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-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 :
geo-intersect: L \/ M,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
geo-line-sep: geo-line-sep(g;l;m),
false: False,
pp-sep: pp-sep(eu;p;l),
not: ¬A,
geo-lsep: a # bc,
or: P ∨ Q,
geo-plsep: p # l,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
geo-line: Line,
top: Top,
pi1: fst(t),
pi2: snd(t),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
geo-perp-in: ab ⊥x cd,
so_lambda: λ2x.t[x],
so_apply: x[s],
right-angle: Rabc,
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
squash: ↓T,
euclidean-parallel-plane: EuclideanParPlane,
and: P ∧ Q,
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
all: ∀x:A. B[x],
implies: P ⇒ Q,
proj-point-sep: proj-point-sep(eu;p;q),
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a
Lemmas referenced :
geo-intersect-irreflexive,
it_wf,
geo-colinear-is-colinear-set,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
equal_wf,
pi2_wf,
subtype_rel_product,
top_wf,
exists_wf,
geo-lsep_wf,
geo-intersect-symmetry,
geo-intersect-lines-iff,
geo-intersect_wf,
Euclid-drop-perp-00,
geo-incident-not-plsep,
unit_wf2,
not_wf,
pp-sep_wf,
geo-lsep_functionality,
euclidean-plane-subtype-oriented,
oriented-plane_wf,
lsep-all-sym2,
geo-eq_inversion,
geo-eq_weakening,
geo-colinear_functionality,
pi1_wf_top,
geo-incident-line,
geo-line-from-points,
geo-sep-sym,
left-implies-sep,
geo-incident_wf,
geoline-subtype1,
geo-plsep_wf,
sq_stable__and,
all_wf,
right-angle_wf,
sq_stable__colinear,
sq_stable__all,
geo-midpoint_wf,
geo-congruent_wf,
sq_stable__geo-congruent,
sq_stable__geo-left,
Euclid-erect-2perp,
geo-sep_wf,
geo-colinear-same,
euclidean-plane-subtype-basic,
basic-geometry_wf,
geo-colinear_wf,
proj-point-sep_wf,
geo-point_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,
geo-line_wf
Rules used in proof :
inrEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
andLevelFunctionality,
existsFunctionality,
addLevel,
dependent_pairEquality,
inlEquality,
independent_pairEquality,
voidElimination,
voidEquality,
dependent_pairFormation,
isect_memberEquality,
productEquality,
lambdaEquality,
functionEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
setElimination,
rename,
dependent_set_memberEquality,
productElimination,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
unionElimination,
thin,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
unionEquality,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
because_Cache
Latex:
\mforall{}e:EuclideanParPlane. \mforall{}p,q:Point + Line.
(proj-point-sep(e;p;q) {}\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_08
Last ObjectModification:
2018_05_21-PM-03_25_02
Theory : euclidean!plane!geometry
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