Nuprl Lemma : AX3
∀e:EuclideanParPlane. ∀l,m:P_line(e).
((P_line-sep(e;l;m) ∧ (∀l,m,n:Line. (l \/ m ⇒ (l \/ n ∨ m \/ n))))
⇒ (∃P:P_point(e). ((¬P_point-line-sep(e;P;l)) ∧ (¬P_point-line-sep(e;P;m)))))
Proof
Definitions occuring in Statement :
P_line-sep: P_line-sep(eu;L;M),
P_point-line-sep: P_point-line-sep(e;P;L),
P_line: P_line(eu),
P_point: P_point(eu),
euclidean-parallel-plane: EuclideanParPlane,
geo-intersect: L \/ M,
geo-line: Line,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
P_line-sep: P_line-sep(eu;L;M),
exists: ∃x:A. B[x],
P_line: P_line(eu),
P_point: P_point(eu),
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
euclidean-parallel-plane: EuclideanParPlane,
so_apply: x[s],
or: P ∨ Q,
P_point-line-sep: P_point-line-sep(e;P;L),
pi2: snd(t),
pi1: fst(t),
not: ¬A,
cand: A c∧ B,
false: False,
iff: P ⇐⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
top: Top,
rev_implies: P ⇐ Q,
geo-Aparallel: l || m,
geo-line-sep: (l # m),
uiff: uiff(P;Q),
geo-plsep: p # l,
geo-lsep: a # bc,
stable: Stable{P}
Lemmas referenced :
P_line-sep_wf,
all_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,
geo-intersect_wf,
geoline-subtype1,
or_wf,
P_line_wf,
common-P_point-intersecting-P_lines,
geo-incident_wf,
geo-plsep_wf,
geo-point_wf,
P_point-line-sep_wf,
not_wf,
geo-intersect-lines-iff,
geo-incident-not-plsep,
geo-Aparallel_weakening2,
subtype_quotient,
geo-Aparallel_wf,
geo-Aparallel-equiv,
and_false_l,
geo-intersect-symmetry,
geo-incident-line,
geo-Aparallel_sym,
stable__false,
false_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
geo-intersect-unique,
geo-eq_inversion,
geo-incident-iff-not-plsep,
geo-incident_functionality,
geo-line-eq_weakening2,
common-P_point-parallel-P_lines
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
productEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
lambdaEquality,
because_Cache,
functionEquality,
setElimination,
rename,
unionElimination,
independent_functionElimination,
dependent_pairEquality,
independent_pairEquality,
setEquality,
independent_pairFormation,
voidElimination,
dependent_set_memberEquality,
dependent_pairFormation,
isect_memberEquality,
voidEquality
Latex:
\mforall{}e:EuclideanParPlane. \mforall{}l,m:P\_line(e).
((P\_line-sep(e;l;m) \mwedge{} (\mforall{}l,m,n:Line. (l \mbackslash{}/ m {}\mRightarrow{} (l \mbackslash{}/ n \mvee{} m \mbackslash{}/ n))))
{}\mRightarrow{} (\mexists{}P:P\_point(e). ((\mneg{}P\_point-line-sep(e;P;l)) \mwedge{} (\mneg{}P\_point-line-sep(e;P;m)))))
Date html generated:
2019_10_16-PM-03_04_52
Last ObjectModification:
2018_08_14-PM-03_26_45
Theory : euclidean!plane!geometry
Home
Index