Nuprl Lemma : common-P_point-intersecting-P_lines2
∀e:EuclideanParPlane. ∀P:P_point(e). ∀L,L1:P_line(e).
(((¬P_point-line-sep(e;P;L)) ∧ (¬P_point-line-sep(e;P;L1)))
⇒ fst(L) \/ fst(L1)
⇒ (∀l,m,n:Line. (l \/ m ⇒ (l \/ n ∨ m \/ n)))
⇒ (∃x:Point. ((x I fst(snd(P)) ∧ x I fst(snd(snd(P)))) ∧ x I fst(L) ∧ x I fst(L1))))
Proof
Definitions occuring in Statement :
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-incident: p I L,
geo-line: Line,
geo-point: Point,
pi1: fst(t),
pi2: snd(t),
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 :
geo-colinear: Colinear(a;b;c),
geo-lsep: a # bc,
basic-geometry-: BasicGeometry-,
geo-incident: p I L,
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
ge: i ≥ j ,
true: True,
less_than: a < b,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
l_member: (x ∈ l),
subtract: n - m,
cons: [a / b],
select: L[n],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
lelt: i ≤ j < k,
int_seg: {i..j-},
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
geo-line-sep: (l # m),
geo-line: Line,
euclidean-plane: EuclideanPlane,
geo-plsep: p # l,
rev_implies: P ⇐ Q,
squash: ↓T,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
geo-eq: a ≡ b,
geo-Aparallel: l || m,
stable: Stable{P},
cand: A c∧ B,
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
false: False,
not: ¬A,
top: Top,
or: P ∨ Q,
prop: ℙ,
euclidean-parallel-plane: EuclideanParPlane,
uimplies: b supposing a,
guard: {T},
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
P_point-line-sep: P_point-line-sep(e;P;L),
pi2: snd(t),
pi1: fst(t),
P_line: P_line(eu),
P_point: P_point(eu),
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
common-P_point-intersecting-P_lines,
geo-line-sep-symmetry,
not-lsep-iff-colinear,
geo-lsep_wf,
geo-colinear_wf,
geo-colinear-iff,
geo-intersect-symmetry,
geo-incident-not-plsep,
geo-colinear-permute,
geo-colinear-cycle,
geo-line-pt-sep,
euclidean-plane-axioms,
geo-colinear-transitivity,
lsep-symmetry,
list_ind_nil_lemma,
list_ind_cons_lemma,
l_member_wf,
int_term_value_var_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformand_wf,
nat_properties,
select_wf,
length_wf,
lsep-implies-sep,
nil_wf,
cons_wf,
geo-colinear-append,
istype-less_than,
istype-le,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-int,
itermConstant_wf,
intformle_wf,
intformnot_wf,
full-omega-unsat,
decidable__le,
length_of_nil_lemma,
length_of_cons_lemma,
geo-colinear-is-colinear-set,
geo-incident-line,
lsep-all-sym,
oriented-plane_wf,
euclidean-plane-subtype-oriented,
colinear-lsep',
geo-point_wf,
geo-sep-or,
geo-plsep_functionality,
sq_stable__incident,
geo-intersect-unique,
geo-line-eq_weakening2,
geo-incident_functionality,
geo-sep_wf,
minimal-not-not-excluded-middle,
minimal-double-negation-hyp-elim,
not_wf,
false_wf,
stable__false,
geo-plsep_wf,
geo-incident_wf,
geo-intersect-lines-iff,
P_point_wf,
P_line_wf,
P_point-line-sep_wf,
istype-void,
geoline_wf,
pi1_wf_top,
geoline-subtype1,
geo-intersect_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-line_wf
Rules used in proof :
dependent_pairEquality_alt,
inlFormation_alt,
equalitySymmetry,
equalityTransitivity,
int_eqEquality,
equalityIstype,
lambdaEquality_alt,
approximateComputation,
natural_numberEquality,
dependent_set_memberEquality_alt,
setIsType,
imageElimination,
baseClosed,
imageMemberEquality,
promote_hyp,
functionEquality,
unionEquality,
independent_pairFormation,
dependent_pairFormation_alt,
unionElimination,
independent_functionElimination,
inhabitedIsType,
productIsType,
voidElimination,
isect_memberEquality_alt,
independent_pairEquality,
unionIsType,
rename,
setElimination,
because_Cache,
independent_isectElimination,
isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
universeIsType,
functionIsType,
sqequalRule,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:EuclideanParPlane. \mforall{}P:P\_point(e). \mforall{}L,L1:P\_line(e).
(((\mneg{}P\_point-line-sep(e;P;L)) \mwedge{} (\mneg{}P\_point-line-sep(e;P;L1)))
{}\mRightarrow{} fst(L) \mbackslash{}/ fst(L1)
{}\mRightarrow{} (\mforall{}l,m,n:Line. (l \mbackslash{}/ m {}\mRightarrow{} (l \mbackslash{}/ n \mvee{} m \mbackslash{}/ n)))
{}\mRightarrow{} (\mexists{}x:Point. ((x I fst(snd(P)) \mwedge{} x I fst(snd(snd(P)))) \mwedge{} x I fst(L) \mwedge{} x I fst(L1))))
Date html generated:
2019_10_29-AM-09_24_17
Last ObjectModification:
2019_10_18-PM-07_33_49
Theory : euclidean!plane!geometry
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