Nuprl Lemma : geo-intersect-points-iff
∀e:EuclideanPlane. ∀a,b,c,d:Point.
(ab \/ cd
⇐⇒ a ≠ b
∧ c ≠ d
∧ (∃a1,b1,c1,d1,v:Point
(a1-v-b1
∧ c1-v-d1
∧ Colinear(a1;a;b)
∧ Colinear(b1;a;b)
∧ Colinear(c1;c;d)
∧ Colinear(d1;c;d)
∧ a1 leftof c1d1
∧ b1 leftof d1c1)))
Proof
Definitions occuring in Statement :
geo-intersect-points: ab \/ cd,
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-strict-between: a-b-c,
geo-left: a leftof bc,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
geo-intersect-points: ab \/ cd,
euclidean-plane: EuclideanPlane,
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
or: P ∨ Q,
basic-geometry: BasicGeometry,
geo-midpoint: a=m=b,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
false: False,
top: Top,
select: L[n],
cons: [a / b],
less_than: a < b,
true: True,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
subtract: n - m,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
geo-lsep: a # bc,
geo-strict-between: a-b-c,
oriented-plane: OrientedPlane,
basic-geometry-: BasicGeometry-,
geo-eq: a ≡ b
Lemmas referenced :
geo-intersect-points_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep_wf,
geo-strict-between_wf,
geo-colinear_wf,
geo-left_wf,
geo-point_wf,
left-implies-sep,
geo-SS_wf,
sq_stable__colinear,
sq_stable__geo-between,
geo-sep-or,
symmetric-point-construction,
geo-sep-sym,
colinear-lsep-cycle,
lsep-all-sym2,
geo-between-sep,
geo-colinear-append,
cons_wf,
nil_wf,
istype-void,
istype-le,
length_of_cons_lemma,
length_of_nil_lemma,
istype-less_than,
length_wf,
select_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
l_member_wf,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
list_ind_cons_lemma,
list_ind_nil_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lsep-all-sym,
geo-colinear-same,
geo-congruent-symmetry,
geo-congruent-sep,
left-between,
geo-between_wf,
not-lsep-iff-colinear,
geo-between-symmetry,
geo-strict-between-implies-colinear,
lsep-colinear-sep,
geo-lsep_wf,
colinear-lsep-general,
geo-strict-between-sep1,
geo-colinear-cases,
false_wf,
stable__false,
geo-eq_wf,
left-between-implies-right1,
geo-strict-between-implies-between,
geo-strict-between-sep3,
not-left-and-right,
geo-colinear_functionality,
geo-eq_weakening,
geo-left_functionality,
geo-sep_functionality,
geo-eq_inversion,
left-between-implies-right2,
between-preserves-left-1,
between-preserves-left-2,
between-preserves-left-3,
geo-strict-between-sep2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
isectElimination,
independent_isectElimination,
sqequalRule,
productElimination,
productIsType,
because_Cache,
inhabitedIsType,
setElimination,
rename,
independent_functionElimination,
dependent_set_memberEquality_alt,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
dependent_pairFormation_alt,
natural_numberEquality,
voidElimination,
isect_memberEquality_alt,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
inlFormation_alt,
functionIsType
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point.
(ab \mbackslash{}/ cd
\mLeftarrow{}{}\mRightarrow{} a \mneq{} b
\mwedge{} c \mneq{} d
\mwedge{} (\mexists{}a1,b1,c1,d1,v:Point
(a1-v-b1
\mwedge{} c1-v-d1
\mwedge{} Colinear(a1;a;b)
\mwedge{} Colinear(b1;a;b)
\mwedge{} Colinear(c1;c;d)
\mwedge{} Colinear(d1;c;d)
\mwedge{} a1 leftof c1d1
\mwedge{} b1 leftof d1c1)))
Date html generated:
2019_10_16-PM-01_45_21
Last ObjectModification:
2019_08_19-PM-01_15_37
Theory : euclidean!plane!geometry
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