Nuprl Lemma : full-Pasch
∀e:EuclideanPlane. ∀a,x,y,d,p:Point.
((((d # xa ∧ x-p-a) ∧ d # py) ∧ a leftof xy) ⇒ (∃p':Point. ((x-p'-y ∨ a-p'-y) ∧ Colinear(d;p;p'))))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-colinear: Colinear(a;b;c),
geo-strict-between: a-b-c,
geo-left: a leftof bc,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q
Definitions unfolded in proof :
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
basic-geometry-: BasicGeometry-,
oriented-plane: OrientedPlane,
exists: ∃x:A. B[x],
basic-geometry: BasicGeometry,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
geo-lsep: a # bc,
or: P ∨ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a
Lemmas referenced :
geo-colinear_wf,
or_wf,
list_ind_nil_lemma,
list_ind_cons_lemma,
exists_wf,
equal_wf,
l_member_wf,
cons_member,
nil_wf,
cons_wf,
oriented-colinear-append,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-strict-between-implies-colinear,
geo-colinear-is-colinear-set,
lsep-all-sym,
colinear-lsep-cycle,
geo-sep_wf,
left-convex2,
geo-sep-sym,
left-between-implies-right1,
geo-proper-extend-exists,
lsep-implies-sep,
geo-between_wf,
geo-strict-between-sep3,
geo-between-symmetry,
geo-strict-between-implies-between,
left-convex,
full-Pasch-lemma,
left-symmetry,
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-strict-between_wf,
geo-left_wf,
geo-point_wf
Rules used in proof :
lambdaEquality,
dependent_pairFormation,
baseClosed,
imageMemberEquality,
natural_numberEquality,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
inlFormation,
rename,
inrFormation,
independent_pairFormation,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
independent_functionElimination,
hypothesis,
unionElimination,
productEquality,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,x,y,d,p:Point.
((((d \# xa \mwedge{} x-p-a) \mwedge{} d \# py) \mwedge{} a leftof xy)
{}\mRightarrow{} (\mexists{}p':Point. ((x-p'-y \mvee{} a-p'-y) \mwedge{} Colinear(d;p;p'))))
Date html generated:
2017_10_04-PM-10_10_20
Last ObjectModification:
2017_10_03-PM-02_53_23
Theory : euclidean!plane!geometry
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