Nuprl Lemma : segment-intersection
∀e:HeytingGeometry. ∀a,b,c:Point. (c # ab ⇒ (∃p,q:Point. (Colinear(a;b;p) ∧ c-p-q)))
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-colinear: Colinear(a;b;c),
geo-strict-between: a-b-c,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
cand: A c∧ B,
and: P ∧ Q,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
heyting-geometry: HeytingGeometry,
basic-geometry-: BasicGeometry-,
heyting-geometry: Error :heyting-geometry,
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),
so_apply: x[s],
so_lambda: λ2x.t[x]
Lemmas referenced :
geo-point_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
heyting-geometry_wf,
subtype_rel_transitivity,
heyting-geometry-subtype,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-triangle_wf,
geo-proper-extend-exists,
geo-sep-sym,
geo-triangle-property,
geo-left-axioms_wf,
basic-geo-axioms_wf,
subtype_rel_self,
geo-strict-between-sym,
geo-inner-pasch-ex,
Error :geo-triangle_wf,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-strict-between-implies-colinear,
geo-colinear-is-colinear-set,
geo-strict-between-sep1,
geo-triangle-symmetry,
geo-triangle-colinear,
exists_wf,
geo-strict-between_wf,
geo-colinear_wf
Rules used in proof :
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
rename,
productElimination,
independent_functionElimination,
dependent_functionElimination,
cumulativity,
productEquality,
setEquality,
setElimination,
dependent_set_memberEquality,
baseClosed,
imageMemberEquality,
independent_pairFormation,
natural_numberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
lambdaEquality,
dependent_pairFormation
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c:Point. (c \# ab {}\mRightarrow{} (\mexists{}p,q:Point. (Colinear(a;b;p) \mwedge{} c-p-q)))
Date html generated:
2017_10_02-PM-07_05_15
Last ObjectModification:
2017_08_06-PM-10_25_59
Theory : euclidean!plane!geometry
Home
Index