Nuprl Lemma : geo-inner-pasch-ex
∀e:HeytingGeometry. ∀a,b:Point. ∀c:{c:Point| c # ab} . ∀p,q:Point. (a-p-c ⇒ b-q-c ⇒ (∃x:Point. (b-x-p ∧ a-x-q)))
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
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,
set: {x:A| B[x]}
Definitions unfolded in proof :
geo-triangle: a # bc,
all: ∀x:A. B[x],
implies: P ⇒ Q,
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],
so_apply: x[s],
heyting-geometry: HeytingGeometry,
euclidean-plane: EuclideanPlane,
oriented-plane: OrientedPlane,
and: P ∧ Q,
sq_exists: ∃x:{A| B[x]},
exists: ∃x:A. B[x],
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
geo-strict-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
heyting-geometry-subtype,
subtype_rel_transitivity,
heyting-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
set_wf,
geo-lsep_wf,
lsep-inner-pasch-strict,
subtype_rel_self,
basic-geo-axioms_wf,
geo-left-axioms_wf,
sq_stable__and,
sq_stable__geo-strict-between
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
setElimination,
rename,
because_Cache,
lambdaEquality,
dependent_functionElimination,
setEquality,
productEquality,
cumulativity,
dependent_set_memberEquality,
dependent_pairFormation,
productElimination,
isect_memberEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b:Point. \mforall{}c:\{c:Point| c \# ab\} . \mforall{}p,q:Point.
(a-p-c {}\mRightarrow{} b-q-c {}\mRightarrow{} (\mexists{}x:Point. (b-x-p \mwedge{} a-x-q)))
Date html generated:
2017_10_02-PM-07_02_46
Last ObjectModification:
2017_08_14-PM-03_59_44
Theory : euclidean!plane!geometry
Home
Index