Nuprl Lemma : lsep-inner-pasch-strict
∀e:OrientedPlane. ∀a,b:Point. ∀c:{c:Point| c # ab} . ∀p:{p:Point| a-p-c} . ∀q:{q:Point| b-q-c} .
(∃x:{Point| (b-x-p ∧ a-x-q)})
Proof
Definitions occuring in Statement :
oriented-plane: OrientedPlane,
geo-lsep: a # bc,
geo-strict-between: a-b-c,
geo-point: Point,
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
oriented-plane: OrientedPlane,
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
guard: {T},
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
basic-geometry: BasicGeometry,
prop: ℙ,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_exists: ∃x:{A| B[x]},
or: P ∨ Q,
geo-strict-between: a-b-c
Lemmas referenced :
lsep-inner-pasch-ext,
sq_stable__geo-lsep,
sq_stable__geo-strict-between,
colinear-lsep-cycle,
lsep-all-sym,
geo-strict-between-sep3,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
oriented-plane-subtype,
subtype_rel_transitivity,
oriented-plane_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-colinear-is-colinear-set,
geo-strict-between-implies-colinear,
subtype_rel_self,
basic-geo-axioms_wf,
geo-left-axioms_wf,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
geo-strict-between-sep2,
set_wf,
geo-point_wf,
geo-strict-between_wf,
geo-lsep_wf,
sq_stable__and,
geo-between_wf,
sq_stable__geo-between,
geo-sep-or,
lsep-implies-sep,
geo-sep_wf,
geo-sep-sym,
geo-between-implies-colinear,
subtype_rel_sets,
colinear-lsep2
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
because_Cache,
productElimination,
applyEquality,
instantiate,
isectElimination,
independent_isectElimination,
setEquality,
productEquality,
cumulativity,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
lambdaEquality,
promote_hyp,
unionElimination,
dependent_set_memberFormation
Latex:
\mforall{}e:OrientedPlane. \mforall{}a,b:Point. \mforall{}c:\{c:Point| c \# ab\} . \mforall{}p:\{p:Point| a-p-c\} . \mforall{}q:\{q:Point| b-q-c\} .
(\mexists{}x:\{Point| (b-x-p \mwedge{} a-x-q)\})
Date html generated:
2017_10_02-PM-04_48_23
Last ObjectModification:
2017_08_13-PM-08_40_37
Theory : euclidean!plane!geometry
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