Nuprl Lemma : colinear-equidistant-points-exist
∀e:EuclideanPlane. ∀a:Point. ∀b:{b:Point| a ≠ b} . ∀c:Point.
∃u,v:Point. (Colinear(a;b;u) ∧ Colinear(a;b;v) ∧ u ≠ v ∧ cu ≅ cv)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-congruent: ab ≅ cd,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
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,
euclidean-plane: EuclideanPlane,
or: P ∨ Q,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
basic-geometry: BasicGeometry,
exists: ∃x:A. B[x],
and: P ∧ Q,
geo-midpoint: a=m=b,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
cand: A c∧ B,
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,
prop: ℙ,
less_than: a < b,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T}
Lemmas referenced :
geo-sep-or,
sq_stable__geo-sep,
geo-sep-sym,
symmetric-point-construction,
use-SC,
geo-congruent-symmetry,
geo-congruent-sep,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
geo-congruent-iff-length,
geo-colinear_wf,
geo-sep_wf,
geo-congruent_wf,
exists_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
unionElimination,
independent_functionElimination,
because_Cache,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
isectElimination,
independent_isectElimination,
dependent_pairFormation,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
productEquality,
applyEquality,
lambdaEquality,
instantiate
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a:Point. \mforall{}b:\{b:Point| a \mneq{} b\} . \mforall{}c:Point.
\mexists{}u,v:Point. (Colinear(a;b;u) \mwedge{} Colinear(a;b;v) \mwedge{} u \mneq{} v \mwedge{} cu \mcong{} cv)
Date html generated:
2018_05_22-PM-00_08_29
Last ObjectModification:
2018_04_04-PM-05_44_13
Theory : euclidean!plane!geometry
Home
Index