Nuprl Lemma : no-three-colinear-on-circle
∀[e:EuclideanPlane]. ∀[a:Point]. ∀[b:{b:Point| b ≠ a} ]. ∀[c:{c:Point| c ≠ a ∧ c ≠ b ∧ Colinear(a;b;c)} ].
∀p:Point. (¬(pa ≅ pb ∧ pb ≅ pc))
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,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
stable: Stable{P},
or: P ∨ Q,
sq_exists: ∃x:A [B[x]],
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),
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
geo-eq: a ≡ b,
oriented-plane: OrientedPlane,
exists: ∃x:A. B[x],
geo-midpoint: a=m=b,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
basic-geometry: BasicGeometry,
geo-perp: ab ⊥ cd
Lemmas referenced :
geo-congruent_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
set_wf,
geo-sep_wf,
geo-colinear_wf,
stable__false,
false_wf,
or_wf,
geo-lsep_wf,
not_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
Euclid-midpoint,
geo-sep-sym,
midpoint-of-equidistant-points-is-perp,
colinear-lsep-cycle,
lsep-all-sym,
geo-colinear-is-colinear-set,
length_of_cons_lemma,
length_of_nil_lemma,
lelt_wf,
not-lsep-iff-colinear,
oriented-colinear-append,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
equal_wf,
exists_wf,
geo-between-implies-colinear,
list_ind_cons_lemma,
list_ind_nil_lemma,
geo-colinear_functionality,
geo-eq_weakening,
geo-perp-unicity,
geo-perp-in_wf,
geo-perp-symmetry,
geo-perp-symmetry2,
geo-perp-colinear,
geo-between_wf,
geo-between_functionality,
geo-eq_inversion,
geo-congruent_functionality,
seg-midpoints-equal-flip,
colinear-implies-midpoint
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
setElimination,
rename,
sqequalHypSubstitution,
productElimination,
hypothesis,
because_Cache,
independent_functionElimination,
voidElimination,
productEquality,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
functionEquality,
unionElimination,
dependent_set_memberEquality,
voidEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
addLevel,
impliesFunctionality,
dependent_pairFormation,
inrFormation,
inlFormation
Latex:
\mforall{}[e:EuclideanPlane]. \mforall{}[a:Point]. \mforall{}[b:\{b:Point| b \mneq{} a\} ]. \mforall{}[c:\{c:Point|
c \mneq{} a \mwedge{} c \mneq{} b \mwedge{} Colinear(a;b;c)\} ].
\mforall{}p:Point. (\mneg{}(pa \mcong{} pb \mwedge{} pb \mcong{} pc))
Date html generated:
2018_05_22-PM-00_10_20
Last ObjectModification:
2018_05_11-PM-06_48_20
Theory : euclidean!plane!geometry
Home
Index