Nuprl Lemma : geo-colinear-cons
∀e:BasicGeometry. ∀L:Point List. ∀A:Point.
(geo-colinear-set(e; [A / L]) ⇐⇒ geo-colinear-set(e; L) ∧ (∀B∈L.(∀C∈L.Colinear(A;B;C))))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-colinear-set: geo-colinear-set(e; L),
geo-colinear: Colinear(a;b;c),
geo-point: Point,
l_all: (∀x∈L.P[x]),
cons: [a / b],
list: T List,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
so_apply: x[s],
basic-geometry: BasicGeometry,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
geo-colinear-set: geo-colinear-set(e; L),
all: ∀x:A. B[x],
cand: A c∧ B,
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])
Lemmas referenced :
list_wf,
iff_wf,
l_all_cons,
geo-colinear_wf,
l_member_wf,
cons_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-point_wf,
l_all_wf2,
l_all_functionality,
l_all_iff,
geo-colinear-same,
all_wf,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-colinear-is-colinear-set,
geo-colinear-permute
Rules used in proof :
independent_functionElimination,
impliesFunctionality,
addLevel,
dependent_functionElimination,
setEquality,
rename,
setElimination,
lambdaEquality,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
productEquality,
thin,
productElimination,
sqequalHypSubstitution,
independent_pairFormation,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
promote_hyp,
functionEquality,
levelHypothesis,
allFunctionality,
baseClosed,
imageMemberEquality,
natural_numberEquality,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality
Latex:
\mforall{}e:BasicGeometry. \mforall{}L:Point List. \mforall{}A:Point.
(geo-colinear-set(e; [A / L]) \mLeftarrow{}{}\mRightarrow{} geo-colinear-set(e; L) \mwedge{} (\mforall{}B\mmember{}L.(\mforall{}C\mmember{}L.Colinear(A;B;C))))
Date html generated:
2017_10_02-PM-06_20_21
Last ObjectModification:
2017_08_05-PM-04_14_46
Theory : euclidean!plane!geometry
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