Nuprl Lemma : eu-colinear-cons
∀e:EuclideanPlane. ∀L:Point List. ∀A:Point.
(eu-colinear-set(e;[A / L]) ⇐⇒ eu-colinear-set(e;L) ∧ (∀B∈L.(∀C∈L.(¬(A = B ∈ Point)) ⇒ Colinear(A;B;C))))
Proof
Definitions occuring in Statement :
eu-colinear-set: eu-colinear-set(e;L),
euclidean-plane: EuclideanPlane,
eu-colinear: Colinear(a;b;c),
eu-point: Point,
l_all: (∀x∈L.P[x]),
cons: [a / b],
list: T List,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
eu-colinear-set: eu-colinear-set(e;L),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
guard: {T},
cand: A c∧ B,
not: ¬A,
false: False,
stable: Stable{P},
uimplies: b supposing a,
or: P ∨ Q
Lemmas referenced :
l_all_wf2,
eu-point_wf,
cons_wf,
l_member_wf,
not_wf,
equal_wf,
eu-colinear_wf,
l_all_cons,
iff_wf,
list_wf,
euclidean-plane_wf,
l_all_functionality,
l_all_iff,
eu-colinear-def,
member_wf,
eu-between_wf,
all_wf,
eu-colinear-swap,
stable__colinear,
false_wf,
or_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
eu-colinear-permute
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
productEquality,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
setEquality,
dependent_functionElimination,
addLevel,
impliesFunctionality,
independent_functionElimination,
promote_hyp,
voidElimination,
allFunctionality,
levelHypothesis,
equalitySymmetry,
independent_isectElimination,
unionElimination,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}L:Point List. \mforall{}A:Point.
(eu-colinear-set(e;[A / L])
\mLeftarrow{}{}\mRightarrow{} eu-colinear-set(e;L) \mwedge{} (\mforall{}B\mmember{}L.(\mforall{}C\mmember{}L.(\mneg{}(A = B)) {}\mRightarrow{} Colinear(A;B;C))))
Date html generated:
2016_10_26-AM-07_43_45
Last ObjectModification:
2016_07_12-AM-08_11_02
Theory : euclidean!geometry
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