Nuprl Lemma : eu-not-not-colinear
∀e:EuclideanStructure
∀[a,b,c:Point].
(¬¬Colinear(a;b;c) ⇐⇒ (¬(a = b ∈ Point)) ∧ (¬¬((c = a ∈ Point) ∨ (c = b ∈ Point) ∨ c-a-b ∨ a-c-b ∨ a-b-c)))
Proof
Definitions occuring in Statement :
eu-colinear: Colinear(a;b;c),
eu-between: a-b-c,
eu-point: Point,
euclidean-structure: EuclideanStructure,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
uimplies: b supposing a,
or: P ∨ Q,
all: ∀x:A. B[x],
guard: {T}
Lemmas referenced :
and_wf,
not_wf,
equal_wf,
eu-point_wf,
eu-between_wf,
or_wf,
not_over_or,
eu-colinear_wf,
iff_wf,
euclidean-structure_wf,
eu-colinear-def
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
lambdaFormation,
cut,
thin,
sqequalHypSubstitution,
hypothesis,
independent_functionElimination,
productElimination,
voidElimination,
lemma_by_obid,
isectElimination,
hypothesisEquality,
addLevel,
impliesFunctionality,
independent_isectElimination,
levelHypothesis,
promote_hyp,
andLevelFunctionality,
sqequalRule,
impliesLevelFunctionality,
productEquality,
because_Cache,
equalityEquality,
isect_memberFormation,
introduction,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
inlFormation,
inrFormation
Latex:
\mforall{}e:EuclideanStructure
\mforall{}[a,b,c:Point].
(\mneg{}\mneg{}Colinear(a;b;c) \mLeftarrow{}{}\mRightarrow{} (\mneg{}(a = b)) \mwedge{} (\mneg{}\mneg{}((c = a) \mvee{} (c = b) \mvee{} c-a-b \mvee{} a-c-b \mvee{} a-b-c)))
Date html generated:
2016_05_18-AM-06_32_50
Last ObjectModification:
2015_12_28-AM-09_28_50
Theory : euclidean!geometry
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