Nuprl Lemma : eu-colinear-def
∀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,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
euclidean-structure: EuclideanStructure,
record+: record+,
member: t ∈ T,
record-select: r.x,
subtype_rel: A ⊆r B,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
btrue: tt,
guard: {T},
prop: ℙ,
spreadn: spread3,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
uimplies: b supposing a,
eu-point: Point,
eu-between: a-b-c,
eu-colinear: Colinear(a;b;c)
Lemmas referenced :
subtype_rel_self,
not_wf,
equal_wf,
uall_wf,
iff_wf,
and_wf,
isect_wf,
eu-point_wf,
euclidean-structure_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
sqequalHypSubstitution,
dependentIntersectionElimination,
sqequalRule,
dependentIntersectionEqElimination,
thin,
cut,
hypothesis,
applyEquality,
tokenEquality,
instantiate,
lemma_by_obid,
isectElimination,
universeEquality,
functionEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
cumulativity,
hypothesisEquality,
because_Cache,
setEquality,
productEquality,
productElimination,
setElimination,
rename,
introduction
Latex:
\mforall{}e:EuclideanStructure
\mforall{}[a,b,c:Point].
(Colinear(a;b;c) \mLeftarrow{}{}\mRightarrow{} (\mneg{}(a = b)) \mwedge{} (\mneg{}((\mneg{}(c = a)) \mwedge{} (\mneg{}(c = b)) \mwedge{} (\mneg{}c-a-b) \mwedge{} (\mneg{}a-c-b) \mwedge{} (\mneg{}a-b-c))))
Date html generated:
2016_05_18-AM-06_32_43
Last ObjectModification:
2015_12_28-AM-09_28_28
Theory : euclidean!geometry
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