### 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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