Nuprl Lemma : eu-between-implies-colinear
∀e:EuclideanStructure. ∀[a,b,c:Point]. (Colinear(a;b;c)) supposing (a-b-c and (¬(a = b ∈ Point)))
Proof
Definitions occuring in Statement :
eu-colinear: Colinear(a;b;c),
eu-between: a-b-c,
eu-point: Point,
euclidean-structure: EuclideanStructure,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
equal: s = t ∈ T
Definitions unfolded in proof :
prop: ℙ,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
false: False,
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
eu-between-eq-implies-colinear,
eu-point_wf,
eu-between-eq-def,
and_wf,
not_wf,
equal_wf,
eu-between_wf,
euclidean-structure_wf
Rules used in proof :
independent_functionElimination,
productElimination,
independent_isectElimination,
rename,
equalityEquality,
voidElimination,
lambdaEquality,
sqequalRule,
introduction,
isectElimination,
isect_memberFormation,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
lemma_by_obid,
cut
Latex:
\mforall{}e:EuclideanStructure. \mforall{}[a,b,c:Point]. (Colinear(a;b;c)) supposing (a-b-c and (\mneg{}(a = b)))
Date html generated:
2016_05_18-AM-06_33_09
Last ObjectModification:
2016_01_03-PM-08_32_58
Theory : euclidean!geometry
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