Nuprl Lemma : eu-colinear-from-between
∀e:EuclideanPlane
∀[A,C,D:Point]. ((¬(A = C ∈ Point)) ⇒ (∃B:Point. ((¬(A = B ∈ Point)) ∧ A_C_B ∧ A_D_B)) ⇒ Colinear(A;C;D))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-between-eq: a_b_c,
eu-colinear: Colinear(a;b;c),
eu-point: Point,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
euclidean-plane: EuclideanPlane,
prop: ℙ,
uimplies: b supposing a,
member: t ∈ T,
and: P ∧ Q,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
eu-colinear-between,
exists_wf,
eu-point_wf,
and_wf,
not_wf,
equal_wf,
eu-between-eq_wf,
euclidean-plane_wf
Rules used in proof :
lambdaEquality,
sqequalRule,
rename,
setElimination,
hypothesis,
independent_isectElimination,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
lemma_by_obid,
cut,
thin,
productElimination,
sqequalHypSubstitution,
isect_memberFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:EuclideanPlane
\mforall{}[A,C,D:Point]. ((\mneg{}(A = C)) {}\mRightarrow{} (\mexists{}B:Point. ((\mneg{}(A = B)) \mwedge{} A\_C\_B \mwedge{} A\_D\_B)) {}\mRightarrow{} Colinear(A;C;D))
Date html generated:
2016_05_18-AM-06_40_06
Last ObjectModification:
2016_01_01-PM-00_40_58
Theory : euclidean!geometry
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