Nuprl Lemma : eu-colinear-between
∀e:EuclideanPlane
∀[A,B,C,D:Point]. (Colinear(A;C;D)) supposing ((¬(A = C ∈ Point)) and (¬(A = B ∈ Point)) and A_C_B and A_D_B)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-between-eq: a_b_c,
eu-colinear: Colinear(a;b;c),
eu-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
squash: ↓T,
prop: ℙ,
and: P ∧ Q,
exists: ∃x:A. B[x]
Lemmas referenced :
eu-point_wf,
sq_stable__colinear,
not_wf,
equal_wf,
eu-between-eq_wf,
euclidean-plane_wf,
eu-proper-extend-exists,
eu-O_wf,
eu-not-colinear-OXY,
eu-X_wf,
eu-colinear-same-side,
eu-between-eq-symmetry,
eu-between-implies-between-eq,
eu-between-eq-exchange3,
eu-between-eq-exchange4,
eu-between_wf,
not-eu-between-same
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
equalityEquality,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesis,
because_Cache,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
equalitySymmetry,
dependent_set_memberEquality,
productElimination,
independent_isectElimination,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}e:EuclideanPlane
\mforall{}[A,B,C,D:Point]. (Colinear(A;C;D)) supposing ((\mneg{}(A = C)) and (\mneg{}(A = B)) and A\_C\_B and A\_D\_B)
Date html generated:
2016_10_26-AM-07_43_18
Last ObjectModification:
2016_07_12-AM-08_09_10
Theory : euclidean!geometry
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