Nuprl Lemma : eu-colinear-transitivity
∀e:EuclideanPlane
∀[A,C,B,D:Point]. (Colinear(A;B;C) ⇒ Colinear(B;C;D) ⇒ {((¬(A = C ∈ Point)) ⇒ Colinear(A;C;D)) ∧ Colinear(A;B;D)})
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-colinear: Colinear(a;b;c),
eu-point: Point,
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
euclidean-plane: EuclideanPlane,
iff: P ⇐⇒ Q,
and: P ∧ Q,
guard: {T},
sq_stable: SqStable(P),
not: ¬A,
false: False,
prop: ℙ,
uimplies: b supposing a,
squash: ↓T
Lemmas referenced :
eu-colinear-def,
sq_stable__colinear,
eu-colinear-cases,
eu-colinear_wf,
stable__colinear,
not_wf,
equal_wf,
eu-point_wf,
eu-between-eq-implies-colinear2,
eu-between-implies-between-eq,
eu-between-eq-symmetry,
eu-between-eq-inner-trans,
eu-between-eq-exchange3,
eu-between-eq-exchange4,
eu-between-eq-outer-trans,
eu-between_wf,
eu-between-eq-implies-colinear,
eu-colinear-permute,
eu-colinear-swap,
not-eu-between-same,
not-eu-between-same2,
euclidean-plane_wf,
eu-between-eq-trivial-right,
eu-colinear-between,
eu-colinear-same-side,
eu-colinear-same-side2,
eu-between-eq_wf,
eu-between-same2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
isectElimination,
hypothesisEquality,
productElimination,
independent_functionElimination,
independent_pairFormation,
equalitySymmetry,
voidElimination,
productEquality,
equalityEquality,
promote_hyp,
hyp_replacement,
Error :applyLambdaEquality,
sqequalRule,
independent_isectElimination,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
universeEquality
Latex:
\mforall{}e:EuclideanPlane
\mforall{}[A,C,B,D:Point].
(Colinear(A;B;C) {}\mRightarrow{} Colinear(B;C;D) {}\mRightarrow{} \{((\mneg{}(A = C)) {}\mRightarrow{} Colinear(A;C;D)) \mwedge{} Colinear(A;B;D)\})
Date html generated:
2016_10_26-AM-07_43_27
Last ObjectModification:
2016_07_12-AM-08_14_11
Theory : euclidean!geometry
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