Nuprl Lemma : geo-colinear-transitivity
∀e:EuclideanPlane
∀[A,C,B,D:Point]. (Colinear(A;B;C) ⇒ Colinear(B;C;D) ⇒ B ≠ C ⇒ {Colinear(A;C;D) ∧ Colinear(A;B;D)})
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-sep: a ≠ b,
geo-point: Point,
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
false: False,
geo-colinear: Colinear(a;b;c),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
not: ¬A,
cand: A c∧ B,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
geo-point_wf,
geo-between_wf,
geo-colinear_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-sep_wf,
not_wf,
geo-lsep_wf,
lsep-implies-sep,
not-lsep-iff-colinear,
colinear-lsep,
lsep-all-sym,
geo-sep-sym
Rules used in proof :
voidElimination,
isect_memberEquality,
independent_pairEquality,
lambdaEquality,
independent_isectElimination,
instantiate,
productEquality,
addLevel,
sqequalRule,
because_Cache,
applyEquality,
isectElimination,
independent_functionElimination,
independent_pairFormation,
productElimination,
hypothesis,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:EuclideanPlane
\mforall{}[A,C,B,D:Point].
(Colinear(A;B;C) {}\mRightarrow{} Colinear(B;C;D) {}\mRightarrow{} B \mneq{} C {}\mRightarrow{} \{Colinear(A;C;D) \mwedge{} Colinear(A;B;D)\})
Date html generated:
2017_10_02-PM-03_29_40
Last ObjectModification:
2017_08_08-PM-00_35_18
Theory : euclidean!plane!geometry
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