Nuprl Lemma : eu-colinear-cycle
∀e:EuclideanPlane. ∀a,b,c:Point. ((¬(c = a ∈ Point)) ⇒ Colinear(a;b;c) ⇒ Colinear(c;a;b))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-colinear: Colinear(a;b;c),
eu-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
prop: ℙ,
false: False,
cand: A c∧ B,
not: ¬A,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a
Lemmas referenced :
eu-between-sym,
eu-colinear-def,
equal_wf,
eu-point_wf,
not_wf,
eu-between_wf,
eu-colinear_wf,
euclidean-plane_wf
Rules used in proof :
because_Cache,
productEquality,
voidElimination,
equalitySymmetry,
introduction,
independent_pairFormation,
independent_functionElimination,
productElimination,
hypothesis,
isectElimination,
hypothesisEquality,
rename,
setElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
lemma_by_obid,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_isectElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point. ((\mneg{}(c = a)) {}\mRightarrow{} Colinear(a;b;c) {}\mRightarrow{} Colinear(c;a;b))
Date html generated:
2016_05_18-AM-06_35_54
Last ObjectModification:
2016_01_01-PM-04_15_18
Theory : euclidean!geometry
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