Nuprl Lemma : eu-colinear-permute
∀e:EuclideanPlane. ∀a,b,c:Point. ((¬(c = b ∈ Point))
⇒ Colinear(a;b;c)
⇒ Colinear(c;b;a))
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 = b)) {}\mRightarrow{} Colinear(a;b;c) {}\mRightarrow{} Colinear(c;b;a))
Date html generated:
2016_05_18-AM-06_35_50
Last ObjectModification:
2016_01_01-PM-01_13_04
Theory : euclidean!geometry
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