Nuprl Lemma : eu-eq-implies-col
∀e:EuclideanPlane. ∀a,b,c:Point. ((¬(a = b ∈ Point)) ⇒ (b = c ∈ Point) ⇒ Colinear(a;b;c))
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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
not: ¬A,
false: False
Lemmas referenced :
equal_wf,
eu-point_wf,
not_wf,
euclidean-plane_wf,
eu-colinear-def,
eu-between_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
productElimination,
independent_functionElimination,
independent_pairFormation,
equalitySymmetry,
voidElimination,
productEquality,
because_Cache
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point. ((\mneg{}(a = b)) {}\mRightarrow{} (b = c) {}\mRightarrow{} Colinear(a;b;c))
Date html generated:
2016_05_18-AM-06_45_43
Last ObjectModification:
2015_12_28-AM-09_22_07
Theory : euclidean!geometry
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