Nuprl Lemma : eu-colinear-switch
∀e:EuclideanPlane. ∀a,b,c:Point. (Colinear(a;b;c) ⇒ (¬(a = c ∈ Point)) ⇒ Colinear(a;c;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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
eu-colinear-set: eu-colinear-set(e;L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
less_than: a < b,
squash: ↓T,
true: True,
uall: ∀[x:A]. B[x],
select: L[n],
cons: [a / b],
subtract: n - m,
euclidean-plane: EuclideanPlane
Lemmas referenced :
euclidean-plane_wf,
eu-colinear_wf,
eu-point_wf,
equal_wf,
not_wf,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
eu-colinear-is-colinear-set
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
introduction,
imageMemberEquality,
baseClosed,
isectElimination,
because_Cache,
setElimination,
rename
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point. (Colinear(a;b;c) {}\mRightarrow{} (\mneg{}(a = c)) {}\mRightarrow{} Colinear(a;c;b))
Date html generated:
2016_05_18-AM-06_44_16
Last ObjectModification:
2016_01_16-PM-10_28_30
Theory : euclidean!geometry
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