Nuprl Lemma : eu-between-eq-implies-colinear3
∀e:EuclideanPlane. ∀[a,b,c:Point]. (Colinear(a;b;c)) supposing (b_c_a and (¬(a = b ∈ Point)))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-between-eq: a_b_c,
eu-colinear: Colinear(a;b;c),
eu-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
euclidean-plane: EuclideanPlane,
stable: Stable{P},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
cand: A c∧ B
Lemmas referenced :
eu-point_wf,
stable__colinear,
eu-between-eq-def,
eu-colinear-def,
not_wf,
equal_wf,
eu-between_wf,
eu-colinear_wf,
eu-between-eq_wf,
euclidean-plane_wf,
eu-between-sym
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
equalityEquality,
lemma_by_obid,
isectElimination,
setElimination,
rename,
hypothesis,
independent_isectElimination,
productElimination,
independent_functionElimination,
addLevel,
impliesFunctionality,
productEquality,
because_Cache,
levelHypothesis,
promote_hyp,
impliesLevelFunctionality,
independent_pairFormation,
equalitySymmetry
Latex:
\mforall{}e:EuclideanPlane. \mforall{}[a,b,c:Point]. (Colinear(a;b;c)) supposing (b\_c\_a and (\mneg{}(a = b)))
Date html generated:
2016_05_18-AM-06_33_57
Last ObjectModification:
2015_12_28-AM-09_28_36
Theory : euclidean!geometry
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