Nuprl Lemma : eu-between-eq-middle
∀e:EuclideanPlane. ∀a,b,c,d:Point. ((¬(a = d ∈ Point)) ⇒ a_b_d ⇒ a_c_d ⇒ (¬((¬b_c_d) ∧ (¬c_b_d))))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-between-eq: a_b_c,
eu-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
exists: ∃x:A. B[x],
uimplies: b supposing a
Lemmas referenced :
not_wf,
eu-between-eq_wf,
equal_wf,
eu-point_wf,
euclidean-plane_wf,
eu-extend-exists,
eu-between-eq-same-side,
eu-between-eq-symmetry,
eu-between-eq-inner-trans,
eu-congruent_wf,
eu-congruence-identity-sym,
false_wf,
eu-between-eq-exchange4,
eu-between-eq-exchange3
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
productEquality,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
because_Cache,
dependent_functionElimination,
equalitySymmetry,
dependent_set_memberEquality,
productElimination,
independent_isectElimination,
hyp_replacement,
Error :applyLambdaEquality,
sqequalRule,
equalityTransitivity,
equalityEquality,
universeEquality,
independent_pairFormation
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point. ((\mneg{}(a = d)) {}\mRightarrow{} a\_b\_d {}\mRightarrow{} a\_c\_d {}\mRightarrow{} (\mneg{}((\mneg{}b\_c\_d) \mwedge{} (\mneg{}c\_b\_d))))
Date html generated:
2016_10_26-AM-07_45_39
Last ObjectModification:
2016_07_12-AM-08_11_56
Theory : euclidean!geometry
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