Nuprl Lemma : geo-between-middle
∀e:BasicGeometry. ∀a,b,c,d:Point. (a ≠ d ⇒ a_b_d ⇒ a_c_d ⇒ (¬((¬b_c_d) ∧ (¬c_b_d))))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
prop: ℙ,
member: t ∈ T,
false: False,
not: ¬A,
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
geo-eq: a ≡ b,
or: P ∨ Q,
stable: Stable{P},
exists: ∃x:A. B[x]
Lemmas referenced :
geo-point_wf,
geo-sep_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-between_wf,
not_wf,
minimal-not-not-excluded-middle,
geo-between-sep,
geo-congruent-sep,
geo-between-outer-trans,
geo-between-exchange3,
geo-between-inner-trans,
geo-between-symmetry,
geo-congruent_functionality,
geo-eq_weakening,
geo-between_functionality,
minimal-double-negation-hyp-elim,
or_wf,
false_wf,
stable__not,
geo-sep-sym,
geo-extend-exists,
geo-between-same-side,
geo-between-exchange4
Rules used in proof :
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
productEquality,
voidElimination,
independent_functionElimination,
sqequalHypSubstitution,
hypothesis,
thin,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
promote_hyp,
independent_pairFormation,
unionElimination,
functionEquality,
productElimination,
dependent_functionElimination
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,d:Point. (a \mneq{} d {}\mRightarrow{} a\_b\_d {}\mRightarrow{} a\_c\_d {}\mRightarrow{} (\mneg{}((\mneg{}b\_c\_d) \mwedge{} (\mneg{}c\_b\_d))))
Date html generated:
2017_10_02-PM-06_37_58
Last ObjectModification:
2017_08_05-PM-04_46_35
Theory : euclidean!plane!geometry
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