Nuprl Lemma : geo-not-not-colinear
∀e:BasicGeometry. ∀[a,b,c:Point]. (¬¬Colinear(a;b;c) ⇐⇒ ¬¬(a_b_c ∨ b_c_a ∨ c_a_b))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-between: a_b_c,
geo-point: Point,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
or: P ∨ Q
Definitions unfolded in proof :
false: False,
or: P ∨ Q,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
not: ¬A,
rev_implies: P ⇐ Q,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
geo-point_wf,
iff_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-between_wf,
or_wf,
geo-not-colinear,
geo-colinear_wf,
not_wf
Rules used in proof :
voidElimination,
isect_memberEquality,
lambdaEquality,
independent_pairEquality,
isect_memberFormation,
impliesLevelFunctionality,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
independent_functionElimination,
dependent_functionElimination,
impliesFunctionality,
productElimination,
addLevel,
because_Cache,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
hypothesis,
lambdaFormation,
independent_pairFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cut
Latex:
\mforall{}e:BasicGeometry. \mforall{}[a,b,c:Point]. (\mneg{}\mneg{}Colinear(a;b;c) \mLeftarrow{}{}\mRightarrow{} \mneg{}\mneg{}(a\_b\_c \mvee{} b\_c\_a \mvee{} c\_a\_b))
Date html generated:
2017_10_02-PM-04_43_11
Last ObjectModification:
2017_08_05-AM-08_42_42
Theory : euclidean!plane!geometry
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