Nuprl Lemma : geo-triangle-property
∀e:HeytingGeometry. ∀a,b,c:Point. (a # bc ⇒ {a ≠ b ∧ b ≠ c ∧ c ≠ a ∧ (¬a-b-c) ∧ (¬b-c-a) ∧ (¬c-a-b)})
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-strict-between: a-b-c,
geo-sep: a ≠ b,
geo-point: Point,
guard: {T},
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
false: False,
geo-strict-between: a-b-c,
uimplies: b supposing a,
heyting-geometry: Error :heyting-geometry,
uall: ∀[x:A]. B[x],
prop: ℙ,
not: ¬A,
subtype_rel: A ⊆r B,
cand: A c∧ B,
and: P ∧ Q,
guard: {T},
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
Error :heyting-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-point_wf,
Error :geo-triangle_wf,
geo-strict-between_wf,
heyting-geometry-subtype,
geo-sep-sym,
geo-triangle-implies
Rules used in proof :
voidElimination,
independent_isectElimination,
instantiate,
rename,
setElimination,
isectElimination,
independent_pairFormation,
sqequalRule,
applyEquality,
because_Cache,
productElimination,
hypothesis,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c:Point.
(a \# bc {}\mRightarrow{} \{a \mneq{} b \mwedge{} b \mneq{} c \mwedge{} c \mneq{} a \mwedge{} (\mneg{}a-b-c) \mwedge{} (\mneg{}b-c-a) \mwedge{} (\mneg{}c-a-b)\})
Date html generated:
2017_10_02-PM-07_02_10
Last ObjectModification:
2017_08_08-PM-00_41_45
Theory : euclidean!plane!geometry
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