Nuprl Lemma : geo-triangle-property2
∀e:HeytingGeometry. ∀a,b,c:Point. (a # bc ⇒ {(¬a_b_c) ∧ (¬b_c_a) ∧ (¬c_a_b)})
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-between: a_b_c,
geo-point: Point,
guard: {T},
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uimplies: b supposing a,
subtype_rel: A ⊆r B,
heyting-geometry: Error :heyting-geometry,
uall: ∀[x:A]. B[x],
prop: ℙ,
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,
heyting-geometry-subtype,
basic-geometry-subtype,
geo-point_wf,
Error :geo-triangle_wf,
geo-triangle-implies
Rules used in proof :
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
rename,
setElimination,
isectElimination,
independent_pairFormation,
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{} \{(\mneg{}a\_b\_c) \mwedge{} (\mneg{}b\_c\_a) \mwedge{} (\mneg{}c\_a\_b)\})
Date html generated:
2017_10_02-PM-07_02_19
Last ObjectModification:
2017_08_06-PM-08_55_32
Theory : euclidean!plane!geometry
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