Nuprl Lemma : geo-orientation_wf
∀[g:EuclideanPlaneStructure]. ∀[a,b:Point]. ∀[c:{c:Point| a # bc} ].
(geo-orientation(g;a;b;c) ∈ a leftof bc ∨ a leftof cb)
Proof
Definitions occuring in Statement :
geo-orientation: geo-orientation(g;a;b;c),
euclidean-plane-structure: EuclideanPlaneStructure,
geo-lsep: a # bc,
geo-left: a leftof bc,
geo-point: Point,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
geo-orientation: geo-orientation(g;a;b;c),
implies: P ⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
sq_exists: ∃x:{A| B[x]},
or: P ∨ Q,
all: ∀x:A. B[x],
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
btrue: tt,
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
subtype_rel: A ⊆r B,
record-select: r.x,
record+: record+,
euclidean-plane-structure: EuclideanPlaneStructure,
member: t ∈ T,
uall: ∀[x:A]. B[x],
geo-lsep: a # bc,
squash: ↓T,
it: ⋅
Lemmas referenced :
euclidean-plane-structure_wf,
geo-lsep_wf,
euclidean-plane-structure-subtype,
set_wf,
geo-gt_wf,
geo-ge_wf,
geo-colinear_wf,
sq_exists_wf,
exists_wf,
or_wf,
geo-left_wf,
geo-sep_wf,
sq_stable_wf,
all_wf,
geo-congruent_wf,
geo-between_wf,
stable_wf,
geo-point_wf,
uall_wf,
subtype_rel_self
Rules used in proof :
isect_memberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
functionExtensionality,
functionEquality,
productElimination,
productEquality,
lambdaFormation,
because_Cache,
setEquality,
lambdaEquality,
isectElimination,
extract_by_obid,
tokenEquality,
applyEquality,
hypothesis,
dependentIntersectionEqElimination,
sqequalRule,
dependentIntersectionElimination,
sqequalHypSubstitution,
hypothesisEquality,
rename,
thin,
setElimination,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
baseClosed,
imageMemberEquality
Latex:
\mforall{}[g:EuclideanPlaneStructure]. \mforall{}[a,b:Point]. \mforall{}[c:\{c:Point| a \# bc\} ].
(geo-orientation(g;a;b;c) \mmember{} a leftof bc \mvee{} a leftof cb)
Date html generated:
2017_10_02-PM-06_49_44
Last ObjectModification:
2017_08_06-PM-07_38_12
Theory : euclidean!plane!geometry
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