Nuprl Lemma : geo-not-left-convex
∀g:OrientedPlane. ∀a,b:Point. IsConvex(x.¬x leftof ab)
Proof
Definitions occuring in Statement :
geo-convex: IsConvex(x.P[x]),
oriented-plane: OrientedPlane,
geo-left: a leftof bc,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
oriented-plane: OrientedPlane,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
geo-convex: IsConvex(x.P[x]),
not: ¬A,
false: False,
or: P ∨ Q,
stable: Stable{P},
iff: P ⇐⇒ Q,
and: P ∧ Q,
geo-lsep: a # bc,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
exists: ∃x:A. B[x],
cand: A c∧ B,
rev_implies: P ⇐ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
geo-eq: a ≡ b
Lemmas referenced :
oriented-plane-axioms,
all_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
oriented-plane-subtype,
subtype_rel_transitivity,
oriented-plane_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-convex_wf,
subtype_rel_self,
basic-geometry-_wf,
not_wf,
geo-left_wf,
geo-left-axioms_wf,
basic-geo-axioms_wf,
istype-void,
geo-between_wf,
stable__false,
false_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
geo-left-convex,
not-left-and-right,
not-lsep-iff-colinear,
geo-lsep_wf,
geo-between-symmetry,
left-convex3,
geo-colinear-is-colinear-set,
length_of_cons_lemma,
length_of_nil_lemma,
istype-false,
istype-le,
istype-less_than,
geo-sep_wf,
oriented-colinear-append,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
left-implies-sep,
list_ind_cons_lemma,
list_ind_nil_lemma,
geo-between-implies-colinear,
geo-colinear_functionality,
geo-eq_weakening,
geo-left_functionality,
geo-between_functionality,
geo-between-same
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
isectElimination,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
because_Cache,
inhabitedIsType,
universeIsType,
independent_functionElimination,
voidElimination,
functionIsType,
unionEquality,
functionEquality,
unionIsType,
unionElimination,
productElimination,
rename,
isect_memberEquality_alt,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
productIsType,
inlFormation_alt,
dependent_pairFormation_alt,
inrFormation_alt,
equalityIsType1,
promote_hyp
Latex:
\mforall{}g:OrientedPlane. \mforall{}a,b:Point. IsConvex(x.\mneg{}x leftof ab)
Date html generated:
2019_10_16-PM-01_40_18
Last ObjectModification:
2018_11_12-PM-04_02_51
Theory : euclidean!plane!geometry
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