Nuprl Lemma : lsep-opposite-iff
∀g:OrientedPlane. ∀a,b,x,y:Point.
(x # ab ⇒ y # ab ⇒ (∃z:Point. (x_z_y ∧ Colinear(z;a;b)) ⇐⇒ x leftof ab ⇐⇒ y leftof ba))
Proof
Definitions occuring in Statement :
oriented-plane: OrientedPlane,
geo-lsep: a # bc,
geo-colinear: Colinear(a;b;c),
geo-left: a leftof bc,
geo-between: a_b_c,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
oriented-plane: Error :oriented-plane,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
or: P ∨ Q,
geo-lsep: a # bc,
exists: ∃x:A. B[x],
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
cand: A c∧ B,
euclidean-geometry: Error :euclidean-geometry
Lemmas referenced :
geo-lsep_wf,
iff_wf,
geo-colinear_wf,
geo-between_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry-_wf,
Error :oriented-plane_wf,
subtype_rel_transitivity,
Error :oriented-plane-subtype,
basic-geometry--subtype,
geo-point_wf,
exists_wf,
not-left-and-right,
geo-left_wf,
left-between-weak,
Error :use-plane-sep,
lsep-symmetry,
lsep-symmetry2,
Error :euclidean-geometry_wf,
subtype_rel_self
Rules used in proof :
productEquality,
lambdaEquality,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
voidElimination,
independent_functionElimination,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
because_Cache,
rename,
setElimination,
isectElimination,
extract_by_obid,
introduction,
cut,
unionElimination,
thin,
productElimination,
sqequalHypSubstitution,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
inlFormation,
inrFormation,
dependent_pairFormation
Latex:
\mforall{}g:OrientedPlane. \mforall{}a,b,x,y:Point.
(x \# ab {}\mRightarrow{} y \# ab {}\mRightarrow{} (\mexists{}z:Point. (x\_z\_y \mwedge{} Colinear(z;a;b)) \mLeftarrow{}{}\mRightarrow{} x leftof ab \mLeftarrow{}{}\mRightarrow{} y leftof ba))
Date html generated:
2017_10_02-PM-04_47_39
Last ObjectModification:
2017_08_05-AM-10_20_20
Theory : euclidean!plane!geometry
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