Nuprl Lemma : geo-lt-angle-in-half-plane-point-exists
∀e:EuclideanPlane. ∀w,x,y,z:Point. (xyz < wyz ⇒ w leftof yz ⇒ x leftof yz ⇒ (∃q:Point. (w-q-z ∧ out(y qx))))
Proof
Definitions occuring in Statement :
geo-lt-angle: abc < xyz,
geo-out: out(p ab),
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-left: a leftof bc,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
basic-geometry: BasicGeometry,
or: P ∨ Q
Lemmas referenced :
geo-lt-angle-in-half-plane-implies-left,
use-plane-sep_strict,
geo-left_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-lt-angle_wf,
geo-point_wf,
left-symmetry,
geo-strict-between_wf,
geo-out_wf,
geo-colinear-left-out2,
left-convex,
geo-strict-between-implies-between,
geo-between-symmetry,
geo-strict-between-sep3,
geo-between_wf,
geo-out_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
inhabitedIsType,
productElimination,
dependent_pairFormation_alt,
independent_pairFormation,
productIsType,
inrFormation_alt
Latex:
\mforall{}e:EuclideanPlane. \mforall{}w,x,y,z:Point.
(xyz < wyz {}\mRightarrow{} w leftof yz {}\mRightarrow{} x leftof yz {}\mRightarrow{} (\mexists{}q:Point. (w-q-z \mwedge{} out(y qx))))
Date html generated:
2019_10_16-PM-02_29_37
Last ObjectModification:
2019_03_19-PM-04_12_51
Theory : euclidean!plane!geometry
Home
Index