Nuprl Lemma : use-plane-sep
∀g:EuclideanPlaneStructure. ∀a,b,u,v:Point. (u leftof ab ⇒ v leftof ba ⇒ (∃x:Point. (Colinear(a;b;x) ∧ u_x_v)))
Proof
Definitions occuring in Statement :
euclidean-plane-structure: EuclideanPlaneStructure,
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],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
squash: ↓T,
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
so_apply: x[s],
and: P ∧ Q,
so_lambda: λ2x.t[x],
prop: ℙ,
subtype_rel: A ⊆r B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
sq_stable__geo-between,
sq_stable__colinear,
sq_stable__and,
euclidean-plane-structure_wf,
equal_wf,
geo-between_wf,
geo-colinear_wf,
euclidean-plane-structure-subtype,
geo-point_wf,
set_wf,
geo-left_wf,
geo-SS_wf
Rules used in proof :
imageElimination,
baseClosed,
imageMemberEquality,
productElimination,
isect_memberEquality,
independent_functionElimination,
dependent_functionElimination,
dependent_pairFormation,
rename,
setElimination,
productEquality,
lambdaEquality,
equalitySymmetry,
equalityTransitivity,
sqequalRule,
because_Cache,
applyEquality,
hypothesis,
dependent_set_memberEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:EuclideanPlaneStructure. \mforall{}a,b,u,v:Point.
(u leftof ab {}\mRightarrow{} v leftof ba {}\mRightarrow{} (\mexists{}x:Point. (Colinear(a;b;x) \mwedge{} u\_x\_v)))
Date html generated:
2017_10_02-PM-03_26_29
Last ObjectModification:
2017_08_13-PM-08_20_30
Theory : euclidean!plane!geometry
Home
Index