Nuprl Lemma : half-plane-point-exists
∀g:EuclideanPlane. ∀x,y:Point. (x ≠ y ⇒ (∃q:Point. q leftof xy))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-left: a leftof bc,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: 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,
prop: ℙ,
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
guard: {T},
uimplies: b supposing a,
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x]
Lemmas referenced :
Euclid-Prop1-left,
geo-sep_wf,
sq_stable__and,
geo-congruent_wf,
geo-left_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
sq_stable__geo-congruent,
sq_stable__geo-left,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality_alt,
hypothesis,
universeIsType,
isectElimination,
applyEquality,
because_Cache,
sqequalRule,
setElimination,
rename,
productEquality,
productElimination,
isect_memberEquality_alt,
instantiate,
independent_isectElimination,
productIsType,
inhabitedIsType,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation_alt
Latex:
\mforall{}g:EuclideanPlane. \mforall{}x,y:Point. (x \mneq{} y {}\mRightarrow{} (\mexists{}q:Point. q leftof xy))
Date html generated:
2019_10_16-PM-01_48_35
Last ObjectModification:
2018_11_05-PM-09_17_41
Theory : euclidean!plane!geometry
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