Nuprl Lemma : Join
∀g:ProjectivePlaneStructure. ∀p,q:Point. (p ≠ q ⇒ (∃l:Line. (p I l ∧ q I l)))
Proof
Definitions occuring in Statement :
projective-plane-structure: ProjectivePlaneStructure,
pgeo-psep: a ≠ b,
pgeo-incident: a I b,
pgeo-line: Line,
pgeo-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,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
exists: ∃x:A. B[x],
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
pgeo-psep_wf,
projective-plane-structure_subtype,
pgeo-point_wf,
projective-plane-structure_wf,
pgeo-join_wf,
set_wf,
pgeo-line_wf,
pgeo-incident_wf,
sq_stable__pgeo-incident,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
rename,
dependent_functionElimination,
lambdaEquality,
productEquality,
dependent_pairFormation,
setElimination,
productElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}g:ProjectivePlaneStructure. \mforall{}p,q:Point. (p \mneq{} q {}\mRightarrow{} (\mexists{}l:Line. (p I l \mwedge{} q I l)))
Date html generated:
2018_05_22-PM-00_30_27
Last ObjectModification:
2017_10_27-PM-01_43_14
Theory : euclidean!plane!geometry
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