Nuprl Lemma : pgeo-three-lines_wf
∀g:ProjectivePlaneStructure. ∀a:Point.
(pgeo-three-line-axiom(a) ∈ ∃l,m,n:Line. (a I l ∧ a I m ∧ a I n ∧ l ≠ m ∧ m ≠ n ∧ n ≠ l))
Proof
Definitions occuring in Statement :
pgeo-three-lines: pgeo-three-line-axiom(p),
projective-plane-structure: ProjectivePlaneStructure,
pgeo-lsep: l ≠ m,
pgeo-incident: a I b,
pgeo-line: Line,
pgeo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
pgeo-three-lines: pgeo-three-line-axiom(p),
projective-plane-structure: ProjectivePlaneStructure,
record+: record+,
record-select: r.x,
subtype_rel: A ⊆r B,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
btrue: tt,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
or: P ∨ Q,
implies: P ⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x]
Lemmas referenced :
subtype_rel_self,
all_wf,
pgeo-line_wf,
pgeo-point_wf,
sq_stable_wf,
pgeo-plsep_wf,
or_wf,
pgeo-lsep_wf,
pgeo-lpsep_wf,
pgeo-psep_wf,
exists_wf,
pgeo-incident_wf,
projective-plane-structure_subtype,
projective-plane-structure_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalRule,
sqequalHypSubstitution,
dependentIntersectionElimination,
dependentIntersectionEqElimination,
thin,
hypothesis,
applyEquality,
tokenEquality,
introduction,
extract_by_obid,
isectElimination,
lambdaEquality,
hypothesisEquality,
setEquality,
setElimination,
rename,
functionEquality,
productEquality,
because_Cache,
functionExtensionality
Latex:
\mforall{}g:ProjectivePlaneStructure. \mforall{}a:Point.
(pgeo-three-line-axiom(a) \mmember{} \mexists{}l,m,n:Line. (a I l \mwedge{} a I m \mwedge{} a I n \mwedge{} l \mneq{} m \mwedge{} m \mneq{} n \mwedge{} n \mneq{} l))
Date html generated:
2018_05_22-PM-00_32_10
Last ObjectModification:
2017_11_03-AM-11_50_07
Theory : euclidean!plane!geometry
Home
Index