Nuprl Lemma : pgeo-join_wf
∀g:ProjectivePlaneStructure. ∀p,q:Point. ∀s:p ≠ q. (p ∨ q ∈ {L:Line| p I L ∧ q I L} )
Proof
Definitions occuring in Statement :
pgeo-join: p ∨ q,
projective-plane-structure: ProjectivePlaneStructure,
pgeo-psep: a ≠ b,
pgeo-incident: a I b,
pgeo-line: Line,
pgeo-point: Point,
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
pgeo-join: p ∨ q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
and: P ∧ Q,
prop: ℙ,
projective-plane-structure: ProjectivePlaneStructure,
record+: record+,
record-select: r.x,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
btrue: tt,
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
top: Top
Lemmas referenced :
pi1_wf_top,
pgeo-line_wf,
projective-plane-structure_subtype,
pgeo-incident_wf,
subtype_rel_self,
all_wf,
pgeo-point_wf,
sq_stable_wf,
pgeo-plsep_wf,
or_wf,
pgeo-lsep_wf,
pgeo-psep_wf,
exists_wf,
sq_exists_wf,
projective-plane-structure_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesisEquality,
applyEquality,
hypothesis,
productEquality,
because_Cache,
dependentIntersectionElimination,
dependentIntersectionEqElimination,
tokenEquality,
lambdaEquality,
setElimination,
rename,
functionEquality,
functionExtensionality,
productElimination,
independent_pairEquality,
dependent_set_memberEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}g:ProjectivePlaneStructure. \mforall{}p,q:Point. \mforall{}s:p \mneq{} q. (p \mvee{} q \mmember{} \{L:Line| p I L \mwedge{} q I L\} )
Date html generated:
2018_05_22-PM-00_30_18
Last ObjectModification:
2017_10_27-PM-01_41_22
Theory : euclidean!plane!geometry
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