Nuprl Lemma : pgeo-order-equiv_rel
∀pg:ProjectivePlane. ∀l:Line. EquivRel({p:Point| p I l} ;p,q.p ≡ q)
Proof
Definitions occuring in Statement :
projective-plane: ProjectivePlane,
pgeo-peq: a ≡ b,
pgeo-incident: a I b,
pgeo-line: Line,
pgeo-point: Point,
equiv_rel: EquivRel(T;x,y.E[x; y]),
all: ∀x:A. B[x],
set: {x:A| B[x]}
Definitions unfolded in proof :
uimplies: b supposing a,
trans: Trans(T;x,y.E[x; y]),
guard: {T},
sym: Sym(T;x,y.E[x; y]),
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
member: t ∈ T,
refl: Refl(T;x,y.E[x; y]),
cand: A c∧ B,
and: P ∧ Q,
equiv_rel: EquivRel(T;x,y.E[x; y]),
all: ∀x:A. B[x]
Lemmas referenced :
pgeo-primitives_wf,
projective-plane-structure_wf,
projective-plane-structure-complete_wf,
projective-plane_wf,
subtype_rel_transitivity,
projective-plane-subtype,
projective-plane-structure-complete_subtype,
projective-plane-structure_subtype,
pgeo-line_wf,
pgeo-peq_transitivity,
pgeo-peq_wf,
pgeo-peq_inversion,
pgeo-incident_wf,
pgeo-point_wf,
set_wf,
pgeo-peq_weakening
Rules used in proof :
independent_isectElimination,
instantiate,
independent_pairFormation,
lambdaEquality,
sqequalRule,
applyEquality,
isectElimination,
independent_functionElimination,
because_Cache,
hypothesis,
rename,
setElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}pg:ProjectivePlane. \mforall{}l:Line. EquivRel(\{p:Point| p I l\} ;p,q.p \mequiv{} q)
Date html generated:
2018_05_22-PM-00_58_06
Last ObjectModification:
2018_01_10-AM-10_33_56
Theory : euclidean!plane!geometry
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