Nuprl Lemma : pgeo-order_incidence-eq
∀pg:ProjectivePlane. ∀n:ℕ. ∀l1,l2:Line. (order(pg) = n ⇒ p,q:{p:Point| p I l1} //p ≡ q ~ p,q:{p:Point| p I l2} //p ≡ q\000C)
Proof
Definitions occuring in Statement :
pgeo-order: order(pg) = n,
projective-plane: ProjectivePlane,
pgeo-peq: a ≡ b,
pgeo-incident: a I b,
pgeo-line: Line,
pgeo-point: Point,
equipollent: A ~ B,
quotient: x,y:A//B[x; y],
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
nat: ℕ,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
prop: ℙ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
guard: {T},
uall: ∀[x:A]. B[x],
member: t ∈ T,
pgeo-order: order(pg) = n,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
nat_wf,
pgeo-line_wf,
pgeo-order_wf,
equipollent_transitivity,
int_seg_wf,
pgeo-order-equiv_rel,
pgeo-peq_wf,
pgeo-incident_wf,
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-point_wf,
quotient_wf,
equipollent_inversion
Rules used in proof :
independent_functionElimination,
addEquality,
natural_numberEquality,
rename,
setElimination,
lambdaEquality,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
setEquality,
isectElimination,
extract_by_obid,
introduction,
hypothesisEquality,
thin,
dependent_functionElimination,
hypothesis,
cut,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}pg:ProjectivePlane. \mforall{}n:\mBbbN{}. \mforall{}l1,l2:Line.
(order(pg) = n {}\mRightarrow{} p,q:\{p:Point| p I l1\} //p \mequiv{} q \msim{} p,q:\{p:Point| p I l2\} //p \mequiv{} q)
Date html generated:
2018_05_22-PM-00_59_10
Last ObjectModification:
2018_05_21-AM-01_31_57
Theory : euclidean!plane!geometry
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