Nuprl Lemma : pgeo-order-implies2
∀pg:ProjectivePlane. ∀n:ℕ. (order(pg) = n ⇒ (∃l:Line. p,q:{p:Point| p I l} //p ≡ q ~ ℕn + 1))
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],
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
nat: ℕ,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
prop: ℙ,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
sq_exists: ∃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-order_wf,
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_wf,
pgeo-non-trivial-dual
Rules used in proof :
addEquality,
natural_numberEquality,
lambdaEquality,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
setEquality,
isectElimination,
because_Cache,
hypothesis,
dependent_pairFormation,
rename,
setElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}pg:ProjectivePlane. \mforall{}n:\mBbbN{}. (order(pg) = n {}\mRightarrow{} (\mexists{}l:Line. p,q:\{p:Point| p I l\} //p \mequiv{} q \msim{} \mBbbN{}n + 1))
Date html generated:
2018_05_22-PM-00_59_00
Last ObjectModification:
2018_01_10-AM-10_36_29
Theory : euclidean!plane!geometry
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