Nuprl Lemma : pgeo-order_2-incidence
∀pg:ProjectivePlane. ∀l:Line. (order(pg) = 2 ⇒ p,q:{p:Point| p I l} //p ≡ q ~ ℕ3)
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-},
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
prop: ℙ,
member: t ∈ T,
pgeo-order: order(pg) = n,
implies: P ⇒ Q,
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,
le_wf,
false_wf,
pgeo-order_wf
Rules used in proof :
independent_isectElimination,
instantiate,
applyEquality,
isectElimination,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality,
extract_by_obid,
introduction,
hypothesisEquality,
thin,
dependent_functionElimination,
hypothesis,
cut,
sqequalRule,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}pg:ProjectivePlane. \mforall{}l:Line. (order(pg) = 2 {}\mRightarrow{} p,q:\{p:Point| p I l\} //p \mequiv{} q \msim{} \mBbbN{}3)
Date html generated:
2018_05_22-PM-00_58_27
Last ObjectModification:
2018_01_10-AM-10_26_51
Theory : euclidean!plane!geometry
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