Nuprl Lemma : pgeo-meet-line-uniqueness
∀g:ProjectivePlane. ∀a,b:Point. ∀l,m:Line. (a ≠ b ⇒ (a I l ∧ a I m) ⇒ (b I l ∧ b I m) ⇒ l ≡ m)
Proof
Definitions occuring in Statement :
projective-plane: ProjectivePlane,
pgeo-leq: a ≡ b,
pgeo-psep: a ≠ b,
pgeo-incident: a I b,
pgeo-line: Line,
pgeo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
false: False,
prop: ℙ,
not: ¬A,
pgeo-leq: a ≡ b,
cand: A c∧ B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
pgeo-peq: a ≡ b
Lemmas referenced :
projective-plane_wf,
pgeo-point_wf,
pgeo-line_wf,
pgeo-psep_wf,
pgeo-incident_wf,
pgeo-lsep_wf,
pgeo-leq_wf,
pgeo-peq_wf,
projective-plane-subtype-basic,
Unique
Rules used in proof :
productEquality,
voidElimination,
because_Cache,
independent_functionElimination,
independent_pairFormation,
independent_isectElimination,
isectElimination,
sqequalRule,
hypothesis,
applyEquality,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:ProjectivePlane. \mforall{}a,b:Point. \mforall{}l,m:Line. (a \mneq{} b {}\mRightarrow{} (a I l \mwedge{} a I m) {}\mRightarrow{} (b I l \mwedge{} b I m) {}\mRightarrow{} l \mequiv{} m)
Date html generated:
2018_05_22-PM-00_52_19
Last ObjectModification:
2018_01_03-PM-05_02_20
Theory : euclidean!plane!geometry
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