Nuprl Lemma : pgeo-join-to-line-2
∀g:BasicProjectivePlane. ∀p,q:Point. ∀l:Line. ∀s:p ≠ q. (p I l ⇒ q I l ⇒ l ≡ p ∨ q)
Proof
Definitions occuring in Statement :
basic-projective-plane: BasicProjectivePlane,
pgeo-join: p ∨ q,
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
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
pgeo-leq: a ≡ b,
not: ¬A,
member: t ∈ T,
uall: ∀[x:A]. B[x],
basic-projective-plane: BasicProjectivePlane,
subtype_rel: A ⊆r B,
and: P ∧ Q,
prop: ℙ,
uimplies: b supposing a,
cand: A c∧ B,
false: False,
guard: {T},
pgeo-peq: a ≡ b
Lemmas referenced :
Unique,
pgeo-join_wf,
pgeo-line_wf,
pgeo-incident_wf,
incident-join-first,
incident-join-second,
pgeo-peq_wf,
pgeo-leq_wf,
pgeo-lsep_wf,
projective-plane-structure_subtype,
basic-projective-plane-subtype,
subtype_rel_transitivity,
basic-projective-plane_wf,
projective-plane-structure_wf,
pgeo-primitives_wf,
pgeo-psep_wf,
pgeo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
setElimination,
rename,
because_Cache,
hypothesis,
applyEquality,
lambdaEquality,
setEquality,
sqequalRule,
productEquality,
independent_isectElimination,
independent_pairFormation,
independent_functionElimination,
voidElimination,
instantiate
Latex:
\mforall{}g:BasicProjectivePlane. \mforall{}p,q:Point. \mforall{}l:Line. \mforall{}s:p \mneq{} q. (p I l {}\mRightarrow{} q I l {}\mRightarrow{} l \mequiv{} p \mvee{} q)
Date html generated:
2018_05_22-PM-00_35_58
Last ObjectModification:
2017_11_10-PM-03_50_11
Theory : euclidean!plane!geometry
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