Nuprl Lemma : pgeo-join-to-line
∀g:BasicProjectivePlane. ∀p,q:Point. ∀l:Line. ∀s:p ≠ q. (p I l ⇒ q I l ⇒ p ∨ q ≡ l)
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 :
pgeo-peq: a ≡ b,
guard: {T},
false: False,
cand: A c∧ B,
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
basic-projective-plane: BasicProjectivePlane,
uall: ∀[x:A]. B[x],
member: t ∈ T,
not: ¬A,
pgeo-leq: a ≡ b,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
pgeo-point_wf,
pgeo-psep_wf,
pgeo-primitives_wf,
projective-plane-structure_wf,
basic-projective-plane_wf,
subtype_rel_transitivity,
basic-projective-plane-subtype,
projective-plane-structure_subtype,
pgeo-lsep_wf,
pgeo-leq_wf,
pgeo-peq_wf,
incident-join-second,
incident-join-first,
pgeo-incident_wf,
pgeo-line_wf,
pgeo-join_wf,
Unique
Rules used in proof :
instantiate,
voidElimination,
independent_functionElimination,
independent_pairFormation,
independent_isectElimination,
productEquality,
sqequalRule,
setEquality,
lambdaEquality,
applyEquality,
hypothesis,
because_Cache,
rename,
setElimination,
isectElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:BasicProjectivePlane. \mforall{}p,q:Point. \mforall{}l:Line. \mforall{}s:p \mneq{} q. (p I l {}\mRightarrow{} q I l {}\mRightarrow{} p \mvee{} q \mequiv{} l)
Date html generated:
2018_05_22-PM-00_35_39
Last ObjectModification:
2017_11_25-AM-09_05_06
Theory : euclidean!plane!geometry
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