Nuprl Lemma : plsep-implies-ptriangle
∀g:ProjectivePlane. ∀p:Point. ∀l:Line. ∀q:Point. ∀s:q ≠ p. (p ≠ l ⇒ q I l ⇒ (∃r:Point. (r I l ∧ r ≠ q ∨ p)))
Proof
Definitions occuring in Statement :
projective-plane: ProjectivePlane,
pgeo-join: p ∨ q,
pgeo-psep: a ≠ b,
pgeo-incident: a I b,
pgeo-plsep: a ≠ b,
pgeo-line: Line,
pgeo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
prop: ℙ,
exists: ∃x:A. B[x],
uimplies: b supposing a,
guard: {T},
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
cand: A c∧ B,
and: P ∧ Q
Lemmas referenced :
pgeo-line_wf,
pgeo-point_wf,
pgeo-psep_wf,
pgeo-plsep_wf,
pgeo-primitives_wf,
projective-plane-structure_subtype,
pgeo-incident_wf,
point-implies-plsep-exists,
projective-plane-structure_wf,
projective-plane-structure-complete_wf,
projective-plane_wf,
subtype_rel_transitivity,
projective-plane-subtype,
projective-plane-structure-complete_subtype,
pgeo-plsep-to-psep,
pgeo-join_wf,
Meet,
pgeo-plsep-to-lsep,
projective-plane-subtype-basic,
pgeo-plsep-implies-join,
use-triangle-axiom1
Rules used in proof :
productElimination,
because_Cache,
independent_functionElimination,
sqequalRule,
independent_isectElimination,
isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
setEquality,
setElimination,
lambdaEquality,
productEquality,
independent_pairFormation,
dependent_pairFormation,
rename
Latex:
\mforall{}g:ProjectivePlane. \mforall{}p:Point. \mforall{}l:Line. \mforall{}q:Point. \mforall{}s:q \mneq{} p.
(p \mneq{} l {}\mRightarrow{} q I l {}\mRightarrow{} (\mexists{}r:Point. (r I l \mwedge{} r \mneq{} q \mvee{} p)))
Date html generated:
2018_05_22-PM-00_42_27
Last ObjectModification:
2017_11_28-PM-05_41_03
Theory : euclidean!plane!geometry
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