Nuprl Lemma : plsep-join-implies
∀g:ProjectivePlane. ∀p,q,r:Point. ∀s:p ≠ q. (r ≠ p ∨ q ⇒ {q ≠ r ∧ p ≠ r})
Proof
Definitions occuring in Statement :
projective-plane: ProjectivePlane,
pgeo-join: p ∨ q,
pgeo-psep: a ≠ b,
pgeo-plsep: a ≠ b,
pgeo-point: Point,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
subtype_rel: A ⊆r B,
and: P ∧ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
cand: A c∧ B,
pgeo-psep: a ≠ b,
false: False,
not: ¬A,
pgeo-leq: a ≡ b,
or: P ∨ Q
Lemmas referenced :
pgeo-point_wf,
pgeo-psep_wf,
pgeo-incident_wf,
pgeo-line_wf,
pgeo-join_wf,
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-plsep_wf,
projective-plane-subtype-basic,
pgeo-join-to-line-2,
Join,
PL-sep-or
Rules used in proof :
productEquality,
setEquality,
rename,
setElimination,
lambdaEquality,
independent_isectElimination,
instantiate,
isectElimination,
independent_pairFormation,
sqequalRule,
applyEquality,
productElimination,
hypothesis,
independent_functionElimination,
hypothesisEquality,
because_Cache,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
dependent_pairFormation,
voidElimination,
unionElimination
Latex:
\mforall{}g:ProjectivePlane. \mforall{}p,q,r:Point. \mforall{}s:p \mneq{} q. (r \mneq{} p \mvee{} q {}\mRightarrow{} \{q \mneq{} r \mwedge{} p \mneq{} r\})
Date html generated:
2018_05_22-PM-00_41_54
Last ObjectModification:
2017_11_28-PM-04_20_49
Theory : euclidean!plane!geometry
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