Nuprl Lemma : pgeo-lsep-implies-plsep
∀g:ProjectivePlane. ∀p:Point. ∀l,m:Line. ∀s:l ≠ m. (p I l ⇒ p ≠ l ∧ m ⇒ p ≠ m)
Proof
Definitions occuring in Statement :
projective-plane: ProjectivePlane,
pgeo-meet: l ∧ m,
pgeo-lsep: l ≠ m,
pgeo-psep: a ≠ b,
pgeo-incident: a I b,
pgeo-plsep: a ≠ b,
pgeo-line: Line,
pgeo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
and: P ∧ Q,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
cand: A c∧ B,
exists: ∃x:A. B[x],
pgeo-lsep: l ≠ m,
or: P ∨ Q,
false: False,
not: ¬A,
pgeo-incident: a I b
Lemmas referenced :
pgeo-line_wf,
pgeo-lsep_wf,
pgeo-incident_wf,
pgeo-point_wf,
pgeo-meet_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-psep_wf,
pgeo-meet-incident,
projective-plane-subtype-basic,
pgeo-plsep-implies-join,
LP-sep-or2,
pgeo-join-implies-plsep,
use-triangle-axiom1
Rules used in proof :
productEquality,
because_Cache,
setEquality,
rename,
setElimination,
lambdaEquality,
dependent_functionElimination,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_pairFormation,
productElimination,
independent_functionElimination,
unionElimination,
voidElimination
Latex:
\mforall{}g:ProjectivePlane. \mforall{}p:Point. \mforall{}l,m:Line. \mforall{}s:l \mneq{} m. (p I l {}\mRightarrow{} p \mneq{} l \mwedge{} m {}\mRightarrow{} p \mneq{} m)
Date html generated:
2018_05_22-PM-00_43_26
Last ObjectModification:
2017_11_28-PM-05_17_48
Theory : euclidean!plane!geometry
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