Nuprl Lemma : pgeo-plsep-iff-all-psep
∀g:ProjectivePlane. ∀p:Point. ∀l:Line. (p ≠ l ⇐⇒ ∀q:Point. (q I l ⇒ p ≠ q))
Proof
Definitions occuring in Statement :
projective-plane: ProjectivePlane,
pgeo-psep: a ≠ b,
pgeo-incident: a I b,
pgeo-plsep: a ≠ b,
pgeo-line: Line,
pgeo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
projective-plane: ProjectivePlane,
exists: ∃x:A. B[x],
or: P ∨ Q
Lemmas referenced :
pgeo-line_wf,
pgeo-psep_wf,
pgeo-incident_wf,
pgeo-primitives_wf,
projective-plane-structure_wf,
basic-projective-plane_wf,
projective-plane_wf,
subtype_rel_transitivity,
projective-plane-subtype,
basic-projective-plane-subtype,
projective-plane-structure_subtype,
pgeo-point_wf,
all_wf,
pgeo-plsep_wf,
Error :pgeo-psep-sym,
pgeo-plsep-to-psep,
pgeo-three-lines-axiom,
pgeo-lsep-or,
pgeo-meet-incident,
pgeo-meet_wf,
pgeo-lsep-implies-plsep,
Meet
Rules used in proof :
functionEquality,
lambdaEquality,
independent_isectElimination,
instantiate,
sqequalRule,
hypothesis,
because_Cache,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
rename,
setElimination,
dependent_functionElimination,
productElimination,
unionElimination,
productEquality,
setEquality
Latex:
\mforall{}g:ProjectivePlane. \mforall{}p:Point. \mforall{}l:Line. (p \mneq{} l \mLeftarrow{}{}\mRightarrow{} \mforall{}q:Point. (q I l {}\mRightarrow{} p \mneq{} q))
Date html generated:
2018_05_22-PM-00_46_38
Last ObjectModification:
2017_11_20-PM-00_31_59
Theory : euclidean!plane!geometry
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