Nuprl Lemma : pgeo-meet-implies-psep2
∀g:ProjectivePlane. ∀l,m:Line. ∀s:l ≠ m. ∀a:Point. ∀c:{c:Point| c I l ∧ c I m} . (a ≠ c ⇒ a ≠ l ∧ 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-line: Line,
pgeo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
pgeo-psep: a ≠ b,
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
cand: A c∧ B,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
prop: ℙ,
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T,
pgeo-peq: a ≡ b,
not: ¬A,
false: False
Lemmas referenced :
LP-sep-or2,
projective-plane-structure-complete_subtype,
projective-plane-subtype,
subtype_rel_transitivity,
projective-plane_wf,
projective-plane-structure-complete_wf,
projective-plane-structure_wf,
pgeo-meet_wf,
pgeo-point_wf,
pgeo-incident_wf,
pgeo-plsep_wf,
pgeo-psep_wf,
projective-plane-structure_subtype,
pgeo-primitives_wf,
set_wf,
pgeo-lsep_wf,
pgeo-line_wf,
pgeo-meet-to-point,
projective-plane-subtype-basic,
sq_stable__pgeo-incident,
pgeo-psep-sym
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
cut,
hypothesis,
independent_pairFormation,
introduction,
extract_by_obid,
dependent_functionElimination,
applyEquality,
instantiate,
isectElimination,
independent_isectElimination,
sqequalRule,
because_Cache,
setElimination,
rename,
lambdaEquality,
setEquality,
productEquality,
independent_functionElimination,
unionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
voidElimination
Latex:
\mforall{}g:ProjectivePlane. \mforall{}l,m:Line. \mforall{}s:l \mneq{} m. \mforall{}a:Point. \mforall{}c:\{c:Point| c I l \mwedge{} c I m\} .
(a \mneq{} c {}\mRightarrow{} a \mneq{} l \mwedge{} m)
Date html generated:
2018_05_22-PM-00_43_06
Last ObjectModification:
2017_12_05-AM-10_45_28
Theory : euclidean!plane!geometry
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