Nuprl Lemma : pgeo-meet-implies-psep
∀g:BasicProjectivePlane. ∀l,m:Line. ∀s:l ≠ m. ∀a:Point. ∀c:{c:Point| c I l ∧ c I m} . (a ≠ l ∧ m ⇒ a ≠ c)
Proof
Definitions occuring in Statement :
basic-projective-plane: BasicProjectivePlane,
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,
uall: ∀[x:A]. B[x],
sq_stable: SqStable(P),
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
basic-projective-plane: BasicProjectivePlane,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
pgeo-meet-implies-plsep,
sq_stable__pgeo-incident,
pgeo-incident_wf,
pgeo-plsep_wf,
pgeo-psep_wf,
projective-plane-structure_subtype,
basic-projective-plane-subtype,
subtype_rel_transitivity,
basic-projective-plane_wf,
projective-plane-structure_wf,
pgeo-primitives_wf,
pgeo-meet_wf,
pgeo-point_wf,
set_wf,
pgeo-lsep_wf,
pgeo-line_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
setElimination,
rename,
isectElimination,
because_Cache,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
dependent_pairFormation,
productEquality,
applyEquality,
instantiate,
independent_isectElimination,
lambdaEquality,
setEquality
Latex:
\mforall{}g:BasicProjectivePlane. \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{} l \mwedge{} m {}\mRightarrow{} a \mneq{} c)
Date html generated:
2018_05_22-PM-00_39_57
Last ObjectModification:
2017_12_05-AM-10_22_48
Theory : euclidean!plane!geometry
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