Nuprl Lemma : point-implies-plsep-exists
∀g:ProjectivePlane. ∀a:Point. ∃l:Line. a ≠ l
Proof
Definitions occuring in Statement :
projective-plane: ProjectivePlane,
pgeo-plsep: a ≠ b,
pgeo-line: Line,
pgeo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x]
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
record-select: r.x,
pgeo-plsep: a ≠ b,
pgeo-lpsep: a ≠ b,
implies: P ⇒ Q,
pgeo-psep: a ≠ b,
pgeo-lsep: l ≠ m,
and: P ∧ Q,
exists: ∃x:A. B[x],
projective-plane: ProjectivePlane,
member: t ∈ T,
all: ∀x:A. B[x],
prop: ℙ,
cand: A c∧ B
Lemmas referenced :
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,
LP-sep-or,
pgeo-three-points-axiom,
pgeo-three-lines-axiom,
pgeo-plsep_wf,
use-triangle-axiom1,
pgeo-plsep-to-psep,
pgeo-plsep-implies-join,
pgeo-incident_wf,
pgeo-line_wf,
pgeo-join_wf
Rules used in proof :
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
isectElimination,
unionElimination,
independent_functionElimination,
because_Cache,
productElimination,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
dependent_pairFormation,
independent_pairFormation,
productEquality,
setEquality,
lambdaEquality
Latex:
\mforall{}g:ProjectivePlane. \mforall{}a:Point. \mexists{}l:Line. a \mneq{} l
Date html generated:
2018_05_22-PM-00_42_04
Last ObjectModification:
2017_11_27-PM-04_08_19
Theory : euclidean!plane!geometry
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