Nuprl Lemma : line-implies-plsep-exists
∀g:ProjectivePlane. ∀l:Line. ∃p:Point. p ≠ 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 :
and: P ∧ Q,
exists: ∃x:A. B[x],
uimplies: b supposing a,
guard: {T},
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
cand: A c∧ B,
or: P ∨ Q,
prop: ℙ
Lemmas referenced :
pgeo-primitives_wf,
projective-plane-structure_subtype,
pgeo-line_wf,
point-implies-plsep-exists,
projective-plane-structure_wf,
projective-plane-structure-complete_wf,
projective-plane_wf,
subtype_rel_transitivity,
projective-plane-subtype,
projective-plane-structure-complete_subtype,
pgeo-three-points-axiom,
pgeo-plsep-to-lsep,
Meet,
use-triangle-axiom1,
pgeo-plsep-to-psep,
projective-plane-subtype-basic,
pgeo-plsep-implies-join,
pgeo-incident_wf,
pgeo-join_wf,
LP-sep-or2,
pgeo-plsep_wf,
pgeo-join-implies-plsep,
incident-join-second
Rules used in proof :
productElimination,
sqequalRule,
independent_isectElimination,
isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
rename,
because_Cache,
independent_pairFormation,
unionElimination,
productEquality,
setEquality,
setElimination,
lambdaEquality,
dependent_pairFormation
Latex:
\mforall{}g:ProjectivePlane. \mforall{}l:Line. \mexists{}p:Point. p \mneq{} l
Date html generated:
2018_05_22-PM-00_42_16
Last ObjectModification:
2017_11_28-PM-05_42_12
Theory : euclidean!plane!geometry
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