Nuprl Lemma : pgeo-peq-preserves-incidence
∀g:ProjectivePlane. ∀a,b:Point. ∀l:Line. (a I l ⇒ a ≡ b ⇒ b I l)
Proof
Definitions occuring in Statement :
projective-plane: ProjectivePlane,
pgeo-peq: a ≡ b,
pgeo-incident: a I b,
pgeo-line: Line,
pgeo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
false: False,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
cand: A c∧ B,
and: P ∧ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
pgeo-psep: a ≠ b,
pgeo-peq: a ≡ b,
not: ¬A,
pgeo-incident: a I b,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
pgeo-point_wf,
pgeo-line_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-peq_wf,
pgeo-plsep_wf,
pgeo-incident_wf
Rules used in proof :
independent_isectElimination,
instantiate,
voidElimination,
sqequalRule,
because_Cache,
applyEquality,
isectElimination,
extract_by_obid,
introduction,
productEquality,
independent_pairFormation,
hypothesis,
cut,
hypothesisEquality,
dependent_pairFormation,
thin,
independent_functionElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:ProjectivePlane. \mforall{}a,b:Point. \mforall{}l:Line. (a I l {}\mRightarrow{} a \mequiv{} b {}\mRightarrow{} b I l)
Date html generated:
2018_05_22-PM-00_43_55
Last ObjectModification:
2017_11_17-PM-00_07_50
Theory : euclidean!plane!geometry
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