Nuprl Lemma : basic-projective-plane-axioms
∀g:BasicProjectivePlane. ∀m,l:Line. ∀p,q:Point. (p I l ⇒ q I l ⇒ p I m ⇒ q I m ⇒ (¬((¬p ≡ q) ∧ (¬l ≡ m))))
Proof
Definitions occuring in Statement :
basic-projective-plane: BasicProjectivePlane,
pgeo-leq: a ≡ b,
pgeo-peq: a ≡ b,
pgeo-incident: a I b,
pgeo-line: Line,
pgeo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
basic-projective-plane: BasicProjectivePlane,
member: t ∈ T,
prop: ℙ,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
basic-pgeo-axioms: BasicProjectiveGeometryAxioms(g)
Lemmas referenced :
not_wf,
pgeo-peq_wf,
projective-plane-structure_subtype,
pgeo-leq_wf,
basic-projective-plane-subtype,
subtype_rel_transitivity,
basic-projective-plane_wf,
projective-plane-structure_wf,
pgeo-primitives_wf,
pgeo-incident_wf,
pgeo-point_wf,
pgeo-line_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
sqequalHypSubstitution,
setElimination,
rename,
hypothesis,
independent_functionElimination,
voidElimination,
productEquality,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
because_Cache,
instantiate,
independent_isectElimination,
dependent_functionElimination
Latex:
\mforall{}g:BasicProjectivePlane. \mforall{}m,l:Line. \mforall{}p,q:Point.
(p I l {}\mRightarrow{} q I l {}\mRightarrow{} p I m {}\mRightarrow{} q I m {}\mRightarrow{} (\mneg{}((\mneg{}p \mequiv{} q) \mwedge{} (\mneg{}l \mequiv{} m))))
Date html generated:
2019_10_16-PM-02_12_05
Last ObjectModification:
2018_08_02-PM-01_17_07
Theory : euclidean!plane!geometry
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