Nuprl Lemma : use-triangle-axiom2
∀g:ProjectivePlane. ∀p,q:Point. ∀l,m:Line. ∀s:p ≠ q. ∀s1:l ≠ m. (p ≠ l ⇒ q ≠ m ⇒ p I m ⇒ q I l ⇒ l ∧ m ≠ p ∨ q)
Proof
Definitions occuring in Statement :
projective-plane: ProjectivePlane,
pgeo-meet: l ∧ m,
pgeo-join: p ∨ q,
pgeo-lsep: l ≠ m,
pgeo-psep: a ≠ b,
pgeo-incident: a I b,
pgeo-plsep: a ≠ b,
pgeo-line: Line,
pgeo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
and: P ∧ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a
Lemmas referenced :
pgeo-incident_wf,
projective-plane-structure_subtype,
basic-projective-plane-subtype,
projective-plane-subtype,
subtype_rel_transitivity,
projective-plane_wf,
basic-projective-plane_wf,
projective-plane-structure_wf,
pgeo-primitives_wf,
pgeo-plsep_wf,
pgeo-lsep_wf,
pgeo-psep_wf,
pgeo-line_wf,
pgeo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
isectElimination,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
independent_functionElimination
Latex:
\mforall{}g:ProjectivePlane. \mforall{}p,q:Point. \mforall{}l,m:Line. \mforall{}s:p \mneq{} q. \mforall{}s1:l \mneq{} m.
(p \mneq{} l {}\mRightarrow{} q \mneq{} m {}\mRightarrow{} p I m {}\mRightarrow{} q I l {}\mRightarrow{} l \mwedge{} m \mneq{} p \mvee{} q)
Date html generated:
2018_05_22-PM-00_41_43
Last ObjectModification:
2017_11_10-PM-03_40_08
Theory : euclidean!plane!geometry
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