Nuprl Lemma : pgeo-triangle-axiom2-dual
∀g:ProjectivePlane. ∀p,q:Line. ∀l,m:Point. ∀s:p ≠ q. ∀s1:l ≠ m. (l ≠ p ⇒ m ≠ q ⇒ m I p ⇒ l I q ⇒ p ∧ q ≠ l ∨ m)
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 :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
pgeo-line_wf,
pgeo-point_wf,
pgeo-lsep_wf,
pgeo-psep_wf,
pgeo-plsep_wf,
pgeo-primitives_wf,
projective-plane-structure_wf,
projective-plane-structure-complete_wf,
basic-projective-plane_wf,
projective-plane_wf,
subtype_rel_transitivity,
projective-plane-subtype,
basic-projective-plane-subtype,
projective-plane-structure-complete_subtype,
projective-plane-structure_subtype,
pgeo-incident_wf,
use-triangle-axiom2
Rules used in proof :
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
isectElimination,
hypothesis,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:ProjectivePlane. \mforall{}p,q:Line. \mforall{}l,m:Point. \mforall{}s:p \mneq{} q. \mforall{}s1:l \mneq{} m.
(l \mneq{} p {}\mRightarrow{} m \mneq{} q {}\mRightarrow{} m I p {}\mRightarrow{} l I q {}\mRightarrow{} p \mwedge{} q \mneq{} l \mvee{} m)
Date html generated:
2018_05_22-PM-00_47_29
Last ObjectModification:
2017_11_27-PM-04_50_34
Theory : euclidean!plane!geometry
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