Nuprl Lemma : pgeo-meet-implies-plsep
∀g:BasicProjectivePlane. ∀a,b,c:Line. ∀s:a ≠ b. (a ∧ b ≠ c ⇒ (∀l:Point. ((l I a ∧ l I b) ⇒ l ≠ c)))
Proof
Definitions occuring in Statement :
basic-projective-plane: BasicProjectivePlane,
pgeo-meet: l ∧ m,
pgeo-lsep: l ≠ m,
pgeo-incident: a I b,
pgeo-plsep: a ≠ b,
pgeo-line: Line,
pgeo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
pgeo-lsep: l ≠ m,
pgeo-meet: l ∧ m,
mk-pgeo-prim: mk-pgeo-prim,
btrue: tt,
bfalse: ff,
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
top: Top,
mk-pgeo: mk-pgeo(p; ss; por; lor; j; m; p3; l3),
pgeo-dual-prim: pg*,
pgeo-point: Point,
pgeo-psep: a ≠ b,
pgeo-join: p ∨ q,
pgeo-line: Line,
pgeo-incident: a I b,
pgeo-plsep: a ≠ b,
pgeo-dual: pg*,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
basic-projective-plane_wf,
rec_select_update_lemma,
pgeo-dual_wf2,
pgeo-join-implies-plsep
Rules used in proof :
voidEquality,
voidElimination,
isect_memberEquality,
sqequalRule,
hypothesis,
hypothesisEquality,
isectElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:BasicProjectivePlane. \mforall{}a,b,c:Line. \mforall{}s:a \mneq{} b.
(a \mwedge{} b \mneq{} c {}\mRightarrow{} (\mforall{}l:Point. ((l I a \mwedge{} l I b) {}\mRightarrow{} l \mneq{} c)))
Date html generated:
2018_05_22-PM-00_36_49
Last ObjectModification:
2017_11_30-AM-10_27_32
Theory : euclidean!plane!geometry
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