Nuprl Lemma : pgeo-join-implies-plsep
∀g:BasicProjectivePlane. ∀a,b,c:Point. ∀s:a ≠ b. (c ≠ a ∨ b ⇒ (∀l:Line. ((a I l ∧ b I l) ⇒ c ≠ l)))
Proof
Definitions occuring in Statement :
basic-projective-plane: BasicProjectivePlane,
pgeo-join: p ∨ q,
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,
and: P ∧ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
or: P ∨ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
basic-projective-plane: BasicProjectivePlane,
member: t ∈ T,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
false: False,
not: ¬A,
pgeo-leq: a ≡ b
Lemmas referenced :
pgeo-point_wf,
pgeo-psep_wf,
pgeo-plsep_wf,
pgeo-primitives_wf,
projective-plane-structure_wf,
basic-projective-plane_wf,
subtype_rel_transitivity,
basic-projective-plane-subtype,
projective-plane-structure_subtype,
pgeo-incident_wf,
pgeo-line_wf,
pgeo-join_wf,
PL-sep-or,
pgeo-join-to-line-2
Rules used in proof :
independent_isectElimination,
instantiate,
unionElimination,
independent_functionElimination,
productEquality,
sqequalRule,
isectElimination,
setEquality,
lambdaEquality,
applyEquality,
hypothesis,
rename,
setElimination,
hypothesisEquality,
because_Cache,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
voidElimination
Latex:
\mforall{}g:BasicProjectivePlane. \mforall{}a,b,c:Point. \mforall{}s:a \mneq{} b.
(c \mneq{} a \mvee{} b {}\mRightarrow{} (\mforall{}l:Line. ((a I l \mwedge{} b I l) {}\mRightarrow{} c \mneq{} l)))
Date html generated:
2018_05_22-PM-00_36_39
Last ObjectModification:
2017_11_15-PM-00_53_58
Theory : euclidean!plane!geometry
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