Nuprl Lemma : pgeo-plsep-implies-join
∀g:BasicProjectivePlane. ∀c:Point. ∀l:Line. (c ≠ l ⇒ (∀a,b:Point. ∀s:a ≠ b. ((a I l ∧ b I l) ⇒ c ≠ a ∨ b)))
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},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
or: P ∨ Q,
basic-projective-plane: BasicProjectivePlane,
false: False,
not: ¬A,
pgeo-leq: a ≡ b
Lemmas referenced :
pgeo-line_wf,
pgeo-plsep_wf,
pgeo-point_wf,
pgeo-psep_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-join_wf,
PL-sep-or,
pgeo-join-to-line
Rules used in proof :
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
cut,
productEquality,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
unionElimination,
independent_functionElimination,
setEquality,
lambdaEquality,
rename,
setElimination,
dependent_functionElimination,
voidElimination
Latex:
\mforall{}g:BasicProjectivePlane. \mforall{}c:Point. \mforall{}l:Line.
(c \mneq{} l {}\mRightarrow{} (\mforall{}a,b:Point. \mforall{}s:a \mneq{} b. ((a I l \mwedge{} b I l) {}\mRightarrow{} c \mneq{} a \mvee{} b)))
Date html generated:
2018_05_22-PM-00_36_08
Last ObjectModification:
2017_11_15-PM-00_54_43
Theory : euclidean!plane!geometry
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