Nuprl Lemma : pgeo-meet-implies-lsep
∀g:BasicProjectivePlane. ∀l,m,n:Line. ∀s:l ≠ m. (l ∧ m ≠ n ⇒ {l ≠ n ∧ m ≠ n})
Proof
Definitions occuring in Statement :
basic-projective-plane: BasicProjectivePlane,
pgeo-meet: l ∧ m,
pgeo-lsep: l ≠ m,
pgeo-plsep: a ≠ b,
pgeo-line: Line,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
prop: ℙ,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
basic-projective-plane: BasicProjectivePlane,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
false: False,
not: ¬A,
pgeo-incident: a I b,
or: P ∨ Q
Lemmas referenced :
pgeo-line_wf,
pgeo-lsep_wf,
pgeo-primitives_wf,
projective-plane-structure_wf,
basic-projective-plane_wf,
subtype_rel_transitivity,
basic-projective-plane-subtype,
projective-plane-structure_subtype,
pgeo-plsep_wf,
pgeo-incident_wf,
pgeo-point_wf,
pgeo-meet_wf,
PL-sep-or,
pgeo-meet-incidence
Rules used in proof :
independent_isectElimination,
instantiate,
independent_pairFormation,
independent_functionElimination,
productEquality,
sqequalRule,
isectElimination,
setEquality,
lambdaEquality,
applyEquality,
hypothesisEquality,
hypothesis,
rename,
setElimination,
because_Cache,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
voidElimination,
productElimination,
unionElimination
Latex:
\mforall{}g:BasicProjectivePlane. \mforall{}l,m,n:Line. \mforall{}s:l \mneq{} m. (l \mwedge{} m \mneq{} n {}\mRightarrow{} \{l \mneq{} n \mwedge{} m \mneq{} n\})
Date html generated:
2018_05_22-PM-00_36_59
Last ObjectModification:
2017_11_29-PM-03_37_01
Theory : euclidean!plane!geometry
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