Nuprl Lemma : pgeo-plsep_functionality
∀g:ProjectivePlane. ∀a,a1:Point. ∀l,l1:Line. (a ≡ a1 ⇒ l ≡ l1 ⇒ {a ≠ l ⇐⇒ a1 ≠ l1})
Proof
Definitions occuring in Statement :
projective-plane: ProjectivePlane,
pgeo-leq: a ≡ b,
pgeo-peq: a ≡ b,
pgeo-plsep: a ≠ b,
pgeo-line: Line,
pgeo-point: Point,
guard: {T},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
guard: {T}
Lemmas referenced :
pgeo-point_wf,
pgeo-line_wf,
pgeo-peq_wf,
pgeo-leq_wf,
pgeo-primitives_wf,
projective-plane-structure_wf,
projective-plane-structure-complete_wf,
projective-plane_wf,
subtype_rel_transitivity,
projective-plane-subtype,
projective-plane-structure-complete_subtype,
projective-plane-structure_subtype,
pgeo-plsep_wf,
pgeo-peq-preserves-plsep,
pgeo-leq-preserves-plsep,
pgeo-leq_inversion,
pgeo-peq-sym
Rules used in proof :
because_Cache,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution,
independent_functionElimination,
dependent_functionElimination,
rename
Latex:
\mforall{}g:ProjectivePlane. \mforall{}a,a1:Point. \mforall{}l,l1:Line. (a \mequiv{} a1 {}\mRightarrow{} l \mequiv{} l1 {}\mRightarrow{} \{a \mneq{} l \mLeftarrow{}{}\mRightarrow{} a1 \mneq{} l1\})
Date html generated:
2018_05_22-PM-00_45_39
Last ObjectModification:
2017_12_01-PM-05_33_39
Theory : euclidean!plane!geometry
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