Nuprl Lemma : basic-pgeo-axioms-imply
∀g:ProjectivePlaneStructure. (BasicProjectiveGeometryAxioms(g) ⇒ ((∀a:Point. a ≡ a) ∧ (∀l:Line. l ≡ l)))
Proof
Definitions occuring in Statement :
projective-plane-structure: ProjectivePlaneStructure,
basic-pgeo-axioms: BasicProjectiveGeometryAxioms(g),
pgeo-leq: a ≡ b,
pgeo-peq: a ≡ b,
pgeo-line: Line,
pgeo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
pgeo-peq: a ≡ b,
not: ¬A,
stable: Stable{P},
uimplies: b supposing a,
or: P ∨ Q,
false: False,
pgeo-psep: a ≠ b,
exists: ∃x:A. B[x],
basic-pgeo-axioms: BasicProjectiveGeometryAxioms(g),
pgeo-incident: a I b,
pgeo-leq: a ≡ b,
pgeo-lsep: l ≠ m
Lemmas referenced :
pgeo-point_wf,
projective-plane-structure_subtype,
pgeo-line_wf,
basic-pgeo-axioms_wf,
projective-plane-structure_wf,
stable__not,
pgeo-psep_wf,
false_wf,
or_wf,
pgeo-peq_wf,
not_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
pgeo-leq_wf,
pgeo-lsep_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
independent_pairFormation,
because_Cache,
functionEquality,
independent_isectElimination,
independent_functionElimination,
unionElimination,
voidElimination,
productElimination,
dependent_functionElimination
Latex:
\mforall{}g:ProjectivePlaneStructure
(BasicProjectiveGeometryAxioms(g) {}\mRightarrow{} ((\mforall{}a:Point. a \mequiv{} a) \mwedge{} (\mforall{}l:Line. l \mequiv{} l)))
Date html generated:
2019_10_16-PM-02_11_49
Last ObjectModification:
2018_08_23-PM-01_56_06
Theory : euclidean!plane!geometry
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