Nuprl Lemma : sq_stable__geo-left-axioms-1
∀g:EuclideanPlaneStructure
(BasicGeometryAxioms(g) ⇒ (∀a,b,c:Point. (a # bc ⇒ (¬Colinear(a;b;c)))) ⇒ SqStable(geo-left-axioms(g)))
Proof
Definitions occuring in Statement :
geo-left-axioms: geo-left-axioms(g),
euclidean-plane-structure: EuclideanPlaneStructure,
geo-lsep: a # bc,
basic-geo-axioms: BasicGeometryAxioms(g),
geo-colinear: Colinear(a;b;c),
geo-point: Point,
sq_stable: SqStable(P),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x],
and: P ∧ Q,
geo-left-axioms: geo-left-axioms(g)
Lemmas referenced :
euclidean-plane-structure_wf,
basic-geo-axioms_wf,
geo-colinear_wf,
not_wf,
geo-lsep_wf,
euclidean-plane-structure-subtype,
geo-point_wf,
all_wf,
sq_stable__geo-left-1,
sq_stable__geo-lsep,
sq_stable__geo-sep,
sq_stable__colinear,
sq_stable__not,
sq_stable__iff,
sq_stable__all,
geo-congruent_wf,
geo-between_wf,
geo-sep_wf,
geo-left_wf,
iff_wf,
sq_stable__and
Rules used in proof :
functionEquality,
because_Cache,
lambdaEquality,
sqequalRule,
applyEquality,
isectElimination,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut,
productEquality,
isect_memberEquality
Latex:
\mforall{}g:EuclideanPlaneStructure
(BasicGeometryAxioms(g)
{}\mRightarrow{} (\mforall{}a,b,c:Point. (a \# bc {}\mRightarrow{} (\mneg{}Colinear(a;b;c))))
{}\mRightarrow{} SqStable(geo-left-axioms(g)))
Date html generated:
2017_10_02-PM-03_27_30
Last ObjectModification:
2017_08_07-PM-04_29_49
Theory : euclidean!plane!geometry
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