Nuprl Lemma : geo-convex-intersection
∀[g:BasicGeometry]. ∀[T:Type]. ∀P:T ⟶ Point ⟶ ℙ. ((∀t:T. IsConvex(x.P[t;x])) ⇒ IsConvex(x.∀t:T. P[t;x]))
Proof
Definitions occuring in Statement :
geo-convex: IsConvex(x.P[x]),
basic-geometry: BasicGeometry,
geo-point: Point,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
so_apply: x[s],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
geo-convex: IsConvex(x.P[x])
Lemmas referenced :
geo-point_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-between_wf,
all_wf
Rules used in proof :
independent_functionElimination,
dependent_functionElimination,
universeEquality,
functionEquality,
because_Cache,
independent_isectElimination,
instantiate,
hypothesis,
functionExtensionality,
applyEquality,
lambdaEquality,
cumulativity,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
hypothesisEquality,
lambdaFormation,
isect_memberFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}[g:BasicGeometry]. \mforall{}[T:Type].
\mforall{}P:T {}\mrightarrow{} Point {}\mrightarrow{} \mBbbP{}. ((\mforall{}t:T. IsConvex(x.P[t;x])) {}\mRightarrow{} IsConvex(x.\mforall{}t:T. P[t;x]))
Date html generated:
2017_10_02-PM-06_49_00
Last ObjectModification:
2017_08_06-PM-07_28_41
Theory : euclidean!plane!geometry
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