Nuprl Lemma : separable-kernel_wf
∀[rv:InnerProductSpace]. ∀[e:Point]. ∀[f:{h:Point| h ⋅ e = r0} ⟶ ℝ ⟶ ℝ]. (separable-kernel(rv;e;f) ∈ ℙ)
Proof
Definitions occuring in Statement :
separable-kernel: separable-kernel(rv;e;f),
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
req: x = y,
int-to-real: r(n),
real: ℝ,
ss-point: Point,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
separable-kernel: separable-kernel(rv;e;f),
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
all: ∀x:A. B[x],
so_apply: x[s]
Lemmas referenced :
exists_wf,
real_wf,
ss-point_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
req_wf,
rv-ip_wf,
int-to-real_wf,
all_wf,
rmul_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
functionEquality,
hypothesis,
because_Cache,
lambdaEquality,
setEquality,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
natural_numberEquality,
lambdaFormation,
setElimination,
rename,
functionExtensionality,
dependent_set_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[e:Point]. \mforall{}[f:\{h:Point| h \mcdot{} e = r0\} {}\mrightarrow{} \mBbbR{} {}\mrightarrow{} \mBbbR{}].
(separable-kernel(rv;e;f) \mmember{} \mBbbP{})
Date html generated:
2017_10_05-AM-00_23_41
Last ObjectModification:
2017_07_01-PM-09_16_16
Theory : inner!product!spaces
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