Nuprl Lemma : separable-kernel-iff
∀rv:InnerProductSpace. ∀e:Point(rv). ∀f:{h:Point(rv)| h ⋅ e = r0} ⟶ ℝ ⟶ ℝ.
(trans-kernel-fun(rv;e;f)
⇒ (separable-kernel(rv;e;f)
⇐⇒ ∃phi:ℝ ⟶ ℝ
∃psi:{h:Point(rv)| h ⋅ e = r0} ⟶ {r:ℝ| r0 < r}
((∀h:{h:Point(rv)| h ⋅ e = r0} . ∀t:ℝ. ((f h t) = ((phi t) * (psi h))))
∧ ((phi r0) = r0)
∧ ((psi 0) = r1)
∧ (∀t,s:ℝ. ((t < s) ⇒ ((phi t) < (phi s))))
∧ (∀t:ℝ. ∃s:ℝ. ((phi s) = t))
∧ (∀h1,h2:{h:Point(rv)| h ⋅ e = r0} . (psi h1 ≠ psi h2 ⇒ h1 # h2))
∧ (∀t,s:ℝ. (phi t ≠ phi s ⇒ t ≠ s)))))
Proof
Definitions occuring in Statement :
separable-kernel: separable-kernel(rv;e;f),
trans-kernel-fun: trans-kernel-fun(rv;e;f),
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rv-0: 0,
rneq: x ≠ y,
rless: x < y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
separable-kernel: separable-kernel(rv;e;f),
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a
Lemmas referenced :
separable-kernel-properties,
separable-kernel_wf,
Error :ss-point_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
req_wf,
rv-ip_wf,
int-to-real_wf,
rmul_wf,
real_wf,
rless_wf,
rv-0ip,
rv-0_wf,
rneq_wf,
Error :ss-sep_wf,
trans-kernel-fun_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
independent_pairFormation,
productElimination,
universeIsType,
isectElimination,
dependent_pairFormation_alt,
functionExtensionality,
applyEquality,
because_Cache,
sqequalRule,
setEquality,
instantiate,
independent_isectElimination,
closedConclusion,
natural_numberEquality,
functionIsType,
productIsType,
setIsType,
inhabitedIsType,
lambdaEquality_alt,
setElimination,
rename,
dependent_set_memberEquality_alt
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}e:Point(rv). \mforall{}f:\{h:Point(rv)| h \mcdot{} e = r0\} {}\mrightarrow{} \mBbbR{} {}\mrightarrow{} \mBbbR{}.
(trans-kernel-fun(rv;e;f)
{}\mRightarrow{} (separable-kernel(rv;e;f)
\mLeftarrow{}{}\mRightarrow{} \mexists{}phi:\mBbbR{} {}\mrightarrow{} \mBbbR{}
\mexists{}psi:\{h:Point(rv)| h \mcdot{} e = r0\} {}\mrightarrow{} \{r:\mBbbR{}| r0 < r\}
((\mforall{}h:\{h:Point(rv)| h \mcdot{} e = r0\} . \mforall{}t:\mBbbR{}. ((f h t) = ((phi t) * (psi h))))
\mwedge{} ((phi r0) = r0)
\mwedge{} ((psi 0) = r1)
\mwedge{} (\mforall{}t,s:\mBbbR{}. ((t < s) {}\mRightarrow{} ((phi t) < (phi s))))
\mwedge{} (\mforall{}t:\mBbbR{}. \mexists{}s:\mBbbR{}. ((phi s) = t))
\mwedge{} (\mforall{}h1,h2:\{h:Point(rv)| h \mcdot{} e = r0\} . (psi h1 \mneq{} psi h2 {}\mRightarrow{} h1 \# h2))
\mwedge{} (\mforall{}t,s:\mBbbR{}. (phi t \mneq{} phi s {}\mRightarrow{} t \mneq{} s)))))
Date html generated:
2020_05_20-PM-01_17_12
Last ObjectModification:
2019_12_09-PM-11_01_49
Theory : inner!product!spaces
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