Nuprl Lemma : separable-kernel-properties
∀rv:InnerProductSpace. ∀e:Point(rv). ∀f:{h:Point(rv)| h ⋅ e = r0} ⟶ ℝ ⟶ ℝ.
(trans-kernel-fun(rv;e;f)
⇒ (∀g:ℝ ⟶ ℝ. ∀k:{h:Point(rv)| h ⋅ e = r0} ⟶ ℝ.
((∀h:{h:Point(rv)| h ⋅ e = r0} . ∀t:ℝ. ((f h t) = ((g t) * (k h))))
⇒ (∃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 :
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],
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],
implies: P ⇒ Q,
trans-kernel-fun: trans-kernel-fun(rv;e;f),
and: P ∧ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
or: P ∨ Q,
rneq: x ≠ y,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
rdiv: (x/y),
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
rat_term_to_real: rat_term_to_real(f;t),
rtermConstant: "const",
rat_term_ind: rat_term_ind,
pi1: fst(t),
rtermDivide: num "/" denom,
rtermVar: rtermVar(var),
pi2: snd(t),
rev_uimplies: rev_uimplies(P;Q),
ss-eq: Error :ss-eq
Lemmas referenced :
rmul_wf,
rv-0ip,
rv-0_wf,
inner-product-space_subtype,
req_wf,
rv-ip_wf,
int-to-real_wf,
real_wf,
rless_wf,
rneq_wf,
Error :ss-sep_wf,
trans-kernel-fun_wf,
Error :ss-point_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
rless-int,
rless-implies-rless,
rsub_wf,
rmul-is-positive,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
rless_functionality,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
assert-rat-term-eq2,
rtermDivide_wf,
rtermVar_wf,
rtermConstant_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
rdiv_wf,
rinv_wf2,
req_functionality,
req_weakening,
req_transitivity,
rmul_functionality,
rmul-rinv3,
req_inversion,
iff_weakening_uiff,
rmul_preserves_rless,
rless_transitivity2,
rleq_weakening_rless,
rless_irreflexivity,
rmul_reverses_rless_iff,
rneq-rmul,
rmul_preserves_rneq_iff2,
rneq_functionality,
rneq_irreflexivity,
Error :ss-eq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation_alt,
lambdaEquality_alt,
cut,
introduction,
extract_by_obid,
isectElimination,
applyEquality,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality_alt,
sqequalRule,
universeIsType,
natural_numberEquality,
productIsType,
functionIsType,
because_Cache,
setIsType,
setElimination,
rename,
inhabitedIsType,
instantiate,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
unionElimination,
inrFormation_alt,
inlFormation_alt,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
equalityElimination,
int_eqReduceTrueSq,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
promote_hyp,
cumulativity,
int_eqReduceFalseSq,
closedConclusion
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{} (\mforall{}g:\mBbbR{} {}\mrightarrow{} \mBbbR{}. \mforall{}k:\{h:Point(rv)| h \mcdot{} e = r0\} {}\mrightarrow{} \mBbbR{}.
((\mforall{}h:\{h:Point(rv)| h \mcdot{} e = r0\} . \mforall{}t:\mBbbR{}. ((f h t) = ((g t) * (k h))))
{}\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_07
Last ObjectModification:
2019_12_09-PM-11_13_20
Theory : inner!product!spaces
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