Nuprl Lemma : kernel-fun-properties
∀rv:InnerProductSpace. ∀e:Point. ∀f:{h:Point| h ⋅ e = r0} ⟶ ℝ ⟶ ℝ.
((e^2 = r1)
⇒ trans-kernel-fun(rv;e;f)
⇒ ((∀h1,h2:{h:Point| h ⋅ e = r0} . ∀t1,t2:ℝ. (h1 ≡ h2 ⇒ (t1 = t2) ⇒ ((f h1 t1) = (f h2 t2))))
∧ (∀g:h:{h:Point| h ⋅ e = r0} ⟶ r:ℝ ⟶ ℝ
((∀h:{h:Point| h ⋅ e = r0} . ∀r:ℝ. ((f h (g h r)) = r))
⇒ ((∀h1,h2:{h:Point| h ⋅ e = r0} . ∀t1,t2:ℝ. (g h1 t1 ≠ g h2 t2 ⇒ (h1 # h2 ∨ t1 ≠ t2)))
∧ (∀h1,h2:{h:Point| h ⋅ e = r0} . ∀t1,t2:ℝ. (h1 ≡ h2 ⇒ (t1 = t2) ⇒ ((g h1 t1) = (g h2 t2)))))))
∧ (∃g:h:{h:Point| h ⋅ e = r0} ⟶ r:ℝ ⟶ ℝ. ∀h:{h:Point| h ⋅ e = r0} . ∀r:ℝ. ((f h (g h r)) = r))))
Proof
Definitions occuring in Statement :
trans-kernel-fun: trans-kernel-fun(rv;e;f),
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rneq: x ≠ y,
req: x = y,
int-to-real: r(n),
real: ℝ,
ss-eq: x ≡ y,
ss-sep: x # y,
ss-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: 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,
cand: A c∧ B,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
uimplies: b supposing a,
not: ¬A,
or: P ∨ Q,
false: False,
ss-eq: x ≡ y,
iff: P ⇐⇒ Q,
rneq: x ≠ y,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
pi1: fst(t)
Lemmas referenced :
req_wf,
ss-eq_wf,
real_wf,
set_wf,
ss-point_wf,
rv-ip_wf,
int-to-real_wf,
rneq_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
all_wf,
trans-kernel-fun_wf,
not-rneq,
rneq_irreflexivity,
rneq_functionality,
req_weakening,
rless-cases,
rless_wf,
ss-sep_wf,
rless_functionality,
exists_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
because_Cache,
sqequalRule,
setElimination,
rename,
lambdaEquality,
natural_numberEquality,
independent_pairFormation,
functionExtensionality,
setEquality,
instantiate,
independent_isectElimination,
dependent_set_memberEquality,
functionEquality,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
voidElimination,
inlFormation,
inrFormation,
promote_hyp,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}e:Point. \mforall{}f:\{h:Point| h \mcdot{} e = r0\} {}\mrightarrow{} \mBbbR{} {}\mrightarrow{} \mBbbR{}.
((e\^{}2 = r1)
{}\mRightarrow{} trans-kernel-fun(rv;e;f)
{}\mRightarrow{} ((\mforall{}h1,h2:\{h:Point| h \mcdot{} e = r0\} . \mforall{}t1,t2:\mBbbR{}. (h1 \mequiv{} h2 {}\mRightarrow{} (t1 = t2) {}\mRightarrow{} ((f h1 t1) = (f h2 t2))))
\mwedge{} (\mforall{}g:h:\{h:Point| h \mcdot{} e = r0\} {}\mrightarrow{} r:\mBbbR{} {}\mrightarrow{} \mBbbR{}
((\mforall{}h:\{h:Point| h \mcdot{} e = r0\} . \mforall{}r:\mBbbR{}. ((f h (g h r)) = r))
{}\mRightarrow{} ((\mforall{}h1,h2:\{h:Point| h \mcdot{} e = r0\} . \mforall{}t1,t2:\mBbbR{}. (g h1 t1 \mneq{} g h2 t2 {}\mRightarrow{} (h1 \# h2 \mvee{} t1 \mneq{} t2)))
\mwedge{} (\mforall{}h1,h2:\{h:Point| h \mcdot{} e = r0\} . \mforall{}t1,t2:\mBbbR{}.
(h1 \mequiv{} h2 {}\mRightarrow{} (t1 = t2) {}\mRightarrow{} ((g h1 t1) = (g h2 t2)))))))
\mwedge{} (\mexists{}g:h:\{h:Point| h \mcdot{} e = r0\} {}\mrightarrow{} r:\mBbbR{} {}\mrightarrow{} \mBbbR{}
\mforall{}h:\{h:Point| h \mcdot{} e = r0\} . \mforall{}r:\mBbbR{}. ((f h (g h r)) = r))))
Date html generated:
2017_10_05-AM-00_23_33
Last ObjectModification:
2017_06_27-PM-00_10_56
Theory : inner!product!spaces
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