Nuprl Lemma : kernel-fun-is-trans-kernel
∀rv:InnerProductSpace. ∀e:{e:Point| e^2 = r1} . ∀f:{h:Point| h ⋅ e = r0} ⟶ ℝ ⟶ ℝ.
(trans-kernel-fun(rv;e;f)
⇒ (∃T:ℝ ⟶ Point ⟶ Point. (translation-group-fun(rv;e;T) ∧ (∀h:{h:Point| h ⋅ e = r0} . ∀t:ℝ. (ρ(h;t) = (f h t))))))
Proof
Definitions occuring in Statement :
trans-kernel-fun: trans-kernel-fun(rv;e;f),
trans-kernel: ρ(h;t),
translation-group-fun: translation-group-fun(rv;e;T),
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
req: x = y,
int-to-real: r(n),
real: ℝ,
ss-point: Point,
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],
member: t ∈ T,
implies: P ⇒ Q,
trans-kernel-fun: trans-kernel-fun(rv;e;f),
and: P ∧ Q,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
pi1: fst(t)
Lemmas referenced :
trans-from-kernel-is-trans,
kernel-trans-from-kernel,
trans-from-kernel_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,
real_wf,
translation-group-fun_wf,
all_wf,
trans-kernel_wf,
trans-kernel-fun_wf,
set_wf,
exists_wf,
equal_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
addLevel,
productElimination,
promote_hyp,
independent_functionElimination,
because_Cache,
dependent_pairFormation,
lambdaEquality,
isectElimination,
setElimination,
rename,
dependent_set_memberEquality,
functionExtensionality,
applyEquality,
setEquality,
instantiate,
independent_isectElimination,
sqequalRule,
natural_numberEquality,
independent_pairFormation,
productEquality,
levelHypothesis,
functionEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}e:\{e:Point| e\^{}2 = r1\} . \mforall{}f:\{h:Point| h \mcdot{} e = r0\} {}\mrightarrow{} \mBbbR{} {}\mrightarrow{} \mBbbR{}.
(trans-kernel-fun(rv;e;f)
{}\mRightarrow{} (\mexists{}T:\mBbbR{} {}\mrightarrow{} Point {}\mrightarrow{} Point
(translation-group-fun(rv;e;T) \mwedge{} (\mforall{}h:\{h:Point| h \mcdot{} e = r0\} . \mforall{}t:\mBbbR{}. (\mrho{}(h;t) = (f h t))))))
Date html generated:
2017_10_05-AM-00_25_51
Last ObjectModification:
2017_06_30-PM-02_27_26
Theory : inner!product!spaces
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