Nuprl Lemma : trans-kernel-increasing
∀rv:InnerProductSpace. ∀e:Point. ∀T:ℝ ⟶ Point ⟶ Point.
((e^2 = r1) ⇒ translation-group-fun(rv;e;T) ⇒ (∀h:{h:Point| h ⋅ e = r0} . ∀t,s:ℝ. ((t < s) ⇒ (ρ(h;t) < ρ(h;s)))))
Proof
Definitions occuring in Statement :
trans-kernel: ρ(h;t),
translation-group-fun: translation-group-fun(rv;e;T),
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rless: x < y,
req: x = y,
int-to-real: r(n),
real: ℝ,
ss-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
trans-kernel-fun: trans-kernel-fun(rv;e;f),
and: P ∧ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
trans-kernel-is-kernel-fun,
rless_wf,
real_wf,
set_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,
translation-group-fun_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
sqequalRule,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
lambdaEquality,
natural_numberEquality,
because_Cache,
functionExtensionality,
functionEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}e:Point. \mforall{}T:\mBbbR{} {}\mrightarrow{} Point {}\mrightarrow{} Point.
((e\^{}2 = r1)
{}\mRightarrow{} translation-group-fun(rv;e;T)
{}\mRightarrow{} (\mforall{}h:\{h:Point| h \mcdot{} e = r0\} . \mforall{}t,s:\mBbbR{}. ((t < s) {}\mRightarrow{} (\mrho{}(h;t) < \mrho{}(h;s)))))
Date html generated:
2017_10_05-AM-00_23_19
Last ObjectModification:
2017_07_02-PM-03_49_55
Theory : inner!product!spaces
Home
Index