Nuprl Lemma : translation-group-point
∀rv:InnerProductSpace. ∀e:Point. ∀T:ℝ ⟶ Point ⟶ Point.
(translation-group-fun(rv;e;T)
⇒ Point ≡ {fg:Point ⟶ Point × (Point ⟶ Point)|
∃t:ℝ. ((∀x:Point. (fst(fg)) x ≡ T t x) ∧ (∀x:Point. (snd(fg)) x ≡ T -(t) x))} )
Proof
Definitions occuring in Statement :
translation-group: translation-group(rv;e;T),
translation-group-fun: translation-group-fun(rv;e;T),
inner-product-space: InnerProductSpace,
rminus: -(x),
real: ℝ,
ss-eq: x ≡ y,
ss-point: Point,
ext-eq: A ≡ B,
pi1: fst(t),
pi2: snd(t),
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],
product: x:A × B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
ss-point: Point,
translation-group: translation-group(rv;e;T),
mk-s-subgroup: mk-s-subgroup(sg;x.P[x]),
mk-s-group: mk-s-group(ss; e; i; o; sep; invsep),
member: t ∈ T,
top: Top,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
bfalse: ff,
set-ss: set-ss(ss;x.P[x]),
mk-ss: mk-ss,
btrue: tt,
uall: ∀[x:A]. B[x],
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
pi1: fst(t),
pi2: snd(t),
exists: ∃x:A. B[x],
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
uimplies: b supposing a,
trans-apply: T_t(x),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
translation-group-fun: translation-group-fun(rv;e;T),
or: P ∨ Q,
false: False
Lemmas referenced :
rec_select_update_lemma,
rv-perm-point,
all_wf,
ss-eq_wf,
real_wf,
rminus_wf,
exists_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,
ss-sep_wf,
translation-group-fun_wf,
trans-apply_wf,
ss-eq_weakening,
ss-eq_functionality,
trans-apply_functionality,
req_weakening,
radd_wf,
int-to-real_wf,
radd-rminus-both,
ss-eq_inversion,
trans-apply-add,
trans-apply-0,
radd-rminus,
ss-eq_transitivity,
ss-sep_functionality,
rneq_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
independent_pairFormation,
lambdaEquality,
setElimination,
rename,
productElimination,
dependent_set_memberEquality,
independent_pairEquality,
hypothesisEquality,
dependent_pairFormation,
because_Cache,
productEquality,
applyEquality,
functionExtensionality,
setEquality,
functionEquality,
instantiate,
independent_isectElimination,
independent_functionElimination,
natural_numberEquality,
addLevel,
allFunctionality,
impliesFunctionality,
levelHypothesis,
andLevelFunctionality,
allLevelFunctionality,
unionElimination
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}e:Point. \mforall{}T:\mBbbR{} {}\mrightarrow{} Point {}\mrightarrow{} Point.
(translation-group-fun(rv;e;T)
{}\mRightarrow{} Point \mequiv{} \{fg:Point {}\mrightarrow{} Point \mtimes{} (Point {}\mrightarrow{} Point)|
\mexists{}t:\mBbbR{}. ((\mforall{}x:Point. (fst(fg)) x \mequiv{} T t x) \mwedge{} (\mforall{}x:Point. (snd(fg)) x \mequiv{} T -(t) x))\} )
Date html generated:
2017_10_05-AM-00_22_19
Last ObjectModification:
2017_06_26-AM-10_19_53
Theory : inner!product!spaces
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