Nuprl Lemma : separable-translation-group_iff
∀rv:InnerProductSpace. ∀e:{e:Point(rv)| e^2 = r1} . ∀T:ℝ ⟶ Point(rv) ⟶ Point(rv).
(translation-group-fun(rv;e;T)
⇒ (separable-translation-group(rv;e;T)
⇐⇒ (∀a:ℝ. ∀h:{h:Point(rv)| h ⋅ e = r0} . (a ≠ r0 ⇒ (r0 < ||T_a(h) - h||)))
∧ (∀a,b:ℝ. ∀h:{h:Point(rv)| h ⋅ e = r0} .
(a ≠ r0 ⇒ b ≠ r0 ⇒ ((||T_a(h) - h||/||T_b(h) - h||) = (||T_a(0)||/||T_b(0)||))))))
Proof
Definitions occuring in Statement :
separable-translation-group: separable-translation-group(rv;e;T),
trans-apply: T_t(x),
translation-group-fun: translation-group-fun(rv;e;T),
rv-norm: ||x||,
rv-sub: x - y,
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rv-0: 0,
rdiv: (x/y),
rneq: x ≠ y,
rless: x < y,
req: x = y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: 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,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
separable-translation-group: separable-translation-group(rv;e;T),
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rv-sub: x - y,
rv-minus: -x,
req_int_terms: t1 ≡ t2,
top: Top,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
false: False,
cand: A c∧ B,
rdiv: (x/y),
separable-kernel: separable-kernel(rv;e;f),
less_than: a < b,
true: True,
stable: Stable{P}
Lemmas referenced :
separable-translation-group_wf,
req_wf,
rv-ip_wf,
int-to-real_wf,
rneq_wf,
rless_wf,
rv-norm_wf,
rv-sub_wf,
inner-product-space_subtype,
trans-apply_wf,
rdiv_wf,
translation-group-fun_wf,
real_wf,
Error :ss-point_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
separable-kernel-iff,
trans-kernel_wf,
trans-kernel-is-kernel-fun,
sq_stable__req,
rv-add_wf,
rv-mul_wf,
rmul_wf,
rabs_wf,
Error :ss-eq_wf,
radd_wf,
itermSubtract_wf,
itermAdd_wf,
itermConstant_wf,
rv-0_wf,
req_functionality,
rv-norm_functionality,
rv-sub_functionality,
trans-kernel-equation,
Error :ss-eq_weakening,
req_weakening,
uiff_transitivity,
Error :ss-eq_functionality,
rv-add_functionality,
rv-add-comm,
rv-mul-1-add-alt,
rv-mul_functionality,
rv-mul0,
rv-0-add,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_add_lemma,
real_term_value_const_lemma,
Error :ss-eq_transitivity,
req_transitivity,
rv-norm-mul,
rmul_functionality,
rabs-rmul,
rnexp_wf,
istype-le,
rleq-int,
istype-false,
itermMultiply_wf,
itermVar_wf,
rabs-of-nonneg,
rleq_weakening_rless,
rv-norm-eq-iff,
rnexp-one,
real_term_value_mul_lemma,
real_term_value_var_lemma,
rmul-is-positive,
rless_functionality,
rabs-positive-iff,
rneq_functionality,
req_inversion,
sq_stable__rless,
rv-0ip,
rv-mul-0,
rmul_preserves_req,
rinv_wf2,
rmul-rinv,
rless-int,
stable_req,
false_wf,
not_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
rdiv_functionality,
Error :ss-eq_inversion,
rmul-rinv3,
rinv-mul-as-rdiv,
rleq_wf,
stable__rleq,
not-rless,
trans-kernel-increasing,
rless_transitivity1,
rless_irreflexivity,
trans-kernel-0,
rleq_antisymmetry,
rleq_weakening_equal,
rleq_functionality,
trans-kernel_functionality,
rminus_wf,
rabs-of-nonpos,
req-implies-req,
rsub_wf,
itermMinus_wf,
real_term_value_minus_lemma,
squash_wf,
true_wf,
rminus-int,
subtype_rel_self,
iff_weakening_equal,
not-rneq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
because_Cache,
hypothesis,
sqequalRule,
productIsType,
functionIsType,
setIsType,
natural_numberEquality,
applyEquality,
dependent_functionElimination,
lambdaEquality_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
inrFormation_alt,
instantiate,
independent_functionElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
minusEquality,
equalityIstype,
approximateComputation,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
int_eqEquality,
inlFormation_alt,
unionElimination,
dependent_pairFormation_alt,
closedConclusion,
unionEquality,
functionEquality,
unionIsType,
universeEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}e:\{e:Point(rv)| e\^{}2 = r1\} . \mforall{}T:\mBbbR{} {}\mrightarrow{} Point(rv) {}\mrightarrow{} Point(rv).
(translation-group-fun(rv;e;T)
{}\mRightarrow{} (separable-translation-group(rv;e;T)
\mLeftarrow{}{}\mRightarrow{} (\mforall{}a:\mBbbR{}. \mforall{}h:\{h:Point(rv)| h \mcdot{} e = r0\} . (a \mneq{} r0 {}\mRightarrow{} (r0 < ||T\_a(h) - h||)))
\mwedge{} (\mforall{}a,b:\mBbbR{}. \mforall{}h:\{h:Point(rv)| h \mcdot{} e = r0\} .
(a \mneq{} r0 {}\mRightarrow{} b \mneq{} r0 {}\mRightarrow{} ((||T\_a(h) - h||/||T\_b(h) - h||) = (||T\_a(0)||/||T\_b(0)||))))))
Date html generated:
2020_05_20-PM-01_17_56
Last ObjectModification:
2019_12_09-PM-11_24_03
Theory : inner!product!spaces
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