Nuprl Lemma : trans-kernel-is-kernel-fun
∀rv:InnerProductSpace. ∀e:Point. ∀T:ℝ ⟶ Point ⟶ Point.
((e^2 = r1) ⇒ translation-group-fun(rv;e;T) ⇒ trans-kernel-fun(rv;e;λh,t. ρ(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],
implies: P ⇒ Q,
lambda: λx.A[x],
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,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
trans-kernel: ρ(h;t),
guard: {T},
uimplies: b supposing a,
or: P ∨ Q,
translation-group-fun: translation-group-fun(rv;e;T),
trans-apply: T_t(x),
not: ¬A,
false: False,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
top: Top,
req_int_terms: t1 ≡ t2,
rge: x ≥ y,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
rneq: x ≠ y
Lemmas referenced :
rneq_wf,
trans-kernel_wf,
real_wf,
req_wf,
rv-ip_wf,
int-to-real_wf,
set_wf,
ss-point_wf,
sq_stable__req,
rless_wf,
translation-group-fun_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-irrefl,
trans-apply_wf,
rv-ip-rneq,
ss-eq_weakening,
trans-apply-0,
rv-ip_functionality,
req_functionality,
uiff_transitivity,
req_weakening,
real_term_value_add_lemma,
trans-apply_functionality,
trans-apply-add,
ss-eq_inversion,
ss-eq_functionality,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
ss-eq_wf,
sq_stable__rleq,
itermAdd_wf,
rv-mul_wf,
rv-add_wf,
radd_wf,
req-iff-rsub-is-0,
itermVar_wf,
itermConstant_wf,
itermSubtract_wf,
rleq_wf,
rleq_weakening,
rleq_weakening_rless,
rsub_functionality_wrt_rleq,
rleq_weakening_equal,
rsub_wf,
rleq_functionality_wrt_implies,
real_term_value_mul_lemma,
real_term_value_minus_lemma,
rv-ip-add,
rmul_functionality,
rv-ip-mul,
req_transitivity,
radd_functionality,
itermMultiply_wf,
itermMinus_wf,
radd-rminus-both,
rless_functionality,
rminus_wf,
rmul_wf,
radd-preserves-rless,
trans-apply-sep,
ss-sep_functionality,
rv-0-add,
rv-0_wf,
rv-add-sep-iff,
rv-mul-sep-zero,
rabs-of-nonneg,
rabs_wf,
req-implies-req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
functionExtensionality,
applyEquality,
hypothesis,
setElimination,
rename,
dependent_set_memberEquality,
natural_numberEquality,
because_Cache,
lambdaEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_functionElimination,
functionEquality,
instantiate,
independent_isectElimination,
unionElimination,
productElimination,
voidElimination,
dependent_pairFormation,
voidEquality,
isect_memberEquality,
intEquality,
int_eqEquality,
approximateComputation,
productEquality,
levelHypothesis,
equalitySymmetry,
equalityTransitivity,
addLevel,
inlFormation,
promote_hyp
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{} trans-kernel-fun(rv;e;\mlambda{}h,t. \mrho{}(h;t)))
Date html generated:
2017_10_05-AM-00_23_09
Last ObjectModification:
2017_08_10-PM-03_38_31
Theory : inner!product!spaces
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