Nuprl Lemma : trans-kernel-equation
∀[rv:InnerProductSpace]. ∀[e:{e:Point| e^2 = r1} ]. ∀[T:ℝ ⟶ Point ⟶ Point].
∀[t:ℝ]. ∀[h:{h:Point| h ⋅ e = r0} ]. T_t(h) ≡ h + ρ(h;t)*e supposing translation-group-fun(rv;e;T)
Proof
Definitions occuring in Statement :
trans-kernel: ρ(h;t),
trans-apply: T_t(x),
translation-group-fun: translation-group-fun(rv;e;T),
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rv-mul: a*x,
rv-add: x + y,
req: x = y,
int-to-real: r(n),
real: ℝ,
ss-eq: x ≡ y,
ss-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
guard: {T},
all: ∀x:A. B[x],
prop: ℙ,
sq_stable: SqStable(P),
implies: P ⇒ Q,
translation-group-fun: translation-group-fun(rv;e;T),
and: P ∧ Q,
trans-apply: T_t(x),
squash: ↓T,
ss-eq: x ≡ y,
not: ¬A,
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
stable: Stable{P},
or: P ∨ Q,
exists: ∃x:A. B[x],
trans-kernel: ρ(h;t),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
top: Top,
cand: A c∧ B
Lemmas referenced :
sq_stable__ss-eq,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
trans-apply_wf,
real_wf,
rv-add_wf,
rv-mul_wf,
trans-kernel_wf,
req_wf,
rv-ip_wf,
int-to-real_wf,
ss-sep_wf,
set_wf,
ss-point_wf,
translation-group-fun_wf,
stable__ss-eq,
false_wf,
or_wf,
rless_wf,
not_wf,
ss-eq_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
rleq_weakening_rless,
rleq_wf,
radd_wf,
rmul_wf,
itermSubtract_wf,
itermAdd_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
req-iff-rsub-is-0,
ss-eq_weakening,
uiff_transitivity,
ss-eq_functionality,
rv-add_functionality,
rv-mul_functionality,
rv-ip_functionality,
rv-ip-add,
radd_functionality,
req_transitivity,
rv-ip-mul,
rmul_functionality,
req_weakening,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
not-rless,
rleq-implies-rleq,
rminus_wf,
rsub_wf,
itermMinus_wf,
exists_wf,
real_term_value_minus_lemma,
ss-eq_inversion,
radd-rminus-both,
trans-apply_functionality,
trans-apply-0,
equal_wf,
radd-rminus,
rv-0_wf,
rv-mul-add-alt,
rv-add-comm,
rv-mul0,
rv-add-0
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
functionExtensionality,
because_Cache,
dependent_set_memberEquality,
natural_numberEquality,
independent_functionElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
voidElimination,
lambdaFormation,
unionElimination,
approximateComputation,
int_eqEquality,
intEquality,
voidEquality,
setEquality,
addLevel,
existsFunctionality,
promote_hyp,
dependent_pairFormation,
independent_pairFormation
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[e:\{e:Point| e\^{}2 = r1\} ]. \mforall{}[T:\mBbbR{} {}\mrightarrow{} Point {}\mrightarrow{} Point].
\mforall{}[t:\mBbbR{}]. \mforall{}[h:\{h:Point| h \mcdot{} e = r0\} ]. T\_t(h) \mequiv{} h + \mrho{}(h;t)*e supposing translation-group-fun(rv;e;T\000C)
Date html generated:
2017_10_05-AM-00_22_46
Last ObjectModification:
2017_07_02-PM-02_04_38
Theory : inner!product!spaces
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