Nuprl Lemma : trans-apply-sep
∀rv:InnerProductSpace. ∀T:ℝ ⟶ Point ⟶ Point.
((∃e:Point. translation-group-fun(rv;e;T)) ⇒ (∀x:Point. ∀t1,t2:ℝ. (t1 ≠ t2 ⇒ T_t2(x) # T_t1(x))))
Proof
Definitions occuring in Statement :
trans-apply: T_t(x),
translation-group-fun: translation-group-fun(rv;e;T),
inner-product-space: InnerProductSpace,
rneq: x ≠ y,
real: ℝ,
ss-sep: x # y,
ss-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
translation-group-fun: translation-group-fun(rv;e;T),
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
not: ¬A,
trans-apply: T_t(x),
ss-eq: x ≡ y,
false: False,
rev_implies: P ⇐ Q,
rneq: x ≠ y,
or: P ∨ Q,
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
trans-apply_wf,
real_wf,
int-to-real_wf,
rneq_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,
exists_wf,
translation-group-fun_wf,
ss-eq_wf,
rv-add_wf,
rv-mul_wf,
rv-0_wf,
uiff_transitivity,
ss-eq_functionality,
rv-add_functionality,
ss-eq_weakening,
rv-mul0,
rv-0-add,
ss-sep_functionality,
not-rneq,
trans-apply-0,
rneq_functionality,
req_weakening,
req_inversion,
rneq-symmetry,
rless-implies-rless,
rsub_wf,
rless_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
radd_wf,
ss-eq_inversion,
trans-apply-add,
itermAdd_wf,
trans-apply_functionality,
real_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
hypothesis,
addLevel,
dependent_functionElimination,
introduction,
extract_by_obid,
hypothesisEquality,
functionExtensionality,
applyEquality,
isectElimination,
natural_numberEquality,
levelHypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
lambdaEquality,
functionEquality,
independent_functionElimination,
allFunctionality,
promote_hyp,
dependent_pairFormation,
voidElimination,
unionElimination,
inlFormation,
inrFormation,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}T:\mBbbR{} {}\mrightarrow{} Point {}\mrightarrow{} Point.
((\mexists{}e:Point. translation-group-fun(rv;e;T))
{}\mRightarrow{} (\mforall{}x:Point. \mforall{}t1,t2:\mBbbR{}. (t1 \mneq{} t2 {}\mRightarrow{} T\_t2(x) \# T\_t1(x))))
Date html generated:
2017_10_05-AM-00_21_50
Last ObjectModification:
2017_06_26-PM-06_56_05
Theory : inner!product!spaces
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