Nuprl Lemma : trans-apply_functionality
∀[rv:InnerProductSpace]. ∀[T:ℝ ⟶ Point ⟶ Point].
∀[x1,x2:Point]. ∀[t1,t2:ℝ]. (T_t1(x1) ≡ T_t2(x2)) supposing ((t1 = t2) and x1 ≡ x2)
supposing ∃e:Point. translation-group-fun(rv;e;T)
Proof
Definitions occuring in Statement :
trans-apply: T_t(x),
translation-group-fun: translation-group-fun(rv;e;T),
inner-product-space: InnerProductSpace,
req: x = y,
real: ℝ,
ss-eq: x ≡ y,
ss-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
exists: ∃x:A. B[x],
translation-group-fun: translation-group-fun(rv;e;T),
and: P ∧ Q,
ss-eq: x ≡ y,
not: ¬A,
implies: P ⇒ Q,
trans-apply: T_t(x),
all: ∀x:A. B[x],
or: P ∨ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
false: False,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q
Lemmas referenced :
ss-sep_wf,
trans-apply_wf,
real_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
req_wf,
ss-eq_wf,
ss-point_wf,
exists_wf,
translation-group-fun_wf,
rneq_irreflexivity,
rneq_functionality,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
lambdaFormation,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
unionElimination,
extract_by_obid,
isectElimination,
applyEquality,
because_Cache,
sqequalRule,
functionExtensionality,
lambdaEquality,
instantiate,
independent_isectElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
voidElimination
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[T:\mBbbR{} {}\mrightarrow{} Point {}\mrightarrow{} Point].
\mforall{}[x1,x2:Point]. \mforall{}[t1,t2:\mBbbR{}]. (T\_t1(x1) \mequiv{} T\_t2(x2)) supposing ((t1 = t2) and x1 \mequiv{} x2)
supposing \mexists{}e:Point. translation-group-fun(rv;e;T)
Date html generated:
2017_10_05-AM-00_21_23
Last ObjectModification:
2017_06_24-PM-04_08_26
Theory : inner!product!spaces
Home
Index