Nuprl Lemma : trans-apply-add
∀rv:InnerProductSpace. ∀T:ℝ ⟶ Point ⟶ Point. ∀t,s:ℝ.
∀x:Point. T_t + s(x) ≡ T_t(T_s(x)) 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,
radd: a + b,
real: ℝ,
ss-eq: x ≡ y,
ss-point: Point,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
exists: ∃x:A. B[x],
translation-group-fun: translation-group-fun(rv;e;T),
and: P ∧ Q,
trans-apply: T_t(x),
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
ss-eq: x ≡ y,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
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,
ss-sep_wf,
trans-apply_wf,
real_wf,
radd_wf,
exists_wf,
translation-group-fun_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
sqequalRule,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
lambdaEquality,
dependent_functionElimination,
voidElimination,
functionExtensionality,
because_Cache,
functionEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}T:\mBbbR{} {}\mrightarrow{} Point {}\mrightarrow{} Point. \mforall{}t,s:\mBbbR{}.
\mforall{}x:Point. T\_t + s(x) \mequiv{} T\_t(T\_s(x)) supposing \mexists{}e:Point. translation-group-fun(rv;e;T)
Date html generated:
2017_10_05-AM-00_21_39
Last ObjectModification:
2017_06_26-AM-10_08_12
Theory : inner!product!spaces
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