Nuprl Lemma : fps-product-reindex
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[T,A:Type]. ∀[g:T ⟶ A]. ∀[h:A ⟶ T].
∀[f:T ⟶ PowerSeries(X;r)]. ∀[b:bag(T)]. (Π(x∈b).f[x] = Π(x∈bag-map(g;b)).f[h x] ∈ PowerSeries(X;r))
supposing ∀x:T. (x = (h (g x)) ∈ T)
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-product: Π(x∈b).f[x],
power-series: PowerSeries(X;r),
bag-map: bag-map(f;bs),
bag: bag(T),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
fps-product: Π(x∈b).f[x],
bag-product: Πx ∈ b. f[x],
and: P ∧ Q,
cand: A c∧ B,
comm: Comm(T;op),
infix_ap: x f y,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
assoc: Assoc(T;op),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
bag-summation-reindex,
power-series_wf,
fps-mul_wf,
fps-one_wf,
equal_wf,
squash_wf,
true_wf,
fps-mul-comm,
iff_weakening_equal,
mul_assoc_fps,
bag_wf,
all_wf,
crng_wf,
valueall-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
lambdaEquality,
independent_isectElimination,
sqequalRule,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
independent_functionElimination,
isect_memberEquality,
axiomEquality,
independent_pairFormation,
functionExtensionality,
functionEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[T,A:Type]. \mforall{}[g:T {}\mrightarrow{} A]. \mforall{}[h:A {}\mrightarrow{} T].
\mforall{}[f:T {}\mrightarrow{} PowerSeries(X;r)]. \mforall{}[b:bag(T)]. (\mPi{}(x\mmember{}b).f[x] = \mPi{}(x\mmember{}bag-map(g;b)).f[h x])
supposing \mforall{}x:T. (x = (h (g x)))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_57_11
Last ObjectModification:
2017_07_26-PM-06_33_15
Theory : power!series
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