Nuprl Lemma : fps-compose-sub
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[x:X]. ∀[g,f,h:PowerSeries(X;r)].
((g-h)(x:=f) = (g(x:=f)-h(x:=f)) ∈ PowerSeries(X;r))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-compose: g(x:=f),
fps-sub: (f-g),
power-series: PowerSeries(X;r),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[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-sub: (f-g),
true: True,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
power-series_wf,
crng_wf,
deq_wf,
valueall-type_wf,
fps-neg_wf,
fps-compose_wf,
fps-add_wf,
equal_wf,
squash_wf,
true_wf,
fps-compose-add,
fps-compose-neg,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_isectElimination,
natural_numberEquality,
applyEquality,
lambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[x:X]. \mforall{}[g,f,h:PowerSeries(X;r)].
((g-h)(x:=f) = (g(x:=f)-h(x:=f)))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_59_45
Last ObjectModification:
2017_07_26-PM-06_34_03
Theory : power!series
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