Nuprl Lemma : fps-set-to-one-sub
∀[r:CRng]. ∀[f,g:PowerSeries(r)]. ∀[y:Atom]. ∀[n:ℕ]. ([(f-g)]_n(y:=1) = ([f]_n(y:=1)-[g]_n(y:=1)) ∈ PowerSeries(r))
Proof
Definitions occuring in Statement :
fps-set-to-one: [f]_n(y:=1),
fps-sub: (f-g),
power-series: PowerSeries(X;r),
nat: ℕ,
uall: ∀[x:A]. B[x],
atom: Atom,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
fps-sub: (f-g),
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
power-series_wf,
fps-set-to-one-add,
fps-neg_wf,
fps-add_wf,
fps-set-to-one_wf,
iff_weakening_equal,
crng_wf,
fps-set-to-one-neg,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
atomEquality,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[f,g:PowerSeries(r)]. \mforall{}[y:Atom]. \mforall{}[n:\mBbbN{}]. ([(f-g)]\_n(y:=1) = ([f]\_n(y:=1)-[g]\_n(y:=1)))
Date html generated:
2018_05_21-PM-10_12_54
Last ObjectModification:
2017_07_26-PM-06_35_12
Theory : power!series
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