### Nuprl Lemma : fps-ext

`∀[X:Type]. ∀[r:CRng]. ∀[f,g:PowerSeries(X;r)].  uiff(f = g ∈ PowerSeries(X;r);∀b:bag(X). (f[b] = g[b] ∈ |r|))`

Proof

Definitions occuring in Statement :  fps-coeff: `f[b]` power-series: `PowerSeries(X;r)` bag: `bag(T)` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` universe: `Type` equal: `s = t ∈ T` crng: `CRng` rng_car: `|r|`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` all: `∀x:A. B[x]` prop: `ℙ` power-series: `PowerSeries(X;r)` fps-coeff: `f[b]` squash: `↓T` crng: `CRng` rng: `Rng` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  and_wf equal_wf power-series_wf fps-coeff_wf squash_wf true_wf rng_car_wf iff_weakening_equal all_wf bag_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation lambdaFormation equalitySymmetry dependent_set_memberEquality hypothesis hypothesisEquality extract_by_obid sqequalHypSubstitution isectElimination thin applyLambdaEquality setElimination rename productElimination because_Cache sqequalRule lambdaEquality dependent_functionElimination axiomEquality cumulativity functionExtensionality applyEquality imageElimination equalityTransitivity natural_numberEquality imageMemberEquality baseClosed universeEquality independent_isectElimination independent_functionElimination independent_pairEquality isect_memberEquality

Latex:
\mforall{}[X:Type].  \mforall{}[r:CRng].  \mforall{}[f,g:PowerSeries(X;r)].    uiff(f  =  g;\mforall{}b:bag(X).  (f[b]  =  g[b]))

Date html generated: 2018_05_21-PM-09_54_44
Last ObjectModification: 2017_07_26-PM-06_32_32

Theory : power!series

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