`∀[X:Type]. ∀[r:CRng]. ∀[n:ℤ]. ∀[f,g:PowerSeries(X;r)].  ([(f+g)]_n = ([f]_n+[g]_n) ∈ PowerSeries(X;r))`

Proof

Definitions occuring in Statement :  fps-slice: `[f]_n` fps-add: `(f+g)` power-series: `PowerSeries(X;r)` uall: `∀[x:A]. B[x]` int: `ℤ` universe: `Type` equal: `s = t ∈ T` crng: `CRng`
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]` fps-slice: `[f]_n` fps-add: `(f+g)` fps-coeff: `f[b]` subtype_rel: `A ⊆r B` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` nat: `ℕ` ifthenelse: `if b then t else f fi ` crng: `CRng` rng: `Rng` power-series: `PowerSeries(X;r)` bfalse: `ff` exists: `∃x:A. B[x]` prop: `ℙ` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` false: `False` true: `True` squash: `↓T` infix_ap: `x f y` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  fps-ext fps-slice_wf fps-add_wf eq_int_wf bag-size_wf bool_wf eqtt_to_assert assert_of_eq_int nat_wf rng_plus_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int bag_wf power-series_wf crng_wf rng_car_wf rng_zero_wf squash_wf true_wf rng_plus_zero iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality cumulativity hypothesis productElimination independent_isectElimination lambdaFormation sqequalRule applyEquality because_Cache unionElimination equalityElimination equalityTransitivity equalitySymmetry lambdaEquality setElimination rename dependent_pairFormation promote_hyp dependent_functionElimination instantiate independent_functionElimination voidElimination isect_memberEquality axiomEquality intEquality universeEquality natural_numberEquality imageElimination imageMemberEquality baseClosed

Latex:
\mforall{}[X:Type].  \mforall{}[r:CRng].  \mforall{}[n:\mBbbZ{}].  \mforall{}[f,g:PowerSeries(X;r)].    ([(f+g)]\_n  =  ([f]\_n+[g]\_n))

Date html generated: 2018_05_21-PM-09_56_05
Last ObjectModification: 2017_07_26-PM-06_32_52

Theory : power!series

Home Index