Nuprl Lemma : rng_times_lsum_r

r:Rng. ∀A:Type. ∀as:A List. ∀f:A ⟶ |r|. ∀u:|r|.  ((Σ{r} x ∈ as. f[x] u) = Σ{r} x ∈ as. (f[x] u) ∈ |r|)


Proof




Definitions occuring in Statement :  rng_lsum: Σ{r} x ∈ as. f[x] list: List infix_ap: y so_apply: x[s] all: x:A. B[x] function: x:A ⟶ B[x] universe: Type equal: t ∈ T rng: Rng rng_times: * rng_car: |r|
Definitions unfolded in proof :  rev_implies:  Q iff: ⇐⇒ Q guard: {T} uimplies: supposing a subtype_rel: A ⊆B true: True infix_ap: y prop: squash: T and: P ∧ Q top: Top implies:  Q so_apply: x[s] rng: Rng so_lambda: λ2x.t[x] member: t ∈ T uall: [x:A]. B[x] all: x:A. B[x]
Lemmas referenced :  rng_wf rng_plus_comm rng_times_over_plus iff_weakening_equal rng_plus_wf true_wf squash_wf rng_lsum_cons_lemma rng_times_zero rng_lsum_nil_lemma list_wf rng_lsum_wf rng_times_wf infix_ap_wf equal_wf rng_car_wf all_wf list_induction
Rules used in proof :  independent_isectElimination baseClosed imageMemberEquality natural_numberEquality universeEquality equalitySymmetry equalityTransitivity imageElimination productElimination voidEquality voidElimination isect_memberEquality dependent_functionElimination independent_functionElimination functionExtensionality applyEquality hypothesis because_Cache rename setElimination cumulativity functionEquality lambdaEquality sqequalRule hypothesisEquality isectElimination sqequalHypSubstitution extract_by_obid introduction thin cut lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}r:Rng.  \mforall{}A:Type.  \mforall{}as:A  List.  \mforall{}f:A  {}\mrightarrow{}  |r|.  \mforall{}u:|r|.
    ((\mSigma{}\{r\}  x  \mmember{}  as.  f[x]  *  u)  =  \mSigma{}\{r\}  x  \mmember{}  as.  (f[x]  *  u))



Date html generated: 2018_05_21-PM-09_32_53
Last ObjectModification: 2017_12_11-PM-04_21_51

Theory : matrices


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