### Nuprl Lemma : rng_times_sum_l

`∀[r:Rng]. ∀[a,b:ℤ].`
`  ∀[E:{a..b-} ⟶ |r|]. ∀[u:|r|].  ((u * (Σ(r) a ≤ j < b. E[j])) = (Σ(r) a ≤ j < b. u * E[j]) ∈ |r|) supposing a ≤ b`

Proof

Definitions occuring in Statement :  rng_sum: rng_sum rng: `Rng` rng_times: `*` rng_car: `|r|` int_seg: `{i..j-}` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` infix_ap: `x f y` so_apply: `x[s]` le: `A ≤ B` function: `x:A ⟶ B[x]` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` rng: `Rng` prop: `ℙ` all: `∀x:A. B[x]` so_lambda: `λ2x.t[x]` int_upper: `{i...}` so_apply: `x[s]` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` implies: `P `` Q` guard: `{T}` int_seg: `{i..j-}` infix_ap: `x f y` decidable: `Dec(P)` or: `P ∨ Q` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` lelt: `i ≤ j < k`
Lemmas referenced :  rng_car_wf int_seg_wf le_wf rng_wf int_le_to_int_upper_uniform uall_wf equal_wf infix_ap_wf rng_times_wf rng_sum_wf int_upper_wf int_upper_ind_uniform decidable__equal_int squash_wf true_wf rng_sum_unroll_base iff_weakening_equal rng_times_zero rng_zero_wf rng_sum_unroll_hi int_upper_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermVar_wf intformeq_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_le_lemma int_formula_prop_wf rng_plus_wf subtract_wf itermSubtract_wf itermConstant_wf int_term_value_subtract_lemma int_term_value_constant_lemma lelt_wf decidable__le rng_times_over_plus
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule sqequalHypSubstitution isect_memberEquality isectElimination thin hypothesisEquality axiomEquality hypothesis extract_by_obid setElimination rename functionEquality equalityTransitivity equalitySymmetry intEquality because_Cache dependent_functionElimination lambdaEquality applyEquality functionExtensionality productElimination independent_functionElimination instantiate lambdaFormation unionElimination imageElimination universeEquality independent_isectElimination natural_numberEquality imageMemberEquality baseClosed dependent_pairFormation int_eqEquality voidElimination voidEquality independent_pairFormation computeAll dependent_set_memberEquality

Latex:
\mforall{}[r:Rng].  \mforall{}[a,b:\mBbbZ{}].
\mforall{}[E:\{a..b\msupminus{}\}  {}\mrightarrow{}  |r|].  \mforall{}[u:|r|].    ((u  *  (\mSigma{}(r)  a  \mleq{}  j  <  b.  E[j]))  =  (\mSigma{}(r)  a  \mleq{}  j  <  b.  u  *  E[j]))
supposing  a  \mleq{}  b

Date html generated: 2017_10_01-AM-08_19_27
Last ObjectModification: 2017_02_28-PM-02_04_15

Theory : rings_1

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