### Nuprl Lemma : rng_sum_single

`∀[r:Rng]. ∀[i,j:ℤ].  ∀[E:{i..j-} ⟶ |r|]. ((Σ(r) i ≤ k < j. E[k]) = E[i] ∈ |r|) supposing j = (i + 1) ∈ ℤ`

Proof

Definitions occuring in Statement :  rng_sum: rng_sum rng: `Rng` rng_car: `|r|` int_seg: `{i..j-}` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  true: `True` lelt: `i ≤ j < k` int_seg: `{i..j-}` so_apply: `x[s]` subtype_rel: `A ⊆r B` rng: `Rng` prop: `ℙ` and: `P ∧ Q` top: `Top` false: `False` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` implies: `P `` Q` not: `¬A` or: `P ∨ Q` decidable: `Dec(P)` all: `∀x:A. B[x]` uimplies: `b supposing a` member: `t ∈ T` uall: `∀[x:A]. B[x]` squash: `↓T` infix_ap: `x f y` so_lambda: `λ2x.t[x]` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  rng_plus_zero lelt_wf int_formula_prop_le_lemma intformle_wf decidable__le rng_plus_wf rng_wf int_subtype_base equal-wf-base rng_car_wf int_seg_wf int_formula_prop_wf int_term_value_constant_lemma int_term_value_add_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermConstant_wf itermAdd_wf intformeq_wf itermVar_wf intformless_wf intformnot_wf intformand_wf full-omega-unsat decidable__lt rng_sum_unroll_lo equal_wf squash_wf true_wf rng_sum_unroll_base iff_weakening_equal
Rules used in proof :  productElimination equalitySymmetry dependent_set_memberEquality functionExtensionality because_Cache baseClosed closedConclusion baseApply applyEquality axiomEquality rename setElimination functionEquality equalityTransitivity independent_pairFormation sqequalRule voidEquality voidElimination isect_memberEquality intEquality int_eqEquality lambdaEquality dependent_pairFormation independent_functionElimination approximateComputation natural_numberEquality unionElimination dependent_functionElimination independent_isectElimination hypothesisEquality thin isectElimination sqequalHypSubstitution hypothesis isect_memberFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution extract_by_obid introduction cut imageElimination universeEquality imageMemberEquality

Latex:
\mforall{}[r:Rng].  \mforall{}[i,j:\mBbbZ{}].    \mforall{}[E:\{i..j\msupminus{}\}  {}\mrightarrow{}  |r|].  ((\mSigma{}(r)  i  \mleq{}  k  <  j.  E[k])  =  E[i])  supposing  j  =  (i  +  1)

Date html generated: 2018_05_21-PM-03_15_01
Last ObjectModification: 2017_12_12-AM-11_39_54

Theory : rings_1

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