### Nuprl Lemma : rng_sum_unroll_hi

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

Proof

Definitions occuring in Statement :  rng_sum: rng_sum rng: `Rng` rng_plus: `+r` rng_car: `|r|` int_seg: `{i..j-}` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` infix_ap: `x f y` so_apply: `x[s]` function: `x:A ⟶ B[x]` subtract: `n - m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` subtype_rel: `A ⊆r B` grp: `Group{i}` mon: `Mon` imon: `IMonoid` prop: `ℙ` rng_sum: rng_sum add_grp_of_rng: `r↓+gp` grp_car: `|g|` pi1: `fst(t)` grp_op: `*` pi2: `snd(t)` uimplies: `b supposing a` rng: `Rng`
Lemmas referenced :  mon_itop_unroll_hi add_grp_of_rng_wf_a grp_sig_wf monoid_p_wf grp_car_wf grp_op_wf grp_id_wf inverse_wf grp_inv_wf int_seg_wf rng_car_wf less_than_wf rng_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis applyEquality sqequalRule lambdaEquality setElimination rename setEquality cumulativity isect_memberEquality axiomEquality functionEquality equalityTransitivity equalitySymmetry intEquality

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

Date html generated: 2016_05_15-PM-00_28_03
Last ObjectModification: 2015_12_26-PM-11_58_38

Theory : rings_1

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