### Nuprl Lemma : rng_sum_plus

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

Proof

Definitions occuring in Statement :  rng_sum: rng_sum rng: `Rng` rng_plus: `+r` 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` subtype_rel: `A ⊆r B` abgrp: `AbGrp` grp: `Group{i}` mon: `Mon` iabmonoid: `IAbMonoid` imon: `IMonoid` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` all: `∀x:A. B[x]` implies: `P `` Q` rng_sum: rng_sum add_grp_of_rng: `r↓+gp` grp_car: `|g|` pi1: `fst(t)` grp_op: `*` pi2: `snd(t)` rng: `Rng`
Lemmas referenced :  mon_itop_op add_grp_of_rng_wf_b subtype_rel_sets grp_sig_wf monoid_p_wf grp_car_wf grp_op_wf grp_id_wf inverse_wf grp_inv_wf comm_wf set_wf int_seg_wf rng_car_wf le_wf rng_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis applyEquality sqequalRule instantiate setEquality cumulativity setElimination rename because_Cache lambdaEquality independent_isectElimination lambdaFormation isect_memberEquality axiomEquality functionEquality equalityTransitivity equalitySymmetry intEquality

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

Date html generated: 2018_05_21-PM-03_15_04
Last ObjectModification: 2018_05_19-AM-08_08_01

Theory : rings_1

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