### Nuprl Lemma : rng_sum_is_0

`∀[r:Rng]. ∀[a,b:ℤ]. ∀[F:{a..b-} ⟶ |r|].`
`  (Σ(r) a ≤ j < b. F[j]) = 0 ∈ |r| supposing (a ≤ b) ∧ (∀j:{a..b-}. (F[j] = 0 ∈ |r|))`

Proof

Definitions occuring in Statement :  rng_sum: rng_sum rng: `Rng` rng_zero: `0` rng_car: `|r|` int_seg: `{i..j-}` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` le: `A ≤ B` all: `∀x:A. B[x]` and: `P ∧ Q` function: `x:A ⟶ B[x]` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  so_apply: `x[s]` implies: `P `` Q` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` guard: `{T}` subtype_rel: `A ⊆r B` true: `True` all: `∀x:A. B[x]` so_lambda: `λ2x.t[x]` rng: `Rng` squash: `↓T` prop: `ℙ` and: `P ∧ Q` uimplies: `b supposing a` member: `t ∈ T` uall: `∀[x:A]. B[x]`
Lemmas referenced :  rng_zero_wf all_wf le_wf iff_weakening_equal rng_wf int_seg_wf true_wf squash_wf rng_sum_wf rng_car_wf equal_wf rng_sum_0
Rules used in proof :  axiomEquality isect_memberEquality functionExtensionality productEquality independent_functionElimination baseClosed imageMemberEquality natural_numberEquality dependent_functionElimination intEquality functionEquality equalityTransitivity rename setElimination because_Cache imageElimination lambdaEquality applyEquality sqequalRule equalitySymmetry hyp_replacement independent_isectElimination productElimination hypothesisEquality thin isectElimination sqequalHypSubstitution hypothesis isect_memberFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution extract_by_obid introduction cut

Latex:
\mforall{}[r:Rng].  \mforall{}[a,b:\mBbbZ{}].  \mforall{}[F:\{a..b\msupminus{}\}  {}\mrightarrow{}  |r|].
(\mSigma{}(r)  a  \mleq{}  j  <  b.  F[j])  =  0  supposing  (a  \mleq{}  b)  \mwedge{}  (\mforall{}j:\{a..b\msupminus{}\}.  (F[j]  =  0))

Date html generated: 2018_05_21-PM-03_15_19
Last ObjectModification: 2018_01_02-PM-02_26_34

Theory : rings_1

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