### Nuprl Lemma : rsum-of-nonneg-zero-iff

`∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ].`
`  uiff(Σ{x[i] | n≤i≤m} = r0;∀i:{n..m + 1-}. (x[i] = r0)) supposing ∀i:{n..m + 1-}. (r0 ≤ x[i])`

Proof

Definitions occuring in Statement :  rsum: `Σ{x[k] | n≤k≤m}` rleq: `x ≤ y` req: `x = y` int-to-real: `r(n)` real: `ℝ` int_seg: `{i..j-}` uiff: `uiff(P;Q)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` all: `∀x:A. B[x]` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` uiff: `uiff(P;Q)` and: `P ∧ Q` all: `∀x:A. B[x]` so_apply: `x[s]` implies: `P `` Q` so_lambda: `λ2x.t[x]` prop: `ℙ` not: `¬A` rneq: `x ≠ y` or: `P ∨ Q` guard: `{T}` false: `False` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` exists: `∃x:A. B[x]`
Lemmas referenced :  int_seg_wf req_witness int-to-real_wf req_wf rsum_wf rleq_wf real_wf istype-int req-iff-not-rneq rneq_wf rless_transitivity1 rless_irreflexivity rsum-of-nonneg-positive-iff rless_wf rleq_weakening rsum-zero-req
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut independent_pairFormation lambdaFormation_alt universeIsType extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality addEquality natural_numberEquality hypothesis sqequalRule lambdaEquality_alt dependent_functionElimination applyEquality independent_functionElimination functionIsTypeImplies inhabitedIsType because_Cache functionIsType productElimination independent_pairEquality isect_memberEquality_alt isectIsTypeImplies independent_isectElimination unionElimination voidElimination dependent_pairFormation_alt

Latex:
\mforall{}[n,m:\mBbbZ{}].  \mforall{}[x:\{n..m  +  1\msupminus{}\}  {}\mrightarrow{}  \mBbbR{}].
uiff(\mSigma{}\{x[i]  |  n\mleq{}i\mleq{}m\}  =  r0;\mforall{}i:\{n..m  +  1\msupminus{}\}.  (x[i]  =  r0))  supposing  \mforall{}i:\{n..m  +  1\msupminus{}\}.  (r0  \mleq{}  x[i])

Date html generated: 2019_10_29-AM-10_12_37
Last ObjectModification: 2019_10_10-PM-10_00_27

Theory : reals

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