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

`∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ.  ((∀i:{n..m + 1-}. (r0 ≤ x[i])) `` (r0 < Σ{x[i] | n≤i≤m} `⇐⇒` ∃i:{n..m + 1-}. (r0 < x[i])))`

Proof

Definitions occuring in Statement :  rsum: `Σ{x[k] | n≤k≤m}` rleq: `x ≤ y` rless: `x < y` int-to-real: `r(n)` real: `ℝ` int_seg: `{i..j-}` so_apply: `x[s]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_implies: `P `` Q` exists: `∃x:A. B[x]` guard: `{T}` uimplies: `b supposing a` int_seg: `{i..j-}` rless: `x < y` sq_exists: `∃x:{A| B[x]}` subtype_rel: `A ⊆r B` real: `ℝ` sq_stable: `SqStable(P)` squash: `↓T` nat_plus: `ℕ+` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` pointwise-rleq: `x[k] ≤ y[k] for k ∈ [n,m]` rge: `x ≥ y` uiff: `uiff(P;Q)`
Lemmas referenced :  rless_wf int-to-real_wf rsum_wf int_seg_wf exists_wf all_wf rleq_wf real_wf rsum-positive-implies rabs_wf rless_functionality req_weakening rabs-of-nonneg sq_stable__less_than nat_plus_properties int_seg_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_wf intformless_wf itermAdd_wf itermConstant_wf int_formula_prop_less_lemma int_term_value_add_lemma int_term_value_constant_lemma radd_wf decidable__lt lelt_wf rsum-split rsum_nonneg le_wf rless_functionality_wrt_implies rleq_weakening_equal radd_functionality_wrt_rleq subtract_wf subtract-add-cancel radd_functionality rsum-split-last trivial-rless-radd itermSubtract_wf int_term_value_subtract_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation independent_pairFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality hypothesis hypothesisEquality sqequalRule lambdaEquality applyEquality functionExtensionality addEquality functionEquality intEquality dependent_functionElimination independent_functionElimination productElimination dependent_pairFormation because_Cache independent_isectElimination setElimination rename imageMemberEquality baseClosed imageElimination unionElimination int_eqEquality isect_memberEquality voidElimination voidEquality computeAll dependent_set_memberEquality equalityTransitivity equalitySymmetry

Latex:
\mforall{}n,m:\mBbbZ{}.  \mforall{}x:\{n..m  +  1\msupminus{}\}  {}\mrightarrow{}  \mBbbR{}.
((\mforall{}i:\{n..m  +  1\msupminus{}\}.  (r0  \mleq{}  x[i]))  {}\mRightarrow{}  (r0  <  \mSigma{}\{x[i]  |  n\mleq{}i\mleq{}m\}  \mLeftarrow{}{}\mRightarrow{}  \mexists{}i:\{n..m  +  1\msupminus{}\}.  (r0  <  x[i])))

Date html generated: 2016_10_26-AM-09_17_28
Last ObjectModification: 2016_09_28-PM-06_09_00

Theory : reals

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