### Nuprl Lemma : rabs-rsum

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

Proof

Definitions occuring in Statement :  rsum: `Σ{x[k] | n≤k≤m}` rleq: `x ≤ y` rabs: `|x|` real: `ℝ` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` so_apply: `x[s]` 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` rsum: `Σ{x[k] | n≤k≤m}` rleq: `x ≤ y` rnonneg: `rnonneg(x)` all: `∀x:A. B[x]` le: `A ≤ B` and: `P ∧ Q` not: `¬A` implies: `P `` Q` false: `False` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` real: `ℝ` prop: `ℙ` uimplies: `b supposing a` int_seg: `{i..j-}` lelt: `i ≤ j < k` callbyvalueall: callbyvalueall has-value: `(a)↓` has-valueall: `has-valueall(a)` top: `Top` compose: `f o g` rev_uimplies: `rev_uimplies(P;Q)` rge: `x ≥ y` guard: `{T}`
Lemmas referenced :  less_than'_wf rsub_wf rsum_wf rabs_wf int_seg_wf real_wf nat_plus_wf value-type-has-value int-value-type valueall-type-has-valueall list_wf list-valueall-type real-valueall-type map_wf le_wf less_than_wf from-upto_wf evalall-reduce valueall-type-real-list radd-list_wf-bag list-subtype-bag map-map subtype_rel_self equal_wf rleq_weakening_equal rleq_functionality_wrt_implies radd-list-rabs
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule sqequalHypSubstitution lambdaEquality dependent_functionElimination thin hypothesisEquality productElimination independent_pairEquality because_Cache extract_by_obid isectElimination applyEquality functionExtensionality addEquality natural_numberEquality hypothesis setElimination rename minusEquality axiomEquality equalityTransitivity equalitySymmetry functionEquality isect_memberEquality intEquality voidElimination independent_isectElimination setEquality productEquality lambdaFormation dependent_set_memberEquality callbyvalueReduce voidEquality independent_functionElimination

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

Date html generated: 2017_10_03-AM-09_00_02
Last ObjectModification: 2017_07_28-AM-07_39_21

Theory : reals

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