### Nuprl Lemma : rsum-triangle-inequality1

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

Proof

Definitions occuring in Statement :  rsum: `Σ{x[k] | n≤k≤m}` rleq: `x ≤ y` rabs: `|x|` rsub: `x - y` radd: `a + b` 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` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uiff: `uiff(P;Q)` and: `P ∧ Q` rev_uimplies: `rev_uimplies(P;Q)` uimplies: `b supposing a` rleq: `x ≤ y` rnonneg: `rnonneg(x)` all: `∀x:A. B[x]` le: `A ≤ B` pointwise-rleq: `x[k] ≤ y[k] for k ∈ [n,m]` implies: `P `` Q` req_int_terms: `t1 ≡ t2` false: `False` not: `¬A` squash: `↓T` int_seg: `{i..j-}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` prop: `ℙ` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` rge: `x ≥ y`

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

Date html generated: 2020_05_20-AM-11_13_14
Last ObjectModification: 2019_12_15-PM-06_48_55

Theory : reals

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