Nuprl Lemma : rsum_functionality_wrt_rleq2

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

Proof

Definitions occuring in Statement :  rsum: `Σ{x[k] | n≤k≤m}` rleq: `x ≤ y` real: `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]` implies: `P `` Q` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  pointwise-rleq: `x[k] ≤ y[k] for k ∈ [n,m]`
Lemmas referenced :  rsum_functionality_wrt_rleq
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep cut lemma_by_obid hypothesis

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

Date html generated: 2016_05_18-AM-07_45_16
Last ObjectModification: 2015_12_28-AM-01_01_33

Theory : reals

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