Nuprl Lemma : rsum_functionality_wrt_rleq3

[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)  (y[k] ≥ x[k]))


Proof




Definitions occuring in Statement :  rsum: Σ{x[k] n≤k≤m} rge: x ≥ y rleq: x ≤ y real: int_seg: {i..j-} uimplies: supposing a uall: [x:A]. B[x] so_apply: x[s] le: A ≤ B all: x:A. B[x] implies:  Q function: x:A ⟶ B[x] add: m natural_number: $n int:
Definitions unfolded in proof :  rge: x ≥ y
Lemmas referenced :  rsum_functionality_wrt_rleq2
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{}  (y[k]  \mgeq{}  x[k]))



Date html generated: 2016_05_18-AM-07_45_21
Last ObjectModification: 2015_12_28-AM-01_01_26

Theory : reals


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