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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