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: 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 :
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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