### Nuprl Lemma : Riemann-sum_wf

`∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f:[a, b] ⟶ℝ]. ∀[k:ℕ+].  (Riemann-sum(f;a;b;k) ∈ ℝ)`

Proof

Definitions occuring in Statement :  Riemann-sum: `Riemann-sum(f;a;b;k)` rfun: `I ⟶ℝ` rccint: `[l, u]` rleq: `x ≤ y` real: `ℝ` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` member: `t ∈ T` set: `{x:A| B[x]} `
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` Riemann-sum: `Riemann-sum(f;a;b;k)` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` uimplies: `b supposing a` has-value: `(a)↓` nat_plus: `ℕ+` so_lambda: `λ2x.t[x]` so_apply: `x[s]` callbyvalueall: callbyvalueall has-valueall: `has-valueall(a)` prop: `ℙ`
Lemmas referenced :  rccint-icompact uniform-partition_wf rccint_wf partition_wf value-type-has-value nat_plus_wf set-value-type less_than_wf int-value-type valueall-type-has-valueall list_wf real_wf list-valueall-type real-valueall-type full-partition_wf evalall-reduce valueall-type-real-list partition-sum_wf default-partition-choice_wf rfun_wf set_wf rleq_wf full-partition-non-dec
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut setElimination thin rename lemma_by_obid sqequalHypSubstitution dependent_functionElimination hypothesisEquality productElimination independent_functionElimination hypothesis isectElimination independent_isectElimination lambdaFormation sqequalRule callbyvalueReduce intEquality lambdaEquality natural_numberEquality equalityTransitivity equalitySymmetry because_Cache equalityEquality axiomEquality isect_memberEquality

Latex:
\mforall{}[a:\mBbbR{}].  \mforall{}[b:\{b:\mBbbR{}|  a  \mleq{}  b\}  ].  \mforall{}[f:[a,  b]  {}\mrightarrow{}\mBbbR{}].  \mforall{}[k:\mBbbN{}\msupplus{}].    (Riemann-sum(f;a;b;k)  \mmember{}  \mBbbR{})

Date html generated: 2016_05_18-AM-10_39_38
Last ObjectModification: 2015_12_27-PM-10_50_01

Theory : reals

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