Nuprl Lemma : rsum_linearity3

[n,m:ℤ]. ∀[x:{n..m 1-} ⟶ ℝ]. ∀[y:ℝ].  {x[k] n≤k≤m} {x[k] n≤k≤m} y))


Proof




Definitions occuring in Statement :  rsum: Σ{x[k] n≤k≤m} req: y rmul: b real: int_seg: {i..j-} uall: [x:A]. B[x] so_apply: x[s] function: x:A ⟶ B[x] add: m natural_number: $n int:
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T rsum: Σ{x[k] n≤k≤m} so_lambda: λ2x.t[x] so_apply: x[s] implies:  Q uimplies: supposing a subtype_rel: A ⊆B int_seg: {i..j-} lelt: i ≤ j < k and: P ∧ Q prop: all: x:A. B[x] callbyvalueall: callbyvalueall has-value: (a)↓ has-valueall: has-valueall(a)
Lemmas referenced :  req_witness rsum_wf rmul_wf int_seg_wf real_wf value-type-has-value int-value-type from-upto_wf list_wf le_wf less_than_wf valueall-type-has-valueall list-valueall-type real-valueall-type map_wf evalall-reduce valueall-type-real-list radd-list-linearity3 equal_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality sqequalRule lambdaEquality applyEquality functionExtensionality addEquality natural_numberEquality hypothesis independent_functionElimination isect_memberEquality because_Cache functionEquality intEquality independent_isectElimination setEquality productEquality lambdaFormation callbyvalueReduce equalityTransitivity equalitySymmetry dependent_functionElimination

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



Date html generated: 2017_10_03-AM-08_59_21
Last ObjectModification: 2017_07_28-AM-07_38_51

Theory : reals


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