Nuprl Lemma : rsum-difference

[k,n,m:ℤ]. ∀[x:{k..m 1-} ⟶ ℝ].
  ((Σ{x[i] k≤i≤m} - Σ{x[i] k≤i≤n}) = Σ{x[i] 1≤i≤m}) supposing ((n ≤ m) and (k ≤ n))


Definitions occuring in Statement :  rsum: Σ{x[k] n≤k≤m} rsub: y req: y real: int_seg: {i..j-} uimplies: supposing a uall: [x:A]. B[x] so_apply: x[s] le: A ≤ B function: x:A ⟶ B[x] add: m natural_number: $n int:
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a so_lambda: λ2x.t[x] so_apply: x[s] int_seg: {i..j-} lelt: i ≤ j < k and: P ∧ Q all: x:A. B[x] decidable: Dec(P) or: P ∨ Q satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] false: False implies:  Q not: ¬A top: Top prop: uiff: uiff(P;Q) rev_uimplies: rev_uimplies(P;Q) rsub: y
Lemmas referenced :  radd-zero-both radd-rminus-both radd_functionality radd-ac radd-assoc req_inversion uiff_transitivity rsub_functionality req_functionality req_weakening int-to-real_wf rminus_wf req_wf radd_wf real_wf le_wf decidable__le lelt_wf int_formula_prop_wf int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma intformle_wf itermConstant_wf itermAdd_wf itermVar_wf intformless_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__lt int_seg_wf rsum_wf rsub_wf req_witness rsum-split
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality independent_isectElimination hypothesis sqequalRule lambdaEquality applyEquality because_Cache addEquality natural_numberEquality setElimination rename dependent_set_memberEquality productElimination independent_pairFormation dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll independent_functionElimination equalityTransitivity equalitySymmetry functionEquality

\mforall{}[k,n,m:\mBbbZ{}].  \mforall{}[x:\{k..m  +  1\msupminus{}\}  {}\mrightarrow{}  \mBbbR{}].
    ((\mSigma{}\{x[i]  |  k\mleq{}i\mleq{}m\}  -  \mSigma{}\{x[i]  |  k\mleq{}i\mleq{}n\})  =  \mSigma{}\{x[i]  |  n  +  1\mleq{}i\mleq{}m\})  supposing  ((n  \mleq{}  m)  and  (k  \mleq{}  n))

Date html generated: 2016_05_18-AM-07_46_32
Last ObjectModification: 2016_01_17-AM-02_11_11

Theory : reals

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