Nuprl Lemma : rsum-split-first

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


Proof




Definitions occuring in Statement :  rsum: Σ{x[k] n≤k≤m} req: y radd: b 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: subtype_rel: A ⊆B uiff: uiff(P;Q)
Lemmas referenced :  req_witness rsum_wf int_seg_wf radd_wf decidable__le satisfiable-full-omega-tt intformnot_wf intformle_wf itermVar_wf int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformand_wf intformless_wf itermAdd_wf itermConstant_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_add_lemma int_term_value_constant_lemma lelt_wf le_wf real_wf equal-wf-base int_subtype_base intformeq_wf int_formula_prop_eq_lemma rsum-split req_functionality req_weakening radd_functionality rsum-single
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 dependent_set_memberEquality because_Cache independent_pairFormation dependent_functionElimination unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll setElimination rename productElimination independent_functionElimination equalityTransitivity equalitySymmetry functionEquality lambdaFormation setEquality

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



Date html generated: 2017_10_03-AM-08_58_30
Last ObjectModification: 2017_07_28-AM-07_38_15

Theory : reals


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