### Nuprl Lemma : rsum-split-first

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

Proof

Definitions occuring in Statement :  rsum: `Σ{x[k] | n≤k≤m}` req: `x = y` radd: `a + b` real: `ℝ` int_seg: `{i..j-}` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` le: `A ≤ B` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b 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: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` subtype_rel: `A ⊆r 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

Home Index