### Nuprl Lemma : exp_difference_factor

`∀[n:ℕ+]. ∀[x,y:ℤ].  ((x^n - y^n) = (Σ(x^n - i + 1 * y^i | i < n) * (x - y)) ∈ ℤ)`

Proof

Definitions occuring in Statement :  exp: `i^n` sum: `Σ(f[x] | x < k)` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` multiply: `n * m` subtract: `n - m` add: `n + m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` squash: `↓T` prop: `ℙ` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` nat: `ℕ` nat_plus: `ℕ+` int_seg: `{i..j-}` guard: `{T}` lelt: `i ≤ j < k` and: `P ∧ Q` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` le: `A ≤ B` less_than': `less_than'(a;b)` so_apply: `x[s]` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtract: `n - m` sq_type: `SQType(T)` uiff: `uiff(P;Q)`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut applyEquality thin lambdaEquality sqequalHypSubstitution imageElimination extract_by_obid isectElimination hypothesisEquality equalityTransitivity hypothesis equalitySymmetry universeEquality because_Cache sqequalRule multiplyEquality dependent_set_memberEquality setElimination rename addEquality natural_numberEquality productElimination dependent_functionElimination unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll lambdaFormation imageMemberEquality baseClosed independent_functionElimination axiomEquality instantiate cumulativity minusEquality pointwiseFunctionality promote_hyp baseApply closedConclusion

Latex:
\mforall{}[n:\mBbbN{}\msupplus{}].  \mforall{}[x,y:\mBbbZ{}].    ((x\^{}n  -  y\^{}n)  =  (\mSigma{}(x\^{}n  -  i  +  1  *  y\^{}i  |  i  <  n)  *  (x  -  y)))

Date html generated: 2017_04_17-AM-09_45_23
Last ObjectModification: 2017_02_27-PM-05_40_27

Theory : num_thy_1

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