### Nuprl Lemma : rpower-greater-one

`∀x,q:ℝ.  ((r0 < x) `` ((r1 + x) < q) `` (∀n:ℕ. (r1 + (r(n) * x)) < q^n supposing 1 < n))`

Proof

Definitions occuring in Statement :  rless: `x < y` rnexp: `x^k1` rmul: `a * b` radd: `a + b` int-to-real: `r(n)` real: `ℝ` nat: `ℕ` less_than: `a < b` uimplies: `b supposing a` all: `∀x:A. B[x]` implies: `P `` Q` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` uimplies: `b supposing a` member: `t ∈ T` uall: `∀[x:A]. B[x]` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` false: `False` and: `P ∧ Q` prop: `ℙ` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` le: `A ≤ B` not: `¬A` uiff: `uiff(P;Q)` itermConstant: `"const"` req_int_terms: `t1 ≡ t2` top: `Top` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` sq_type: `SQType(T)` so_lambda: `λ2x.t[x]` nat: `ℕ` rless: `x < y` sq_exists: `∃x:{A| B[x]}` subtype_rel: `A ⊆r B` real: `ℝ` sq_stable: `SqStable(P)` nat_plus: `ℕ+` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` so_apply: `x[s]` cand: `A c∧ B` rge: `x ≥ y` true: `True` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T ` rev_uimplies: `rev_uimplies(P;Q)`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut thin isect_memberFormation introduction extract_by_obid sqequalHypSubstitution isectElimination sqequalRule imageElimination productElimination hypothesis voidElimination independent_isectElimination rename natural_numberEquality setElimination hypothesisEquality dependent_functionElimination because_Cache independent_functionElimination independent_pairFormation computeAll lambdaEquality int_eqEquality intEquality isect_memberEquality voidEquality unionElimination instantiate cumulativity dependent_set_memberEquality addEquality applyEquality imageMemberEquality baseClosed dependent_pairFormation productEquality inlFormation equalitySymmetry equalityTransitivity multiplyEquality levelHypothesis minusEquality addLevel promote_hyp equalityElimination

Latex:
\mforall{}x,q:\mBbbR{}.    ((r0  <  x)  {}\mRightarrow{}  ((r1  +  x)  <  q)  {}\mRightarrow{}  (\mforall{}n:\mBbbN{}.  (r1  +  (r(n)  *  x))  <  q\^{}n  supposing  1  <  n))

Date html generated: 2017_10_03-AM-08_34_00
Last ObjectModification: 2017_07_28-AM-07_28_33

Theory : reals

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