### Nuprl Lemma : rnexp-nonneg

`∀x:ℝ. ((r0 ≤ x) `` (∀n:ℕ. (r0 ≤ x^n)))`

Proof

Definitions occuring in Statement :  rleq: `x ≤ y` rnexp: `x^k1` int-to-real: `r(n)` real: `ℝ` nat: `ℕ` all: `∀x:A. B[x]` implies: `P `` Q` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` rleq: `x ≤ y` rnonneg: `rnonneg(x)` le: `A ≤ B` nat_plus: `ℕ+` subtype_rel: `A ⊆r B` decidable: `Dec(P)` or: `P ∨ Q` less_than': `less_than'(a;b)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T ` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtract: `n - m` rev_uimplies: `rev_uimplies(P;Q)` itermConstant: `"const"` req_int_terms: `t1 ≡ t2`
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf less_than'_wf rsub_wf rnexp_wf nat_plus_properties int-to-real_wf nat_plus_wf le_wf decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma nat_wf rleq_wf real_wf false_wf eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int rmul_wf intformeq_wf int_formula_prop_eq_lemma rleq-int rleq_functionality req_weakening rnexp-req rmul_preserves_rleq2 rleq-implies-rleq real_term_polynomial itermMultiply_wf real_term_value_const_lemma real_term_value_sub_lemma real_term_value_mul_lemma real_term_value_var_lemma req-iff-rsub-is-0 req_transitivity rmul_functionality rmul-identity1
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination productElimination independent_pairEquality applyEquality because_Cache minusEquality axiomEquality equalityTransitivity equalitySymmetry dependent_set_memberEquality unionElimination equalityElimination promote_hyp instantiate cumulativity isect_memberFormation

Latex:
\mforall{}x:\mBbbR{}.  ((r0  \mleq{}  x)  {}\mRightarrow{}  (\mforall{}n:\mBbbN{}.  (r0  \mleq{}  x\^{}n)))

Date html generated: 2017_10_03-AM-08_32_45
Last ObjectModification: 2017_07_28-AM-07_27_54

Theory : reals

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