Nuprl Lemma : rnexp-rleq

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


Proof




Definitions occuring in Statement :  rleq: x ≤ y rnexp: x^k1 int-to-real: r(n) real: nat: all: x:A. B[x] implies:  Q natural_number: $n
Definitions unfolded in proof :  all: x:A. B[x] implies:  Q uall: [x:A]. B[x] member: t ∈ T nat: false: False ge: i ≥  uimplies: supposing a satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] not: ¬A top: Top and: P ∧ Q prop: rleq: x ≤ y rnonneg: rnonneg(x) le: A ≤ B nat_plus: + subtype_rel: A ⊆B decidable: Dec(P) or: P ∨ Q less_than': less_than'(a;b) bool: 𝔹 unit: Unit it: btrue: tt ifthenelse: if then else fi  uiff: uiff(P;Q) bfalse: ff sq_type: SQType(T) guard: {T} bnot: ¬bb assert: b nequal: a ≠ b ∈  eq_int: (i =z j) subtract: m rev_uimplies: rev_uimplies(P;Q) cand: c∧ B rge: x ≥ y 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 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 int-to-real_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_weakening_equal rleq_functionality rnexp_unroll rnexp-nonneg rleq_functionality_wrt_implies rmul_functionality_wrt_rleq2 rmul_preserves_rleq2 rleq_transitivity 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
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 inlFormation productEquality isect_memberFormation

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



Date html generated: 2017_10_03-AM-08_33_15
Last ObjectModification: 2017_07_28-AM-07_28_13

Theory : reals


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