### Nuprl Lemma : rmin-rnexp

`∀[n:ℕ]. ∀[x,y:ℝ].  ((r0 ≤ x) `` (r0 ≤ y) `` (rmin(x^n;y^n) = rmin(x;y)^n))`

Proof

Definitions occuring in Statement :  rleq: `x ≤ y` rmin: `rmin(x;y)` rnexp: `x^k1` req: `x = y` int-to-real: `r(n)` real: `ℝ` nat: `ℕ` uall: `∀[x:A]. B[x]` implies: `P `` Q` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` implies: `P `` Q` uimplies: `b supposing a` prop: `ℙ` nat: `ℕ` false: `False` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` all: `∀x:A. B[x]` top: `Top` and: `P ∧ Q` rleq: `x ≤ y` rnonneg: `rnonneg(x)` le: `A ≤ B` nat_plus: `ℕ+` subtype_rel: `A ⊆r B` decidable: `Dec(P)` or: `P ∨ Q` 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 ` cand: `A c∧ B` iff: `P `⇐⇒` Q` rev_uimplies: `rev_uimplies(P;Q)` rge: `x ≥ y` rev_implies: `P `` Q` less_than': `less_than'(a;b)` true: `True` subtract: `n - m` rless: `x < y` sq_exists: `∃x:{A| B[x]}` real: `ℝ` sq_stable: `SqStable(P)` squash: `↓T`
Lemmas referenced :  rleq_antisymmetry rmin_wf rnexp_wf rleq_wf int-to-real_wf req_witness real_wf nat_wf 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 nat_plus_properties nat_plus_wf rnexp_zero_lemma le_wf decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma rleq_weakening_equal rmin_lb 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 rmin_ub rnexp-nonneg rleq_functionality rmin_functionality rnexp_unroll rleq_functionality_wrt_implies rmul_functionality_wrt_rleq2 rmul_comm req_weakening rmul-rmin rmin-rleq not-rless rless_wf rmin_strict_ub not_wf rnexp-rleq-iff decidable__lt false_wf not-lt-2 not-equal-2 less-iff-le add_functionality_wrt_le add-associates zero-add add-zero le-add-cancel condition-implies-le add-commutes minus-add add-swap le-add-cancel2 minus-minus minus-one-mul minus-one-mul-top rleq_weakening_rless rmul_preserves_rleq2 sq_stable__less_than rless_transitivity1 rless_irreflexivity rnexp-rleq
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis independent_isectElimination natural_numberEquality sqequalRule lambdaEquality dependent_functionElimination independent_functionElimination isect_memberEquality because_Cache setElimination rename intWeakElimination dependent_pairFormation int_eqEquality intEquality voidElimination voidEquality independent_pairFormation computeAll productElimination independent_pairEquality applyEquality minusEquality axiomEquality equalityTransitivity equalitySymmetry dependent_set_memberEquality unionElimination inlFormation equalityElimination promote_hyp instantiate cumulativity productEquality addLevel impliesFunctionality addEquality imageMemberEquality baseClosed imageElimination

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[x,y:\mBbbR{}].    ((r0  \mleq{}  x)  {}\mRightarrow{}  (r0  \mleq{}  y)  {}\mRightarrow{}  (rmin(x\^{}n;y\^{}n)  =  rmin(x;y)\^{}n))

Date html generated: 2017_10_03-AM-08_46_20
Last ObjectModification: 2017_07_28-AM-07_32_31

Theory : reals

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