### Nuprl Lemma : rpositive-iff

`∀[x:ℝ]. (rpositive(x) `⇐⇒` rpositive2(x))`

Proof

Definitions occuring in Statement :  rpositive2: `rpositive2(x)` rpositive: `rpositive(x)` real: `ℝ` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q`
Definitions unfolded in proof :  rpositive2: `rpositive2(x)` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` real: `ℝ` rev_implies: `P `` Q` exists: `∃x:A. B[x]` so_lambda: `λ2x.t[x]` nat_plus: `ℕ+` so_apply: `x[s]` rpositive: `rpositive(x)` sq_exists: `∃x:{A| B[x]}` all: `∀x:A. B[x]` uimplies: `b supposing a` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` le: `A ≤ B` decidable: `Dec(P)` or: `P ∨ Q` less_than: `a < b` squash: `↓T` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` regular-int-seq: `k-regular-seq(f)` subtype_rel: `A ⊆r B` nat: `ℕ` true: `True` less_than': `less_than'(a;b)`
Lemmas referenced :  rpositive_wf exists_wf nat_plus_wf all_wf le_wf real_wf rnonzero-lemma1 absval_ifthenelse lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int nat_plus_properties decidable__le less_than_wf satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_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 eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot less_than'_wf assert_wf bnot_wf not_wf bool_cases iff_transitivity iff_weakening_uiff assert_of_bnot mul_preserves_le nat_plus_subtype_nat multiply-is-int-iff int_subtype_base minus-is-int-iff false_wf mul_preserves_lt squash_wf true_wf absval_pos subtract_wf decidable__lt itermSubtract_wf itermMultiply_wf int_term_value_subtract_lemma int_term_value_mul_lemma mul_nat_plus itermAdd_wf int_term_value_add_lemma iff_weakening_equal mul_cancel_in_lt
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation independent_pairFormation lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis productElimination lambdaEquality functionEquality because_Cache multiplyEquality applyEquality dependent_pairFormation independent_isectElimination natural_numberEquality unionElimination equalityElimination equalityTransitivity equalitySymmetry dependent_functionElimination dependent_set_memberEquality imageElimination int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll promote_hyp instantiate cumulativity independent_functionElimination independent_pairEquality axiomEquality impliesFunctionality functionExtensionality baseClosed baseApply closedConclusion pointwiseFunctionality addEquality imageMemberEquality universeEquality dependent_set_memberFormation

Latex:
\mforall{}[x:\mBbbR{}].  (rpositive(x)  \mLeftarrow{}{}\mRightarrow{}  rpositive2(x))

Date html generated: 2017_10_03-AM-08_23_14
Last ObjectModification: 2017_07_28-AM-07_22_46

Theory : reals

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