### Nuprl Lemma : rpositive-implies-rnonneg

`∀[x:ℝ]. (rpositive(x) `` rnonneg(x))`

Proof

Definitions occuring in Statement :  rnonneg: `rnonneg(x)` rpositive: `rpositive(x)` real: `ℝ` uall: `∀[x:A]. B[x]` implies: `P `` Q`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` implies: `P `` Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` prop: `ℙ` real: `ℝ` rnonneg: `rnonneg(x)` all: `∀x:A. B[x]` le: `A ≤ B` not: `¬A` false: `False` rpositive2: `rpositive2(x)` exists: `∃x:A. B[x]` rnonneg2: `rnonneg2(x)` nat_plus: `ℕ+` so_lambda: `λ2x.t[x]` int_upper: `{i...}` guard: `{T}` uimplies: `b supposing a` so_apply: `x[s]` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` subtype_rel: `A ⊆r B` rev_uimplies: `rev_uimplies(P;Q)` ge: `i ≥ j ` nat: `ℕ`
Lemmas referenced :  rnonneg-iff rpositive-iff rpositive_wf less_than'_wf nat_plus_wf real_wf int_upper_wf all_wf le_wf less_than_transitivity1 less_than_wf int_upper_properties nat_plus_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_wf mul_preserves_le nat_plus_subtype_nat mul_cancel_in_le mul-swap mul-commutes le_functionality le_weakening mul_bounds_1a int_upper_subtype_nat nat_wf nat_properties itermMultiply_wf itermConstant_wf intformeq_wf int_term_value_mul_lemma int_term_value_constant_lemma int_formula_prop_eq_lemma equal_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality productElimination independent_functionElimination because_Cache hypothesis setElimination rename sqequalRule lambdaEquality dependent_functionElimination independent_pairEquality voidElimination applyEquality minusEquality natural_numberEquality axiomEquality equalityTransitivity equalitySymmetry dependent_pairFormation multiplyEquality dependent_set_memberEquality independent_isectElimination unionElimination int_eqEquality intEquality isect_memberEquality voidEquality independent_pairFormation computeAll applyLambdaEquality

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

Date html generated: 2017_10_03-AM-08_23_41
Last ObjectModification: 2017_07_28-AM-07_23_00

Theory : reals

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