Nuprl Lemma : rnonneg-iff

`∀[x:ℝ]. (rnonneg(x) `⇐⇒` rnonneg2(x))`

Proof

Definitions occuring in Statement :  rnonneg2: `rnonneg2(x)` rnonneg: `rnonneg(x)` real: `ℝ` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q`
Definitions unfolded in proof :  rnonneg2: `rnonneg2(x)` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` all: `∀x:A. B[x]` member: `t ∈ T` prop: `ℙ` real: `ℝ` rev_implies: `P `` Q` rnonneg: `rnonneg(x)` le: `A ≤ B` not: `¬A` false: `False` so_lambda: `λ2x.t[x]` nat_plus: `ℕ+` int_upper: `{i...}` guard: `{T}` uimplies: `b supposing a` so_apply: `x[s]` exists: `∃x:A. B[x]` subtype_rel: `A ⊆r B` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` rev_uimplies: `rev_uimplies(P;Q)` ge: `i ≥ j ` sq_stable: `SqStable(P)` regular-int-seq: `k-regular-seq(f)` nat: `ℕ` less_than': `less_than'(a;b)` uiff: `uiff(P;Q)` squash: `↓T` pi1: `fst(t)` subtract: `n - m` less_than: `a < b` true: `True`
Lemmas referenced :  mul_preserves_lt imax_strict_ub mul_nat_plus imax_ub imax_wf add-swap add-commutes mul-commutes mul-swap minus-one-mul minus-add mul-associates mul-distributes equal_wf int_term_value_add_lemma itermAdd_wf subtract-is-int-iff add-is-int-iff multiply-is-int-iff int_subtype_base minus-is-int-iff int_upper_subtype_nat add_nat_wf false_wf mul_bounds_1a subtract_wf absval_ubound int_formula_prop_less_lemma intformless_wf decidable__lt subtype_rel_sets sq_stable__le le_weakening le_functionality int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_term_value_mul_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermConstant_wf itermMultiply_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le nat_plus_properties int_upper_properties nat_plus_subtype_nat mul_preserves_le real_wf less_than_wf less_than_transitivity1 le_wf int_upper_wf exists_wf all_wf less_than'_wf rnonneg_wf nat_plus_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation independent_pairFormation lambdaFormation cut lemma_by_obid hypothesis sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality introduction lambdaEquality dependent_functionElimination productElimination independent_pairEquality voidElimination applyEquality minusEquality natural_numberEquality axiomEquality equalityTransitivity equalitySymmetry because_Cache multiplyEquality dependent_set_memberEquality independent_isectElimination dependent_pairFormation unionElimination int_eqEquality intEquality isect_memberEquality voidEquality computeAll promote_hyp addEquality independent_functionElimination setEquality imageMemberEquality baseClosed imageElimination baseApply closedConclusion pointwiseFunctionality inrFormation inlFormation

Latex:
\mforall{}[x:\mBbbR{}].  (rnonneg(x)  \mLeftarrow{}{}\mRightarrow{}  rnonneg2(x))

Date html generated: 2016_05_18-AM-07_01_40
Last ObjectModification: 2016_01_17-AM-01_49_41

Theory : reals

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