### Nuprl Lemma : rabs-nonneg

`∀[x:ℝ]. rnonneg(|x|)`

Proof

Definitions occuring in Statement :  rnonneg: `rnonneg(x)` rabs: `|x|` real: `ℝ` uall: `∀[x:A]. B[x]`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` implies: `P `` Q` rnonneg2: `rnonneg2(x)` all: `∀x:A. B[x]` rabs: `|x|` exists: `∃x:A. B[x]` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` prop: `ℙ` so_lambda: `λ2x.t[x]` int_upper: `{i...}` real: `ℝ` le: `A ≤ B` guard: `{T}` uimplies: `b supposing a` subtype_rel: `A ⊆r B` so_apply: `x[s]` rnonneg: `rnonneg(x)` not: `¬A` false: `False` decidable: `Dec(P)` or: `P ∨ Q` uiff: `uiff(P;Q)` top: `Top` satisfiable_int_formula: `satisfiable_int_formula(fmla)`
Lemmas referenced :  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 le-add-cancel zero-add add-commutes add_functionality_wrt_le not-lt-2 false_wf decidable__lt absval-non-neg real_wf less_than'_wf nat_plus_wf less_than_transitivity1 absval_wf le_wf all_wf int_upper_wf less_than_wf rabs_wf rnonneg-iff
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis productElimination independent_functionElimination lambdaFormation sqequalRule dependent_pairFormation dependent_set_memberEquality natural_numberEquality independent_pairFormation imageMemberEquality baseClosed setElimination rename lambdaEquality multiplyEquality minusEquality applyEquality because_Cache independent_isectElimination dependent_functionElimination independent_pairEquality voidElimination axiomEquality equalityTransitivity equalitySymmetry unionElimination isect_memberEquality voidEquality intEquality int_eqEquality computeAll

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

Date html generated: 2016_05_18-AM-07_02_41
Last ObjectModification: 2016_01_17-AM-01_49_48

Theory : reals

Home Index