### Nuprl Lemma : not-rpositive

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

Proof

Definitions occuring in Statement :  rnonneg: `rnonneg(x)` rpositive: `rpositive(x)` rminus: `-(x)` real: `ℝ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` not: `¬A`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` real: `ℝ` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` implies: `P `` Q` rpositive: `rpositive(x)` sq_exists: `∃x:{A| B[x]}` nat_plus: `ℕ+` prop: `ℙ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` and: `P ∧ Q` rnonneg: `rnonneg(x)` le: `A ≤ B` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` rnonneg2: `rnonneg2(x)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` so_lambda: `λ2x.t[x]` int_upper: `{i...}` guard: `{T}` so_apply: `x[s]` rminus: `-(x)`
Lemmas referenced :  int_term_value_minus_lemma itermMinus_wf nat_plus_subtype_nat mul_preserves_le int_term_value_mul_lemma itermMultiply_wf int_upper_properties subtype_rel_sets less_than_transitivity1 le_wf all_wf int_upper_wf mul_nat_plus rnonneg-iff rpositive_wf not_wf real_wf rminus_wf less_than'_wf nat_plus_wf int_formula_prop_wf int_formula_prop_le_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma intformle_wf itermVar_wf itermConstant_wf intformless_wf intformnot_wf intformand_wf satisfiable-full-omega-tt less_than_wf decidable__lt nat_plus_properties decidable__le
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation lemma_by_obid sqequalHypSubstitution dependent_functionElimination thin applyEquality setElimination rename hypothesisEquality natural_numberEquality hypothesis unionElimination independent_functionElimination dependent_set_memberFormation isectElimination dependent_set_memberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll productElimination independent_pairEquality because_Cache minusEquality axiomEquality equalityTransitivity equalitySymmetry imageMemberEquality baseClosed multiplyEquality setEquality

Latex:
\mforall{}[x:\mBbbR{}].  rnonneg(-(x))  supposing  \mneg{}rpositive(x)

Date html generated: 2016_05_18-AM-07_13_02
Last ObjectModification: 2016_01_17-AM-01_52_39

Theory : reals

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