### Nuprl Lemma : rmax-nonneg

`∀[x,y:ℝ].  rnonneg(rmax(x;y)) supposing rnonneg(x) ∨ rnonneg(y)`

Proof

Definitions occuring in Statement :  rnonneg: `rnonneg(x)` rmax: `rmax(x;y)` real: `ℝ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` or: `P ∨ Q`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` implies: `P `` Q` rnonneg: `rnonneg(x)` le: `A ≤ B` and: `P ∧ Q` not: `¬A` false: `False` subtype_rel: `A ⊆r B` real: `ℝ` prop: `ℙ` or: `P ∨ Q` rnonneg2: `rnonneg2(x)` exists: `∃x:A. B[x]` rmax: `rmax(x;y)` squash: `↓T` nat_plus: `ℕ+` true: `True` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` int_upper: `{i...}` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` so_lambda: `λ2x.t[x]` so_apply: `x[s]` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top`
Lemmas referenced :  less_than'_wf rmax_wf real_wf nat_plus_wf or_wf rnonneg_wf le_wf squash_wf true_wf imax_unfold iff_weakening_equal le_int_wf less_than_transitivity1 less_than_wf bool_wf eqtt_to_assert assert_of_le_int eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot int_upper_wf all_wf rnonneg2_wf rnonneg-iff mul_preserves_le nat_plus_subtype_nat int_upper_properties nat_plus_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermMultiply_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_mul_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf mul_cancel_in_le multiply-is-int-iff false_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin sqequalHypSubstitution dependent_functionElimination hypothesisEquality hypothesis independent_functionElimination sqequalRule lambdaEquality productElimination independent_pairEquality because_Cache extract_by_obid isectElimination applyEquality setElimination rename minusEquality natural_numberEquality axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality voidElimination lambdaFormation unionElimination dependent_pairFormation imageElimination intEquality multiplyEquality imageMemberEquality baseClosed universeEquality independent_isectElimination dependent_set_memberEquality equalityElimination promote_hyp instantiate cumulativity addLevel impliesFunctionality orFunctionality orLevelFunctionality functionEquality int_eqEquality voidEquality independent_pairFormation computeAll pointwiseFunctionality baseApply closedConclusion

Latex:
\mforall{}[x,y:\mBbbR{}].    rnonneg(rmax(x;y))  supposing  rnonneg(x)  \mvee{}  rnonneg(y)

Date html generated: 2017_10_03-AM-08_24_26
Last ObjectModification: 2017_07_28-AM-07_23_20

Theory : reals

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