### Nuprl Lemma : rmax-positive

`∀x,y:ℝ.  ((rpositive(x) ∨ rpositive(y)) `` rpositive(rmax(x;y)))`

Proof

Definitions occuring in Statement :  rpositive: `rpositive(x)` rmax: `rmax(x;y)` real: `ℝ` all: `∀x:A. B[x]` implies: `P `` Q` or: `P ∨ Q`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` or: `P ∨ Q` rpositive2: `rpositive2(x)` exists: `∃x:A. B[x]` member: `t ∈ T` rmax: `rmax(x;y)` squash: `↓T` uall: `∀[x:A]. B[x]` prop: `ℙ` nat_plus: `ℕ+` real: `ℝ` true: `True` subtype_rel: `A ⊆r B` uimplies: `b supposing a` guard: `{T}` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` 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` false: `False` not: `¬A` so_lambda: `λ2x.t[x]` so_apply: `x[s]` le: `A ≤ B` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top`
Lemmas referenced :  le_wf squash_wf true_wf imax_unfold iff_weakening_equal le_int_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 nat_plus_wf all_wf rmax_wf or_wf rpositive2_wf rpositive-iff rpositive_wf real_wf mul_preserves_le nat_plus_subtype_nat nat_plus_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermVar_wf itermMultiply_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_term_value_mul_lemma int_formula_prop_wf mul_cancel_in_le less_than_wf multiply-is-int-iff false_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut sqequalHypSubstitution unionElimination thin productElimination dependent_pairFormation hypothesisEquality hypothesis dependent_functionElimination independent_functionElimination sqequalRule applyEquality lambdaEquality imageElimination introduction extract_by_obid isectElimination equalityTransitivity equalitySymmetry intEquality setElimination rename multiplyEquality because_Cache natural_numberEquality imageMemberEquality baseClosed universeEquality independent_isectElimination equalityElimination promote_hyp instantiate cumulativity voidElimination functionEquality addLevel impliesFunctionality orFunctionality orLevelFunctionality int_eqEquality isect_memberEquality voidEquality independent_pairFormation computeAll dependent_set_memberEquality pointwiseFunctionality baseApply closedConclusion

Latex:
\mforall{}x,y:\mBbbR{}.    ((rpositive(x)  \mvee{}  rpositive(y))  {}\mRightarrow{}  rpositive(rmax(x;y)))

Date html generated: 2017_10_03-AM-08_24_20
Last ObjectModification: 2017_07_28-AM-07_23_16

Theory : reals

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