### Nuprl Lemma : rpositive-rmul

`∀x,y:ℝ.  (rpositive(x) `` rpositive(y) `` rpositive(x * y))`

Proof

Definitions occuring in Statement :  rpositive: `rpositive(x)` rmul: `a * b` real: `ℝ` all: `∀x:A. B[x]` implies: `P `` Q`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` rmul: `a * b` has-value: `(a)↓` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat_plus: `ℕ+` subtype_rel: `A ⊆r B` int_upper: `{i...}` so_lambda: `λ2x.t[x]` real: `ℝ` so_apply: `x[s]` prop: `ℙ` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` rpositive2: `rpositive2(x)` exists: `∃x:A. B[x]` nat: `ℕ` true: `True` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` uimplies: `b supposing a` ifthenelse: `if b then t else f fi ` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` less_than: `a < b` squash: `↓T` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` cand: `A c∧ B` reg-seq-mul: `reg-seq-mul(x;y)` nequal: `a ≠ b ∈ T ` int_nzero: `ℤ-o` sq_stable: `SqStable(P)` rev_uimplies: `rev_uimplies(P;Q)` ge: `i ≥ j ` gt: `i > j`
Lemmas referenced :  real-has-value rpositive2_functionality accelerate_wf imax_wf canonical-bound_wf int_upper_wf all_wf nat_plus_wf le_wf absval_wf less_than_wf reg-seq-mul_wf2 reg-seq-mul_wf accelerate-bdd-diff rpositive2_wf rpositive-iff rmul_wf rpositive_wf real_wf nat_wf ifthenelse_wf le_int_wf bool_wf eqtt_to_assert assert_of_le_int add_nat_plus multiply_nat_wf subtype_rel_set int_upper_subtype_nat false_wf nat_plus_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf itermAdd_wf itermMultiply_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_mul_lemma int_formula_prop_eq_lemma int_formula_prop_wf equal_wf eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot add-is-int-iff multiply-is-int-iff squash_wf true_wf add_functionality_wrt_eq imax_unfold iff_weakening_equal imax_nat_plus multiply_nat_plus imax_ub decidable__le intformle_wf int_formula_prop_le_lemma mul_nat_plus mul_preserves_le nat_plus_subtype_nat mul_preserves_lt mul_cancel_in_le equal-wf-base mul-swap div_rem_sum2 nequal_wf left_mul_subtract_distrib rem_bounds_absval set_wf int_subtype_base decidable__equal_int sq_stable__less_than le_functionality le_weakening multiply_functionality_wrt_le sq_stable__le absval_pos neg_mul_arg_bounds gt_wf mul_bounds_1a subtract_wf itermSubtract_wf int_term_value_subtract_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut sqequalRule callbyvalueReduce introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis dependent_functionElimination dependent_set_memberEquality addEquality multiplyEquality natural_numberEquality applyEquality lambdaEquality setElimination rename setEquality because_Cache independent_functionElimination productElimination addLevel impliesFunctionality functionEquality intEquality unionElimination equalityElimination equalityTransitivity equalitySymmetry independent_isectElimination independent_pairFormation imageMemberEquality baseClosed applyLambdaEquality dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality computeAll promote_hyp instantiate cumulativity pointwiseFunctionality baseApply closedConclusion imageElimination universeEquality inlFormation inrFormation productEquality divideEquality remainderEquality

Latex:
\mforall{}x,y:\mBbbR{}.    (rpositive(x)  {}\mRightarrow{}  rpositive(y)  {}\mRightarrow{}  rpositive(x  *  y))

Date html generated: 2017_10_03-AM-08_23_29
Last ObjectModification: 2017_07_28-AM-07_22_53

Theory : reals

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