### Nuprl Lemma : int-rmul_wf

`∀[k:ℤ]. ∀[a:ℝ].  (k * a ∈ ℝ)`

Proof

Definitions occuring in Statement :  int-rmul: `k1 * a` real: `ℝ` uall: `∀[x:A]. B[x]` member: `t ∈ T` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` real: `ℝ` int-rmul: `k1 * a` has-value: `(a)↓` uimplies: `b supposing a` all: `∀x:A. B[x]` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` less_than: `a < b` less_than': `less_than'(a;b)` top: `Top` true: `True` squash: `↓T` not: `¬A` false: `False` nat_plus: `ℕ+` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` prop: `ℙ` bfalse: `ff` subtype_rel: `A ⊆r B` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` regular-int-seq: `k-regular-seq(f)` nat: `ℕ` le: `A ≤ B` int_lower: `{...i}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` absval: `|i|` subtract: `n - m`
Lemmas referenced :  value-type-has-value int-value-type lt_int_wf eqtt_to_assert assert_of_lt_int istype-top istype-void mul_nat_plus nat_plus_properties decidable__lt full-omega-unsat intformand_wf intformnot_wf intformless_wf itermConstant_wf itermMinus_wf itermVar_wf istype-int int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_minus_lemma int_term_value_var_lemma int_formula_prop_wf istype-less_than nat_plus_wf eqff_to_assert int_subtype_base bool_subtype_base bool_cases_sqequal subtype_base_sq bool_wf assert-bnot iff_weakening_uiff assert_wf less_than_wf regular-int-seq_wf real_wf mul_cancel_in_le absval_wf subtract_wf absval_nat_plus intformeq_wf int_formula_prop_eq_lemma le_wf squash_wf true_wf absval_mul subtype_rel_self iff_weakening_equal decidable__equal_int multiply-is-int-iff add-is-int-iff itermMultiply_wf itermSubtract_wf int_term_value_mul_lemma int_term_value_subtract_lemma istype-nat mul-swap mul-distributes equal_wf istype-universe add_functionality_wrt_eq absval_neg decidable__le intformle_wf int_formula_prop_le_lemma istype-le mul-commutes set_subtype_base left_mul_subtract_distrib mul_assoc absval_pos itermAdd_wf int_term_value_add_lemma minus-one-mul mul-associates zero-mul zero-add add-commutes
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut sqequalHypSubstitution setElimination thin rename sqequalRule callbyvalueReduce extract_by_obid isectElimination intEquality independent_isectElimination hypothesis hypothesisEquality dependent_set_memberEquality_alt closedConclusion natural_numberEquality because_Cache inhabitedIsType lambdaFormation_alt unionElimination equalityElimination productElimination lessCases axiomSqEquality isect_memberEquality_alt isectIsTypeImplies independent_pairFormation voidElimination imageMemberEquality baseClosed imageElimination independent_functionElimination lambdaEquality_alt minusEquality applyEquality dependent_functionElimination approximateComputation dependent_pairFormation_alt int_eqEquality universeIsType equalityTransitivity equalitySymmetry equalityIsType4 baseApply promote_hyp instantiate cumulativity equalityIsType1 axiomEquality multiplyEquality addEquality universeEquality hyp_replacement

Latex:
\mforall{}[k:\mBbbZ{}].  \mforall{}[a:\mBbbR{}].    (k  *  a  \mmember{}  \mBbbR{})

Date html generated: 2019_10_29-AM-09_32_16
Last ObjectModification: 2018_11_10-PM-01_33_05

Theory : reals

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