### Nuprl Lemma : int-rmul-req

`∀[k:ℤ]. ∀[a:ℝ].  (k * a = (r(k) * a))`

Proof

Definitions occuring in Statement :  int-rmul: `k1 * a` req: `x = y` rmul: `a * b` int-to-real: `r(n)` real: `ℝ` uall: `∀[x:A]. B[x]` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` implies: `P `` Q` subtype_rel: `A ⊆r B` real: `ℝ` int-to-real: `r(n)` reg-seq-mul: `reg-seq-mul(x;y)` int-rmul: `k1 * a` bdd-diff: `bdd-diff(f;g)` has-value: `(a)↓` exists: `∃x:A. B[x]` nat: `ℕ` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` prop: `ℙ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` guard: `{T}` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` so_lambda: `λ2x.t[x]` less_than: `a < b` true: `True` squash: `↓T` nat_plus: `ℕ+` nequal: `a ≠ b ∈ T ` so_apply: `x[s]` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` assert: `↑b` ifthenelse: `if b then t else f fi ` bnot: `¬bb` sq_type: `SQType(T)` bfalse: `ff` btrue: `tt` it: `⋅` unit: `Unit` bool: `𝔹` int_nzero: `ℤ-o` regular-int-seq: `k-regular-seq(f)` sq_stable: `SqStable(P)` subtract: `n - m` rev_uimplies: `rev_uimplies(P;Q)` absval: `|i|`
Lemmas referenced :  req-iff-bdd-diff int-rmul_wf rmul_wf int-to-real_wf req_witness real_wf reg-seq-mul_wf value-type-has-value int-value-type absval_wf mul-non-neg1 false_wf decidable__le nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformnot_wf itermAdd_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_not_lemma int_term_value_add_lemma int_formula_prop_wf le_wf nat_plus_wf all_wf subtract_wf less_than_wf mul_nat_plus nat_plus_properties intformeq_wf itermMultiply_wf intformless_wf int_formula_prop_eq_lemma int_term_value_mul_lemma int_formula_prop_less_lemma equal-wf-base bdd-diff_functionality bdd-diff_weakening rmul-bdd-diff-reg-seq-mul assert-bnot bool_subtype_base subtype_base_sq bool_cases_sqequal equal_wf eqff_to_assert top_wf assert_of_lt_int eqtt_to_assert bool_wf lt_int_wf iff_weakening_equal subtype_rel_self absval_mul true_wf squash_wf equal-wf-T-base absval_nat_plus nat_wf int_term_value_minus_lemma itermMinus_wf decidable__lt mul_cancel_in_le nequal_wf div-cancel2 decidable__equal_int int_subtype_base sq_stable__le mul-associates mul-distributes minus-one-mul mul-swap one-mul add-commutes absval_sym add_functionality_wrt_eq minus-add mul-commutes le_functionality le_weakening absval_unfold add-is-int-iff set_subtype_base multiply-is-int-iff nat_plus_subtype_nat absval_pos zero-add zero-mul zero-div-rem
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis productElimination independent_isectElimination independent_functionElimination sqequalRule isect_memberEquality because_Cache intEquality applyEquality lambdaEquality setElimination rename callbyvalueReduce dependent_pairFormation dependent_set_memberEquality multiplyEquality natural_numberEquality addEquality independent_pairFormation lambdaFormation dependent_functionElimination unionElimination equalityTransitivity equalitySymmetry applyLambdaEquality approximateComputation int_eqEquality voidElimination voidEquality lessCases baseClosed imageMemberEquality axiomSqEquality imageElimination minusEquality divideEquality baseApply closedConclusion cumulativity instantiate promote_hyp equalityElimination universeEquality

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

Date html generated: 2019_10_29-AM-09_32_26
Last ObjectModification: 2018_08_23-PM-01_45_09

Theory : reals

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