### Nuprl Lemma : rmul-int

`∀[a,b:ℤ].  ((r(a) * r(b)) = r(a * b))`

Proof

Definitions occuring in Statement :  req: `x = y` rmul: `a * b` int-to-real: `r(n)` uall: `∀[x:A]. B[x]` multiply: `n * m` 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)` bdd-diff: `bdd-diff(f;g)` 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]` so_lambda: `λ2x.t[x]` nat_plus: `ℕ+` nequal: `a ≠ b ∈ T ` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` so_apply: `x[s]` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` decidable: `Dec(P)` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` int_nzero: `ℤ-o` absval: `|i|` subtract: `n - m`
Lemmas referenced :  zero-mul mul-distributes-right add-commutes mul-associates mul-commutes mul-swap minus-one-mul nat_wf nequal_wf div-cancel int_formula_prop_not_lemma intformnot_wf decidable__equal_int int_subtype_base subtype_base_sq bdd-diff_weakening rmul-bdd-diff-reg-seq-mul bdd-diff_functionality equal_wf int_formula_prop_wf int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_constant_lemma int_term_value_mul_lemma int_formula_prop_eq_lemma int_formula_prop_and_lemma intformless_wf itermVar_wf itermConstant_wf itermMultiply_wf intformeq_wf intformand_wf satisfiable-full-omega-tt nat_properties nat_plus_properties subtract_wf absval_wf all_wf nat_plus_wf le_wf false_wf reg-seq-mul_wf real_wf req_witness int-to-real_wf rmul_wf req-iff-bdd-diff
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis multiplyEquality productElimination independent_isectElimination independent_functionElimination intEquality sqequalRule isect_memberEquality because_Cache applyEquality lambdaEquality setElimination rename dependent_pairFormation dependent_set_memberEquality natural_numberEquality independent_pairFormation lambdaFormation divideEquality int_eqEquality dependent_functionElimination voidElimination voidEquality computeAll instantiate unionElimination equalityTransitivity equalitySymmetry

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

Date html generated: 2016_05_18-AM-06_51_58
Last ObjectModification: 2016_01_17-AM-01_47_12

Theory : reals

Home Index