### Nuprl Lemma : int-to-ring-mul

`∀[r:Rng]. ∀[a1,a2:ℤ].  (int-to-ring(r;a1 * a2) = (int-to-ring(r;a1) * int-to-ring(r;a2)) ∈ |r|)`

Proof

Definitions occuring in Statement :  int-to-ring: `int-to-ring(r;n)` rng: `Rng` rng_times: `*` rng_car: `|r|` uall: `∀[x:A]. B[x]` infix_ap: `x f y` multiply: `n * m` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` subtype_rel: `A ⊆r B` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` le: `A ≤ B` less_than': `less_than'(a;b)` uiff: `uiff(P;Q)` squash: `↓T` rng: `Rng` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` sq_type: `SQType(T)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` less_than: `a < b` bfalse: `ff` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` subtract: `n - m` infix_ap: `x f y`
Lemmas referenced :  rng_wf nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf absval_wf decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma nat_wf add_nat_wf false_wf le_wf add-is-int-iff itermAdd_wf intformeq_wf int_term_value_add_lemma int_formula_prop_eq_lemma equal_wf decidable__lt squash_wf true_wf rng_car_wf infix_ap_wf rng_times_wf int-to-ring_wf iff_weakening_equal subtype_base_sq int_subtype_base lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf decidable__equal_int equal-wf-base not_wf eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot itermMinus_wf int_term_value_minus_lemma absval_unfold zero-mul rng_times_zero int-to-ring-zero itermMultiply_wf int_term_value_mul_lemma rng_plus_wf int-to-ring-add int-to-ring-one rng_one_wf rng_times_over_plus rng_times_one minus-zero int-to-ring-minus int-to-ring-minus-one rng_minus_wf rng_times_over_minus
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis intEquality sqequalRule sqequalHypSubstitution isect_memberEquality isectElimination thin hypothesisEquality axiomEquality because_Cache extract_by_obid lambdaFormation setElimination rename intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality dependent_functionElimination voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination applyEquality equalityTransitivity equalitySymmetry applyLambdaEquality unionElimination dependent_set_memberEquality addEquality pointwiseFunctionality promote_hyp baseApply closedConclusion baseClosed productElimination imageElimination universeEquality imageMemberEquality instantiate cumulativity minusEquality equalityElimination lessCases sqequalAxiom inlFormation inrFormation multiplyEquality

Latex:
\mforall{}[r:Rng].  \mforall{}[a1,a2:\mBbbZ{}].    (int-to-ring(r;a1  *  a2)  =  (int-to-ring(r;a1)  *  int-to-ring(r;a2)))

Date html generated: 2017_10_01-AM-08_19_22
Last ObjectModification: 2017_02_28-PM-02_04_37

Theory : rings_1

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