Nuprl Lemma : int-to-ring-add

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


Proof



Definitions occuring in Statement :  int-to-ring: int-to-ring(r;n),  rng: Rng,  rng_plus: +r,  rng_car: |r|,  uall: ∀[x:A]. B[x],  infix_ap: x f y,  add: 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],  decidable: Dec(P),  or: P ∨ Q,  uimplies: b supposing a,  sq_type: SQType(T),  implies: P ⇒ Q,  guard: {T},  squash: ↓T,  prop: ℙ,  rng: Rng,  top: Top,  infix_ap: x f y,  true: True,  subtype_rel: A ⊆r B,  iff: P ⇐⇒ Q,  and: P ∧ Q,  rev_implies: P ⇐ Q,  int-to-ring: int-to-ring(r;n),  bool: 𝔹,  unit: Unit,  it: ⋅,  btrue: tt,  uiff: uiff(P;Q),  ifthenelse: if b then t else f fi ,  bfalse: ff,  exists: ∃x:A. B[x],  bnot: ¬bb,  assert: ↑b,  false: False,  not: ¬A,  satisfiable_int_formula: satisfiable_int_formula(fmla),  nat: ℕ,  le: A ≤ B,  less_than': less_than'(a;b),  ge: i ≥ j ,  less_than: a < b,  subtract: n - m
Lemmas referenced :  rng_wf,  decidable__equal_int,  subtype_base_sq,  int_subtype_base,  equal_wf,  squash_wf,  true_wf,  rng_car_wf,  int-to-ring-zero,  rng_zero_wf,  rng_plus_wf,  int-to-ring-minus-one,  rng_one_wf,  subtype_rel_self,  iff_weakening_equal,  rng_plus_inv,  lt_int_wf,  bool_wf,  eqtt_to_assert,  assert_of_lt_int,  eqff_to_assert,  bool_cases_sqequal,  bool_subtype_base,  assert-bnot,  less_than_wf,  full-omega-unsat,  intformand_wf,  intformless_wf,  itermAdd_wf,  itermVar_wf,  itermConstant_wf,  intformnot_wf,  int_formula_prop_and_lemma,  int_formula_prop_less_lemma,  int_term_value_add_lemma,  int_term_value_var_lemma,  int_term_value_constant_lemma,  int_formula_prop_not_lemma,  int_formula_prop_wf,  intformeq_wf,  int_formula_prop_eq_lemma,  rng_nat_op_add,  decidable__le,  intformle_wf,  int_formula_prop_le_lemma,  le_wf,  false_wf,  infix_ap_wf,  rng_nat_op_wf,  rng_nat_op_one,  itermMinus_wf,  int_term_value_minus_lemma,  rng_minus_wf,  rng_minus_over_plus,  rng_plus_assoc,  rng_plus_comm,  rng_plus_inv_assoc,  decidable__lt,  subtract_wf,  subtract-add-cancel,  int-to-ring_wf,  rng_plus_zero,  nat_properties,  ge_wf,  absval_wf,  itermSubtract_wf,  int_term_value_subtract_lemma,  nat_wf,  add_nat_wf,  add-is-int-iff,  absval_unfold,  top_wf,  add-associates,  add-swap,  add-commutes,  zero-add,  rng_plus_ac_1
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,  dependent_functionElimination,  minusEquality,  natural_numberEquality,  unionElimination,  instantiate,  cumulativity,  independent_isectElimination,  equalityTransitivity,  equalitySymmetry,  independent_functionElimination,  applyEquality,  lambdaEquality,  imageElimination,  universeEquality,  setElimination,  rename,  voidElimination,  voidEquality,  imageMemberEquality,  baseClosed,  productElimination,  addEquality,  equalityElimination,  dependent_pairFormation,  promote_hyp,  approximateComputation,  int_eqEquality,  independent_pairFormation,  dependent_set_memberEquality,  equalityUniverse,  levelHypothesis,  hyp_replacement,  applyLambdaEquality,  intWeakElimination,  pointwiseFunctionality,  baseApply,  closedConclusion,  lessCases,  sqequalAxiom
Latex:
\mforall{}[r:Rng].  \mforall{}[a1,a2:\mBbbZ{}].    (int-to-ring(r;a1  +  a2)  =  (int-to-ring(r;a1)  +r  int-to-ring(r;a2)))


Date html generated: 2018_05_21-PM-03_14_58
Last ObjectModification: 2018_05_19-AM-08_08_28

Theory : rings_1


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