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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