Nuprl Lemma : int_ring_wf
ℤ-rng ∈ IntegDom{i}
Proof
Definitions occuring in Statement :
int_ring: ℤ-rng,
integ_dom: IntegDom{i},
member: t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
integ_dom: IntegDom{i},
crng: CRng,
rng: Rng,
int_ring: ℤ-rng,
rng_sig: RngSig,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
not: ¬A,
prop: ℙ,
rev_implies: P ⇐ Q,
nequal: a ≠ b ∈ T ,
ring_p: IsRing(T;plus;zero;neg;times;one),
bilinear: BiLinear(T;pl;tm),
monoid_p: IsMonoid(T;op;id),
group_p: IsGroup(T;op;id;inv),
ident: Ident(T;op;id),
assoc: Assoc(T;op),
inverse: Inverse(T;op;id;inv),
rng_car: |r|,
pi1: fst(t),
rng_plus: +r,
pi2: snd(t),
rng_zero: 0,
rng_minus: -r,
rng_times: *,
rng_one: 1,
infix_ap: x f y,
cand: A c∧ B,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
comm: Comm(T;op),
integ_dom_p: IsIntegDom(r),
true: True,
sq_type: SQType(T),
guard: {T}
Lemmas referenced :
eq_int_wf,
le_int_wf,
bool_wf,
uiff_transitivity,
equal-wf-base,
int_subtype_base,
assert_wf,
eqtt_to_assert,
assert_of_eq_int,
it_wf,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
unit_wf2,
equal_wf,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
itermConstant_wf,
int_term_value_constant_lemma,
itermMinus_wf,
int_term_value_minus_lemma,
itermMultiply_wf,
int_term_value_mul_lemma,
ring_p_wf,
rng_car_wf,
rng_plus_wf,
rng_zero_wf,
rng_minus_wf,
rng_times_wf,
rng_one_wf,
comm_wf,
subtype_base_sq,
true_wf,
int_entire,
integ_dom_p_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_set_memberEquality,
dependent_pairEquality,
intEquality,
lambdaEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
addEquality,
natural_numberEquality,
minusEquality,
multiplyEquality,
lambdaFormation,
unionElimination,
equalityElimination,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
independent_functionElimination,
because_Cache,
productElimination,
independent_isectElimination,
inrEquality,
independent_pairFormation,
impliesFunctionality,
equalityTransitivity,
equalitySymmetry,
inlEquality,
divideEquality,
dependent_functionElimination,
functionEquality,
unionEquality,
productEquality,
cumulativity,
isect_memberFormation,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
axiomEquality,
independent_pairEquality,
setElimination,
rename,
addLevel,
instantiate
Latex:
\mBbbZ{}-rng \mmember{} IntegDom\{i\}
Date html generated:
2017_10_01-AM-08_18_36
Last ObjectModification:
2017_02_28-PM-02_03_29
Theory : rings_1
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