Nuprl Lemma : int_pi_det_fun_wf
∀[i:ℤ]. ((i)ℤ-det-fun ∈ detach_fun(|ℤ-rng|;(i)ℤ-rng))
Proof
Definitions occuring in Statement :
int_pi_det_fun: (i)ℤ-det-fun,
int_ring: ℤ-rng,
princ_ideal: (a)r,
rng_car: |r|,
detach_fun: detach_fun(T;A),
uall: ∀[x:A]. B[x],
member: t ∈ T,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_ring: ℤ-rng,
rng_car: |r|,
pi1: fst(t),
int_pi_det_fun: (i)ℤ-det-fun,
princ_ideal: (a)r,
detach_fun: detach_fun(T;A),
rng_times: *,
pi2: snd(t),
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
top: Top,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
squash: ↓T,
true: True
Lemmas referenced :
eq_int_wf,
bool_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
eqtt_to_assert,
assert_of_eq_int,
all_wf,
iff_wf,
exists_wf,
equal-wf-base,
int_subtype_base,
assert_wf,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
div_rem_sum,
nequal_wf,
mul-commutes,
squash_wf,
true_wf,
divide-exact,
iff_weakening_equal,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
multiply-is-int-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
dependent_set_memberEquality,
lambdaEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
hypothesis,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
because_Cache,
voidElimination,
remainderEquality,
intEquality,
applyEquality,
baseApply,
closedConclusion,
baseClosed,
functionExtensionality,
axiomEquality,
independent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
addLevel,
impliesFunctionality,
divideEquality,
multiplyEquality,
imageElimination,
universeEquality,
equalityUniverse,
levelHypothesis,
imageMemberEquality,
hyp_replacement,
applyLambdaEquality,
pointwiseFunctionality,
rename
Latex:
\mforall{}[i:\mBbbZ{}]. ((i)\mBbbZ{}-det-fun \mmember{} detach\_fun(|\mBbbZ{}-rng|;(i)\mBbbZ{}-rng))
Date html generated:
2017_10_01-AM-08_18_40
Last ObjectModification:
2017_02_28-PM-02_03_32
Theory : rings_1
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