Nuprl Lemma : rng_nexp-int
∀[n:ℕ]. ∀[a:ℤ]. ((a ↑ℤ-rng n) = a^n ∈ ℤ)
Proof
Definitions occuring in Statement :
rng_nexp: e ↑r n,
int_ring: ℤ-rng,
exp: i^n,
nat: ℕ,
uall: ∀[x:A]. B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
exp: i^n,
rng_nexp: e ↑r n,
mon_nat_op: n ⋅ e,
mul_mon_of_rng: r↓xmn,
grp_op: *,
pi2: snd(t),
pi1: fst(t),
grp_id: e,
int_ring: ℤ-rng,
rng_times: *,
rng_one: 1,
nat_op: n x(op;id) e,
uall: ∀[x:A]. B[x],
member: t ∈ T,
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,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
itop: Π(op,id) lb ≤ i < ub. E[i],
ycomb: Y,
lt_int: i <z j,
infix_ap: x f y,
subtract: n - m,
ifthenelse: if b then t else f fi ,
bfalse: ff,
decidable: Dec(P),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
nequal: a ≠ b ∈ T ,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
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,
primrec0_lemma,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
primrec-unroll,
lt_int_wf,
bool_wf,
uiff_transitivity,
equal-wf-base,
int_subtype_base,
assert_wf,
eqtt_to_assert,
assert_of_lt_int,
eq_int_wf,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
mul-commutes,
itop_wf,
int_seg_wf,
squash_wf,
true_wf,
primrec_wf,
le_wf,
iff_weakening_equal,
le_int_wf,
bnot_wf,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
unionElimination,
equalityElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
productElimination,
promote_hyp,
instantiate,
cumulativity,
multiplyEquality,
imageElimination,
universeEquality,
dependent_set_memberEquality,
imageMemberEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a:\mBbbZ{}]. ((a \muparrow{}\mBbbZ{}-rng n) = a\^{}n)
Date html generated:
2017_10_01-AM-08_18_52
Last ObjectModification:
2017_02_28-PM-02_03_52
Theory : rings_1
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