Nuprl Lemma : gcd-exp
∀x,y:ℤ. ∀n:ℕ. (gcd(x^n;y^n) ~ gcd(x;y)^n)
Proof
Definitions occuring in Statement :
assoced: a ~ b,
exp: i^n,
gcd: gcd(a;b),
nat: ℕ,
all: ∀x:A. B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
gcd_p: GCD(a;b;y),
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
top: Top,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A
Lemmas referenced :
divides-mul,
divides-add,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermMultiply_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
nat_properties,
iff_weakening_equal,
exp-of-mul,
add_functionality_wrt_eq,
true_wf,
squash_wf,
one-mul,
mul-commutes,
mul-swap,
mul-distributes,
int_subtype_base,
subtype_base_sq,
coprime_bezout_id,
coprime-exp,
equal_wf,
coprime_wf,
exists_wf,
gcd-property,
nat_wf,
divides_wf,
and_wf,
gcd_is_divisor_2,
gcd_is_divisor_1,
exp-divides,
gcd_sat_pred,
gcd_wf,
exp_wf2,
gcd_unique
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
independent_pairFormation,
because_Cache,
productElimination,
intEquality,
sqequalRule,
lambdaEquality,
multiplyEquality,
equalityEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
equalityUniverse,
levelHypothesis,
isect_memberEquality,
voidElimination,
voidEquality,
instantiate,
cumulativity,
independent_isectElimination,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
setElimination,
rename,
unionElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
inrFormation
Latex:
\mforall{}x,y:\mBbbZ{}. \mforall{}n:\mBbbN{}. (gcd(x\^{}n;y\^{}n) \msim{} gcd(x;y)\^{}n)
Date html generated:
2016_05_15-PM-04_50_22
Last ObjectModification:
2016_01_16-AM-11_27_17
Theory : general
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