Nuprl Lemma : exp-divides-exp2
∀x,y:ℤ. (x | y ⇐⇒ ∃n:ℕ+. (x^n | y^n))
Proof
Definitions occuring in Statement :
divides: b | a,
exp: i^n,
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
exp: i^n,
top: Top,
divides: b | a,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
false: False,
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T
Lemmas referenced :
exp-equal-minusone,
int_term_value_minus_lemma,
itermMinus_wf,
minus-is-int-iff,
int_formula_prop_less_lemma,
intformless_wf,
exp-equal-one,
nequal_wf,
exp_wf3,
mul_cancel_in_eq,
not_wf,
iff_weakening_equal,
exp-of-mul,
assoced_elim,
gcd-exp,
divides_transitivity,
gcd_wf,
equal_wf,
gcd_is_divisor_1,
false_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
multiply-is-int-iff,
nat_plus_properties,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
gcd_is_divisor_2,
divides-iff-gcd,
one-mul,
mul-commutes,
primrec1_lemma,
less_than_wf,
nat_plus_subtype_nat,
exp_wf2,
nat_plus_wf,
exists_wf,
divides_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
sqequalRule,
lambdaEquality,
applyEquality,
because_Cache,
intEquality,
dependent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality,
introduction,
imageMemberEquality,
baseClosed,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_functionElimination,
unionElimination,
instantiate,
cumulativity,
independent_isectElimination,
setElimination,
rename,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
baseApply,
closedConclusion,
int_eqEquality,
computeAll,
minusEquality,
multiplyEquality,
equalityEquality
Latex:
\mforall{}x,y:\mBbbZ{}. (x | y \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}\msupplus{}. (x\^{}n | y\^{}n))
Date html generated:
2016_05_15-PM-04_51_23
Last ObjectModification:
2016_01_16-AM-11_28_16
Theory : general
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