Nuprl Lemma : divides-iff-gcd
∀x,y:ℤ. (x | y ⇐⇒ gcd(y;x) = x ∈ ℤ)
Proof
Definitions occuring in Statement :
divides: b | a,
gcd: gcd(a;b),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
int: ℤ,
equal: s = t ∈ T
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,
subtype_rel: A ⊆r B,
gcd: gcd(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
not: ¬A,
divides: b | a,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
squash: ↓T,
true: True
Lemmas referenced :
divides_wf,
equal-wf-base,
int_subtype_base,
eq_int_wf,
bool_wf,
assert_wf,
bnot_wf,
not_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal_wf,
subtype_base_sq,
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,
divides_iff_rem_zero,
nequal_wf,
gcd_is_divisor_1,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
intEquality,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
because_Cache,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
equalityElimination,
independent_functionElimination,
productElimination,
independent_isectElimination,
impliesFunctionality,
dependent_functionElimination,
instantiate,
cumulativity,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
remainderEquality,
dependent_set_memberEquality,
imageElimination,
imageMemberEquality,
universeEquality
Latex:
\mforall{}x,y:\mBbbZ{}. (x | y \mLeftarrow{}{}\mRightarrow{} gcd(y;x) = x)
Date html generated:
2018_05_21-PM-01_10_19
Last ObjectModification:
2018_01_28-PM-02_03_46
Theory : num_thy_1
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