Nuprl Lemma : coprime_divisors_prod
∀a1,a2,b:ℤ. (CoPrime(a1,a2) ⇒ (a1 | b) ⇒ (a2 | b) ⇒ ((a1 * a2) | b))
Proof
Definitions occuring in Statement :
coprime: CoPrime(a,b),
divides: b | a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
multiply: n * m,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
divides: b | a,
top: Top,
uimplies: b supposing a,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False
Lemmas referenced :
divides_wf,
coprime_wf,
istype-int,
coprime_bezout_id,
istype-void,
mul_preserves_eq,
mul-distributes,
mul-swap,
mul-commutes,
one-mul,
equal_wf,
squash_wf,
true_wf,
add_functionality_wrt_eq,
subtype_rel_self,
iff_weakening_equal,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermMultiply_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
Error :inhabitedIsType,
dependent_functionElimination,
productElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
Error :isect_memberEquality_alt,
voidElimination,
independent_isectElimination,
sqequalRule,
intEquality,
multiplyEquality,
because_Cache,
natural_numberEquality,
applyEquality,
Error :lambdaEquality_alt,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
instantiate,
Error :dependent_pairFormation_alt,
addEquality,
unionElimination,
approximateComputation,
int_eqEquality,
independent_pairFormation,
Error :equalityIsType4
Latex:
\mforall{}a1,a2,b:\mBbbZ{}. (CoPrime(a1,a2) {}\mRightarrow{} (a1 | b) {}\mRightarrow{} (a2 | b) {}\mRightarrow{} ((a1 * a2) | b))
Date html generated:
2019_06_20-PM-02_23_48
Last ObjectModification:
2018_10_03-AM-00_12_58
Theory : num_thy_1
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