Nuprl Lemma : coprime_divides_prod
∀a1,a2,b:ℤ. ((b | (a1 * a2)) ⇒ CoPrime(b,a1) ⇒ (b | a2))
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 :
coprime: CoPrime(a,b),
sq_type: SQType(T),
guard: {T},
subtract: n - m,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
top: Top,
subtype_rel: A ⊆r B,
divides: b | a,
prop: ℙ,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
subtype_base_sq,
one-mul,
mul-commutes,
minus-one-mul,
add-associates,
equal-wf-base,
int_formula_prop_wf,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
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,
itermAdd_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
itermMultiply_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__equal_int,
mul-swap,
istype-void,
mul-distributes,
int_subtype_base,
istype-int,
divides_wf,
coprime_wf,
coprime_bezout_id
Rules used in proof :
minusEquality,
equalityTransitivity,
cumulativity,
instantiate,
promote_hyp,
intEquality,
applyLambdaEquality,
hyp_replacement,
independent_pairFormation,
int_eqEquality,
Error :lambdaEquality_alt,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
unionElimination,
because_Cache,
voidElimination,
Error :isect_memberEquality_alt,
equalitySymmetry,
sqequalBase,
baseClosed,
closedConclusion,
baseApply,
sqequalRule,
applyEquality,
Error :equalityIstype,
addEquality,
Error :dependent_pairFormation_alt,
Error :inhabitedIsType,
multiplyEquality,
isectElimination,
Error :universeIsType,
hypothesis,
independent_functionElimination,
productElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
Error :lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a1,a2,b:\mBbbZ{}. ((b | (a1 * a2)) {}\mRightarrow{} CoPrime(b,a1) {}\mRightarrow{} (b | a2))
Date html generated:
2019_06_20-PM-02_23_51
Last ObjectModification:
2019_06_18-PM-10_47_49
Theory : num_thy_1
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