Nuprl Lemma : coprime_prod
∀a,b1,b2:ℤ. (CoPrime(a,b1) ⇒ CoPrime(a,b2) ⇒ CoPrime(a,b1 * b2))
Proof
Definitions occuring in Statement :
coprime: CoPrime(a,b),
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,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
top: Top,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
subtract: n - m,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False
Lemmas referenced :
coprime_wf,
istype-int,
coprime_bezout_id,
istype-void,
add-associates,
minus-one-mul,
add-commutes,
add-swap,
add-mul-special,
zero-mul,
zero-add,
add_mono_wrt_eq,
subtract_wf,
int_subtype_base,
mul-distributes,
mul-associates,
mul-distributes-right,
mul-swap,
mul-commutes,
one-mul,
mul_functionality_wrt_eq,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf
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,
because_Cache,
multiplyEquality,
rename,
equalityTransitivity,
equalitySymmetry,
minusEquality,
Error :isect_memberEquality_alt,
voidElimination,
natural_numberEquality,
sqequalRule,
independent_isectElimination,
Error :dependent_pairFormation_alt,
addEquality,
Error :equalityIsType4,
applyEquality,
Error :productIsType,
unionElimination,
approximateComputation,
Error :lambdaEquality_alt,
int_eqEquality,
independent_pairFormation
Latex:
\mforall{}a,b1,b2:\mBbbZ{}. (CoPrime(a,b1) {}\mRightarrow{} CoPrime(a,b2) {}\mRightarrow{} CoPrime(a,b1 * b2))
Date html generated:
2019_06_20-PM-02_23_44
Last ObjectModification:
2018_10_03-AM-00_12_51
Theory : num_thy_1
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