Nuprl Lemma : gcd-non-zero
∀a,b:ℤ.  ((a ≠ 0 ∨ b ≠ 0) 
⇒ gcd(a;b) ≠ 0)
Proof
Definitions occuring in Statement : 
gcd: gcd(a;b)
, 
all: ∀x:A. B[x]
, 
nequal: a ≠ b ∈ T 
, 
implies: P 
⇒ Q
, 
or: P ∨ Q
, 
natural_number: $n
, 
int: ℤ
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
nequal: a ≠ b ∈ T 
, 
not: ¬A
, 
false: False
, 
member: t ∈ T
, 
exists: ∃x:A. B[x]
, 
and: P ∧ Q
, 
uall: ∀[x:A]. B[x]
, 
uimplies: b supposing a
, 
sq_type: SQType(T)
, 
guard: {T}
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
top: Top
, 
prop: ℙ
Lemmas referenced : 
nequal_wf, 
or_wf, 
gcd_wf, 
equal_wf, 
int_formula_prop_wf, 
int_formula_prop_not_lemma, 
int_formula_prop_or_lemma, 
int_term_value_constant_lemma, 
int_term_value_mul_lemma, 
int_term_value_var_lemma, 
int_formula_prop_eq_lemma, 
int_formula_prop_and_lemma, 
intformnot_wf, 
intformor_wf, 
itermConstant_wf, 
itermMultiply_wf, 
itermVar_wf, 
intformeq_wf, 
intformand_wf, 
satisfiable-full-omega-tt, 
int_subtype_base, 
subtype_base_sq, 
gcd-property
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
cut, 
thin, 
lemma_by_obid, 
sqequalHypSubstitution, 
dependent_functionElimination, 
hypothesisEquality, 
productElimination, 
instantiate, 
isectElimination, 
cumulativity, 
intEquality, 
independent_isectElimination, 
hypothesis, 
equalityTransitivity, 
equalitySymmetry, 
independent_functionElimination, 
natural_numberEquality, 
dependent_pairFormation, 
lambdaEquality, 
int_eqEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
sqequalRule, 
independent_pairFormation, 
computeAll, 
because_Cache
Latex:
\mforall{}a,b:\mBbbZ{}.    ((a  \mneq{}  0  \mvee{}  b  \mneq{}  0)  {}\mRightarrow{}  gcd(a;b)  \mneq{}  0)
Date html generated:
2016_05_14-PM-09_24_31
Last ObjectModification:
2016_01_14-PM-11_32_58
Theory : num_thy_1
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