Nuprl Lemma : divides-mul
∀x,y,z:ℤ. (((z | x) ∨ (z | y)) ⇒ (z | (x * y)))
Proof
Definitions occuring in Statement :
divides: b | a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
multiply: n * m,
int: ℤ
Definitions unfolded in proof :
divides: b | a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top
Lemmas referenced :
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
itermMultiply_wf,
intformeq_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
equal_wf,
exists_wf,
or_wf,
int_subtype_base,
subtype_base_sq
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
thin,
productElimination,
cut,
instantiate,
lemma_by_obid,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
lambdaEquality,
multiplyEquality,
dependent_pairFormation,
because_Cache,
natural_numberEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}x,y,z:\mBbbZ{}. (((z | x) \mvee{} (z | y)) {}\mRightarrow{} (z | (x * y)))
Date html generated:
2016_05_14-PM-09_23_38
Last ObjectModification:
2016_01_14-PM-11_33_51
Theory : num_thy_1
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