Nuprl Lemma : divides_mul
∀a,b:ℤ. ((a | b) ⇒ (∀n:ℤ. ((n * a) | (n * b))))
Proof
Definitions occuring in Statement :
divides: b | a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
multiply: n * m,
int: ℤ
Definitions unfolded in proof :
prop: ℙ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
divides: b | a,
exists: ∃x:A. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
subtype_rel: A ⊆r B
Lemmas referenced :
istype-int,
divides_wf,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf
Rules used in proof :
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
Error :universeIsType,
hypothesisEquality,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productElimination,
Error :dependent_pairFormation_alt,
promote_hyp,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
multiplyEquality,
unionElimination,
natural_numberEquality,
approximateComputation,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
Error :equalityIstype,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
sqequalBase
Latex:
\mforall{}a,b:\mBbbZ{}. ((a | b) {}\mRightarrow{} (\mforall{}n:\mBbbZ{}. ((n * a) | (n * b))))
Date html generated:
2019_06_20-PM-02_20_18
Last ObjectModification:
2019_01_12-AM-10_36_27
Theory : num_thy_1
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