Nuprl Lemma : divides_iff_div_exact
∀a:ℤ. ∀n:ℤ-o. (n | a ⇐⇒ ((a ÷ n) * n) = a ∈ ℤ)
Proof
Definitions occuring in Statement :
divides: b | a,
int_nzero: ℤ-o,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
divide: n ÷ m,
multiply: n * m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_nzero: ℤ-o,
prop: ℙ,
rev_implies: P ⇐ Q,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
decidable: Dec(P),
or: P ∨ Q,
divides: b | a
Lemmas referenced :
divides_wf,
int_nzero_properties,
full-omega-unsat,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_subtype_base,
int_nzero_wf,
divides_iff_rem_zero,
add_mono_wrt_eq,
div_rem_sum,
decidable__equal_int,
add-is-int-iff,
multiply-is-int-iff,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
int_term_value_mul_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
independent_pairFormation,
cut,
hypothesis,
Error :universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
Error :equalityIsType4,
Error :inhabitedIsType,
multiplyEquality,
divideEquality,
because_Cache,
independent_functionElimination,
voidElimination,
independent_isectElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
natural_numberEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
sqequalRule,
applyEquality,
productElimination,
equalityTransitivity,
equalitySymmetry,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed
Latex:
\mforall{}a:\mBbbZ{}. \mforall{}n:\mBbbZ{}\msupminus{}\msupzero{}. (n | a \mLeftarrow{}{}\mRightarrow{} ((a \mdiv{} n) * n) = a)
Date html generated:
2019_06_20-PM-02_20_31
Last ObjectModification:
2018_10_03-AM-00_35_42
Theory : num_thy_1
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