Nuprl Lemma : modulus-equal
∀x,y:ℤ. ∀m:ℕ+. ((x mod m) = (y mod m) ∈ ℤ ⇐⇒ m | (x - y))
Proof
Definitions occuring in Statement :
divides: b | a,
modulus: a mod n,
nat_plus: ℕ+,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
subtract: n - m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
guard: {T},
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
sq_type: SQType(T),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
divides: b | a,
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
nat_plus_wf,
mod_bounds,
div_floor_mod_sum,
modulus_wf,
subtype_rel_sets,
less_than_wf,
nequal_wf,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
equal_wf,
subtype_base_sq,
equal-wf-base-T,
div_floor_wf,
le_wf,
divides_wf,
subtract_wf,
decidable__equal_int,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
decidable__lt,
mul_preserves_le,
nat_plus_subtype_nat,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
intEquality,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
applyEquality,
sqequalRule,
lambdaEquality,
natural_numberEquality,
independent_isectElimination,
setElimination,
rename,
setEquality,
applyLambdaEquality,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
baseClosed,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
cumulativity,
productElimination,
addEquality,
multiplyEquality,
productEquality,
unionElimination,
baseApply,
closedConclusion,
minusEquality
Latex:
\mforall{}x,y:\mBbbZ{}. \mforall{}m:\mBbbN{}\msupplus{}. ((x mod m) = (y mod m) \mLeftarrow{}{}\mRightarrow{} m | (x - y))
Date html generated:
2017_04_17-AM-09_43_03
Last ObjectModification:
2017_02_27-PM-05_38_08
Theory : num_thy_1
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