Nuprl Lemma : modulus-idempotent
∀x:ℤ. ∀m:ℕ+. (((x mod m) mod m) = (x mod m) ∈ ℤ)
Proof
Definitions occuring in Statement :
modulus: a mod n,
nat_plus: ℕ+,
all: ∀x:A. B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
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,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B
Lemmas referenced :
mod_bounds,
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,
nat_wf,
modulus_base,
lelt_wf,
le_wf,
equal_wf,
nat_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
intEquality,
because_Cache,
lambdaEquality,
natural_numberEquality,
hypothesis,
independent_isectElimination,
setElimination,
rename,
setEquality,
applyLambdaEquality,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
baseClosed,
independent_functionElimination,
productElimination,
dependent_set_memberEquality,
productEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}x:\mBbbZ{}. \mforall{}m:\mBbbN{}\msupplus{}. (((x mod m) mod m) = (x mod m))
Date html generated:
2017_04_17-AM-09_42_56
Last ObjectModification:
2017_02_27-PM-05_37_38
Theory : num_thy_1
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