Nuprl Lemma : rem-eqmod
∀a:ℤ. ∀m:ℤ-o. ((a rem m) ≡ a mod m)
Proof
Definitions occuring in Statement :
eqmod: a ≡ b mod m,
int_nzero: ℤ-o,
all: ∀x:A. B[x],
remainder: n rem m,
int: ℤ
Definitions unfolded in proof :
uiff: uiff(P;Q),
or: P ∨ Q,
decidable: Dec(P),
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
implies: P ⇒ Q,
not: ¬A,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
exists: ∃x:A. B[x],
divides: b | a,
eqmod: a ≡ b mod m,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
int_nzero_wf,
subtract_wf,
equal_wf,
false_wf,
int_term_value_add_lemma,
int_term_value_minus_lemma,
int_term_value_mul_lemma,
int_term_value_subtract_lemma,
itermAdd_wf,
itermMinus_wf,
itermMultiply_wf,
itermSubtract_wf,
multiply-is-int-iff,
add-is-int-iff,
decidable__equal_int,
int_subtype_base,
equal-wf-base,
int_formula_prop_wf,
int_formula_prop_not_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformnot_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
full-omega-unsat,
int_nzero_properties,
div_rem_sum
Rules used in proof :
multiplyEquality,
remainderEquality,
productElimination,
closedConclusion,
baseApply,
promote_hyp,
equalitySymmetry,
equalityTransitivity,
pointwiseFunctionality,
unionElimination,
baseClosed,
applyEquality,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
hypothesis,
because_Cache,
rename,
setElimination,
divideEquality,
minusEquality,
dependent_pairFormation,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a:\mBbbZ{}. \mforall{}m:\mBbbZ{}\msupminus{}\msupzero{}. ((a rem m) \mequiv{} a mod m)
Date html generated:
2018_05_21-PM-00_55_53
Last ObjectModification:
2018_01_09-PM-03_49_50
Theory : num_thy_1
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