Nuprl Lemma : rem-mul
∀[a:ℤ]. ∀[n,m:ℤ-o]. ((a * n rem m * n) = ((a rem m) * n) ∈ ℤ)
Proof
Definitions occuring in Statement :
int_nzero: ℤ-o,
uall: ∀[x:A]. B[x],
remainder: n rem m,
multiply: n * m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_nzero: ℤ-o,
uimplies: b supposing a,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q)
Lemmas referenced :
false_wf,
int_term_value_add_lemma,
int_term_value_mul_lemma,
itermAdd_wf,
itermMultiply_wf,
multiply-is-int-iff,
add-is-int-iff,
decidable__equal_int,
mul-swap,
mul-distributes,
mul-commutes,
mul_preserves_eq,
nequal_wf,
equal_wf,
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,
satisfiable-full-omega-tt,
int_nzero_properties,
int_entire_a,
div_rem_sum,
int_nzero_wf,
div-mul-cancel
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
intEquality,
multiplyEquality,
setElimination,
rename,
dependent_set_memberEquality,
independent_isectElimination,
lambdaFormation,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
equalityTransitivity,
equalitySymmetry,
divideEquality,
because_Cache,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
productElimination
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[n,m:\mBbbZ{}\msupminus{}\msupzero{}]. ((a * n rem m * n) = ((a rem m) * n))
Date html generated:
2016_05_14-AM-07_24_55
Last ObjectModification:
2016_01_14-PM-10_01_39
Theory : int_2
Home
Index