Nuprl Lemma : rem_mul2
∀[x,y:ℕ]. ∀[m:ℕ+]. ((x * y rem m) = ((x rem m) * y rem m) ∈ ℤ)
Proof
Definitions occuring in Statement :
nat_plus: ℕ+,
nat: ℕ,
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,
nequal: a ≠ b ∈ T ,
nat_plus: ℕ+,
nat: ℕ,
ge: i ≥ j ,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
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,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
rem_rem_to_rem,
iff_weakening_equal,
rem_mul,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
nat_plus_properties,
remainder_wf,
nat_wf,
nat_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
lemma_by_obid,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
because_Cache,
intEquality,
remainderEquality,
multiplyEquality,
setElimination,
rename,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
equalityEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_functionElimination
Latex:
\mforall{}[x,y:\mBbbN{}]. \mforall{}[m:\mBbbN{}\msupplus{}]. ((x * y rem m) = ((x rem m) * y rem m))
Date html generated:
2016_05_15-PM-04_48_36
Last ObjectModification:
2016_01_16-AM-11_25_41
Theory : general
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