Nuprl Lemma : rem_mul
∀[x,y:ℕ]. ∀[m:ℕ+]. ((x * y rem m) = ((x rem m) * (y rem m) 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,
nat: ℕ,
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,
all: ∀x:A. B[x],
implies: P ⇒ Q,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q)
Lemmas referenced :
false_wf,
int_term_value_mul_lemma,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
itermMultiply_wf,
itermAdd_wf,
intformle_wf,
intformnot_wf,
multiply-is-int-iff,
add-is-int-iff,
decidable__le,
nat_plus_subtype_nat,
divide_wf,
multiply_nat_wf,
add_nat_wf,
le_wf,
remainder_wf,
mul_bounds_1a,
rem_invariant,
add-commutes,
add-swap,
mul-commutes,
mul-swap,
add-associates,
mul-associates,
mul-distributes-right,
mul-distributes,
nat_wf,
nat_plus_wf,
iff_weakening_equal,
equal_wf,
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,
nequal_wf,
less_than_wf,
subtype_rel_sets,
div_rem_sum
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
applyEquality,
sqequalRule,
because_Cache,
lambdaEquality,
natural_numberEquality,
hypothesis,
intEquality,
independent_isectElimination,
setEquality,
lambdaFormation,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
equalityEquality,
remainderEquality,
multiplyEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
axiomEquality,
divideEquality,
dependent_set_memberEquality,
addEquality,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed
Latex:
\mforall{}[x,y:\mBbbN{}]. \mforall{}[m:\mBbbN{}\msupplus{}]. ((x * y rem m) = ((x rem m) * (y rem m) rem m))
Date html generated:
2016_05_15-PM-04_48_28
Last ObjectModification:
2016_01_16-AM-11_26_42
Theory : general
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