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,
true: True,
top: Top,
nat_plus: ℕ+,
squash: ↓T,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
false: False,
uiff: uiff(P;Q),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ
Lemmas referenced :
div_rem_sum,
nat_plus_inc_int_nzero,
nat_plus_wf,
istype-nat,
remainder_wfa,
istype-void,
divide_wfa,
equal_wf,
iff_weakening_equal,
mul-distributes,
mul-distributes-right,
mul-associates,
add-associates,
mul-swap,
mul-commutes,
add-swap,
add-commutes,
rem_invariant,
mul_bounds_1a,
remainder_wf,
istype-le,
add_nat_wf,
multiply_nat_wf,
divide_wf,
nat_plus_subtype_nat,
nat_properties,
nat_plus_properties,
decidable__le,
add-is-int-iff,
multiply-is-int-iff,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
intformeq_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
Error :universeIsType,
Error :isect_memberEquality_alt,
axiomEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
intEquality,
because_Cache,
natural_numberEquality,
multiplyEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
Error :lambdaEquality_alt,
imageElimination,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
Error :dependent_set_memberEquality_alt,
addEquality,
Error :lambdaFormation_alt,
applyLambdaEquality,
dependent_functionElimination,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
Error :equalityIstype
Latex:
\mforall{}[x,y:\mBbbN{}]. \mforall{}[m:\mBbbN{}\msupplus{}]. ((x * y rem m) = ((x rem m) * (y rem m) rem m))
Date html generated:
2019_06_20-PM-02_32_01
Last ObjectModification:
2019_03_06-AM-10_53_18
Theory : num_thy_1
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