Nuprl Lemma : residue-mul-assoc
∀[n:ℕ+]. ∀[a,b,c:{i:ℤ| CoPrime(n,i)} ]. (((ab mod n)c mod n) = (a(bc mod n) mod n) ∈ ℤ)
Proof
Definitions occuring in Statement :
residue-mul: (ai mod n),
coprime: CoPrime(a,b),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
residue-mul: (ai mod n),
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top
Lemmas referenced :
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
itermMultiply_wf,
intformeq_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
nat_plus_properties,
eqmod_inversion,
multiply_functionality_wrt_eqmod,
eqmod_weakening,
eqmod_functionality_wrt_eqmod,
int_mod_wf,
subtype_rel_set,
mod-eqmod,
coprime_wf,
set_wf,
int-subtype-int_mod,
modulus_wf_int_mod,
modulus-equal-iff-eqmod
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
multiplyEquality,
isectElimination,
hypothesisEquality,
setElimination,
rename,
hypothesis,
applyEquality,
sqequalRule,
because_Cache,
productElimination,
independent_functionElimination,
intEquality,
lambdaEquality,
isect_memberEquality,
axiomEquality,
independent_isectElimination,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[a,b,c:\{i:\mBbbZ{}| CoPrime(n,i)\} ]. (((ab mod n)c mod n) = (a(bc mod n) mod n))
Date html generated:
2016_05_15-PM-07_29_53
Last ObjectModification:
2016_01_16-AM-09_39_59
Theory : general
Home
Index