Nuprl Lemma : mprime_divs_list_el
∀g:IAbMonoid. ∀p:|g|. (IsPrime(p) ⇒ (∀as:|g| List. ((p | (Π as)) ⇒ (∃i:ℕ||as||. (p | as[i])))))
Proof
Definitions occuring in Statement :
mprime: IsPrime(a),
mdivides: b | a,
mon_reduce: mon_reduce,
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
natural_number: $n,
iabmonoid: IAbMonoid,
grp_car: |g|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
iabmonoid: IAbMonoid,
imon: IMonoid,
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
less_than: a < b,
squash: ↓T,
so_apply: x[s],
mon_reduce: mon_reduce,
mprime: IsPrime(a),
munit: g-unit(u),
infix_ap: x f y,
le: A ≤ B,
less_than': less_than'(a;b),
nat_plus: ℕ+,
true: True,
uiff: uiff(P;Q),
select: L[n],
cons: [a / b],
subtract: n - m,
ge: i ≥ j ,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
mdivides_wf,
mon_reduce_wf,
exists_wf,
int_seg_wf,
length_wf,
grp_car_wf,
select_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
reduce_nil_lemma,
nil_wf,
reduce_cons_lemma,
cons_wf,
list_wf,
mprime_wf,
iabmonoid_wf,
length_of_cons_lemma,
istype-false,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_properties,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
le_wf,
add-member-int_seg2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
non_neg_length,
squash_wf,
true_wf,
grp_sig_wf,
select_cons_tl,
subtype_rel_self,
iff_weakening_equal,
add-associates,
add-swap,
add-commutes,
zero-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality_alt,
functionEquality,
dependent_functionElimination,
setElimination,
rename,
hypothesis,
hypothesisEquality,
natural_numberEquality,
independent_isectElimination,
productElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
imageElimination,
functionIsType,
productIsType,
inhabitedIsType,
dependent_set_memberEquality_alt,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
equalityIsType1,
addEquality,
applyEquality,
instantiate,
universeEquality
Latex:
\mforall{}g:IAbMonoid. \mforall{}p:|g|. (IsPrime(p) {}\mRightarrow{} (\mforall{}as:|g| List. ((p | (\mPi{} as)) {}\mRightarrow{} (\mexists{}i:\mBbbN{}||as||. (p | as[i])))))
Date html generated:
2019_10_16-PM-01_05_48
Last ObjectModification:
2018_10_08-PM-00_11_58
Theory : factor_1
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