Nuprl Lemma : prime_ideals_in_int_ring
∀i:ℕ+. (ℤ-rng-Prime(i) ⇐⇒ prime(i))
Proof
Definitions occuring in Statement :
int_ring: ℤ-rng,
rprime: r-Prime(u),
prime: prime(a),
nat_plus: ℕ+,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
int_ring: ℤ-rng,
rprime: r-Prime(u),
rng_one: 1,
pi2: snd(t),
pi1: fst(t),
rng_car: |r|,
rng_times: *,
infix_ap: x f y,
ring_divs: a | b in r,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
so_apply: x[s],
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
or: P ∨ Q,
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
prime: prime(a),
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
cand: A c∧ B,
assoced: a ~ b,
divides: b | a,
decidable: Dec(P),
guard: {T}
Lemmas referenced :
one_divs_any,
assoced_wf,
int_term_value_mul_lemma,
int_formula_prop_not_lemma,
itermMultiply_wf,
intformnot_wf,
decidable__equal_int,
divides_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_plus_properties,
nat_plus_wf,
prime_wf,
equal_wf,
or_wf,
int_subtype_base,
all_wf,
equal-wf-T-base,
exists_wf,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
productEquality,
cut,
lemma_by_obid,
isectElimination,
intEquality,
lambdaEquality,
multiplyEquality,
hypothesisEquality,
setElimination,
rename,
baseClosed,
hypothesis,
because_Cache,
functionEquality,
baseApply,
closedConclusion,
applyEquality,
independent_functionElimination,
voidElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidEquality,
computeAll,
introduction,
unionElimination,
inlFormation,
inrFormation
Latex:
\mforall{}i:\mBbbN{}\msupplus{}. (\mBbbZ{}-rng-Prime(i) \mLeftarrow{}{}\mRightarrow{} prime(i))
Date html generated:
2016_05_15-PM-00_26_25
Last ObjectModification:
2016_01_15-AM-08_52_21
Theory : rings_1
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