Nuprl Lemma : equipollent-primes
ℕ ~ {p:ℕ| prime(p)}
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
prime: prime(a),
equipollent: A ~ B,
nat: ℕ,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_apply: x[s],
implies: P ⇒ Q,
int_upper: {i...},
subtype_rel: A ⊆r B,
ge: i ≥ j ,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
nat_plus: ℕ+,
guard: {T},
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
sq_exists: ∃x:{A| B[x]},
cand: A c∧ B,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
sq_stable: SqStable(P),
squash: ↓T,
int_seg: {i..j-},
lelt: i ≤ j < k,
divides: b | a,
mul-list: Π(ns) ,
reduce: reduce(f;k;as),
list_ind: list_ind,
prime: prime(a),
assoced: a ~ b
Lemmas referenced :
one_divs_any,
int_term_value_subtract_lemma,
int_term_value_mul_lemma,
itermSubtract_wf,
itermMultiply_wf,
decidable__equal_int,
int_upper_properties,
subtract_wf,
nat_plus_wf,
equal_wf,
subtype_rel_list,
mul-list_wf,
lelt_wf,
decidable__lt,
divides-fact,
sq_stable__le,
sq_stable_from_decidable,
false_wf,
int_upper_subtype_nat,
subtype_rel_set,
product_subtype_list,
and_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
mul_list_nil_lemma,
list-cases,
int_upper_wf,
set_wf,
sq_stable__equal,
int_formula_prop_wf,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
itermAdd_wf,
intformle_wf,
intformnot_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformand_wf,
satisfiable-full-omega-tt,
less_than_wf,
nat_plus_properties,
le_wf,
decidable__le,
nat_properties,
fact_wf,
prime-factors,
decidable__prime,
nat_wf,
prime_wf,
equipollent-nat-decidable-subset
Rules used in proof :
cut,
lemma_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
sqequalRule,
lambdaEquality,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
independent_functionElimination,
lambdaFormation,
because_Cache,
dependent_set_memberEquality,
addEquality,
natural_numberEquality,
applyEquality,
unionElimination,
equalityTransitivity,
equalitySymmetry,
setEquality,
intEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
promote_hyp,
hypothesis_subsumption,
productElimination,
introduction,
imageMemberEquality,
baseClosed,
imageElimination,
multiplyEquality
Latex:
\mBbbN{} \msim{} \{p:\mBbbN{}| prime(p)\}
Date html generated:
2016_05_15-PM-05_28_44
Last ObjectModification:
2016_01_16-PM-00_31_20
Theory : general
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