Nuprl Lemma : prime_elim
∀p:ℤ. (prime(p) ⇒ ((¬(p = 0 ∈ ℤ)) ∧ (¬(p ~ 1)) ∧ (∀a:ℤ. ((a | p) ⇒ ((a ~ 1) ∨ (a ~ p))))))
Proof
Definitions occuring in Statement :
prime: prime(a),
assoced: a ~ b,
divides: b | a,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
atomic: atomic(a),
and: P ∧ Q,
not: ¬A,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
int_nzero: ℤ-o,
so_apply: x[s],
or: P ∨ Q,
reducible: reducible(a),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
divides: b | a,
exists: ∃x:A. B[x],
decidable: Dec(P),
squash: ↓T,
true: True,
guard: {T},
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
nequal: a ≠ b ∈ T
Lemmas referenced :
prime_imp_atomic,
equal-wf-base,
int_subtype_base,
assoced_wf,
divides_wf,
prime_wf,
not_wf,
exists_wf,
int_nzero_wf,
equal-wf-base-T,
all_wf,
or_wf,
decidable__not,
decidable__assoced,
iff_transitivity,
iff_weakening_uiff,
not_over_exists,
not_over_and,
dneg_elim_a,
decidable__equal_int,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
nequal_wf,
assoced_weakening,
mul-commutes,
one-mul,
assoced_functionality_wrt_assoced,
multiply_functionality_wrt_assoced,
assoced_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
independent_pairFormation,
productElimination,
independent_functionElimination,
voidElimination,
intEquality,
applyEquality,
sqequalRule,
baseClosed,
natural_numberEquality,
lambdaEquality,
productEquality,
setElimination,
rename,
multiplyEquality,
because_Cache,
dependent_functionElimination,
allFunctionality,
addLevel,
orFunctionality,
unionElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
equalityUniverse,
levelHypothesis,
imageMemberEquality,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
dependent_set_memberEquality,
inlFormation,
inrFormation
Latex:
\mforall{}p:\mBbbZ{}. (prime(p) {}\mRightarrow{} ((\mneg{}(p = 0)) \mwedge{} (\mneg{}(p \msim{} 1)) \mwedge{} (\mforall{}a:\mBbbZ{}. ((a | p) {}\mRightarrow{} ((a \msim{} 1) \mvee{} (a \msim{} p))))))
Date html generated:
2017_04_17-AM-09_42_27
Last ObjectModification:
2017_02_27-PM-05_37_32
Theory : num_thy_1
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