Nuprl Lemma : assoced-prime
∀p,q:ℤ. ((p ~ q) ⇒ prime(p) ⇒ prime(q))
Proof
Definitions occuring in Statement :
prime: prime(a),
assoced: a ~ b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
prop: ℙ,
prime: prime(a),
cand: A c∧ B,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
subtype_rel: A ⊆r B,
assoced: a ~ b,
rev_implies: P ⇐ Q
Lemmas referenced :
assoced_elim,
subtype_base_sq,
int_subtype_base,
prime_wf,
assoced_wf,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformnot_wf,
itermMinus_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_term_value_minus_lemma,
int_formula_prop_wf,
equal-wf-base,
divides_wf,
divides_invar_1,
minus-minus,
divides_invar_2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
approximateComputation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
applyEquality,
baseClosed,
multiplyEquality,
promote_hyp,
minusEquality,
inlFormation,
inrFormation
Latex:
\mforall{}p,q:\mBbbZ{}. ((p \msim{} q) {}\mRightarrow{} prime(p) {}\mRightarrow{} prime(q))
Date html generated:
2019_06_20-PM-02_23_05
Last ObjectModification:
2018_09_22-PM-05_52_04
Theory : num_thy_1
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