Nuprl Lemma : gcd-prime
∀[p:ℕ]. ∀[n:{1..p-}]. (gcd(n;p) = 1 ∈ ℤ) supposing prime(p)
Proof
Definitions occuring in Statement :
prime: prime(a),
gcd: gcd(a;b),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
and: P ∧ Q,
int_seg: {i..j-},
gcd_p: GCD(a;b;y),
prop: ℙ,
iff: P ⇐⇒ Q,
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
guard: {T},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top
Lemmas referenced :
int_seg_subtype_nat_plus,
le_wf,
divisor_bound,
int_term_value_minus_lemma,
itermMinus_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__le,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_properties,
int_seg_properties,
false_wf,
int_seg_subtype_nat,
gcd-positive,
assoced_wf,
equal_wf,
or_wf,
assoced_elim,
gcd_wf,
nat_wf,
prime_wf,
int_seg_wf,
gcd_sat_gcd_p,
prime_elim
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
because_Cache,
isectElimination,
natural_numberEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
addLevel,
orFunctionality,
intEquality,
minusEquality,
promote_hyp,
applyEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
voidElimination,
voidEquality,
computeAll,
dependent_set_memberEquality
Latex:
\mforall{}[p:\mBbbN{}]. \mforall{}[n:\{1..p\msupminus{}\}]. (gcd(n;p) = 1) supposing prime(p)
Date html generated:
2016_05_15-PM-06_21_09
Last ObjectModification:
2016_01_16-PM-00_54_33
Theory : general
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