Nuprl Lemma : p-mul-int-cancelation-2
∀[p:{p:{2...}| prime(p)} ]. ∀[k:ℕ]. ∀[a,b:p-adics(p)].
(a = b ∈ p-adics(p)) supposing ((k(p) * a = k(p) * b ∈ p-adics(p)) and CoPrime(k,p))
Proof
Definitions occuring in Statement :
p-int: k(p),
p-mul: x * y,
p-adics: p-adics(p),
prime: prime(a),
coprime: CoPrime(a,b),
int_upper: {i...},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
natural_number: $n,
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],
prop: ℙ,
int_upper: {i...},
nat_plus: ℕ+,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
p-units: p-units(p),
top: Top,
p-int: k(p),
p-adics: p-adics(p),
less_than: a < b,
squash: ↓T,
int_seg: {i..j-},
sq_type: SQType(T),
guard: {T},
coprime: CoPrime(a,b),
gcd_p: GCD(a;b;y),
eqmod: a ≡ b mod m,
divides: b | a,
exists: ∃x:A. B[x],
cand: A c∧ B,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
crng: CRng,
rng: Rng,
p-adic-ring: ℤ(p),
ring_p: IsRing(T;plus;zero;neg;times;one),
rng_car: |r|,
pi1: fst(t),
rng_plus: +r,
pi2: snd(t),
rng_zero: 0,
rng_minus: -r,
rng_times: *,
rng_one: 1,
monoid_p: IsMonoid(T;op;id),
group_p: IsGroup(T;op;id;inv),
bilinear: BiLinear(T;pl;tm),
ident: Ident(T;op;id),
assoc: Assoc(T;op),
inverse: Inverse(T;op;id;inv),
infix_ap: x f y,
comm: Comm(T;op)
Lemmas referenced :
p-inv_wf,
equal_wf,
p-adics_wf,
p-mul_wf,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
p-int_wf,
coprime_wf,
nat_wf,
set_wf,
int_upper_wf,
prime_wf,
not_wf,
equal-wf-T-base,
int_seg_wf,
exp_wf2,
le_wf,
p-reduce-eqmod,
p-reduce_wf,
subtype_base_sq,
int_subtype_base,
nat_plus_wf,
set_subtype_base,
exp-positive,
exp1,
nat_properties,
int_upper_properties,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermMultiply_wf,
itermMinus_wf,
itermSubtract_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_minus_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
divides_reflexivity,
divisor_bound,
subtype_rel_set,
upper_subtype_nat,
intformle_wf,
int_formula_prop_le_lemma,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
p-adic-ring_wf,
crng_properties,
rng_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
isectElimination,
setElimination,
rename,
dependent_set_memberEquality,
because_Cache,
productElimination,
natural_numberEquality,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
sqequalRule,
applyEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
voidEquality,
intEquality,
imageMemberEquality,
baseClosed,
instantiate,
cumulativity,
dependent_pairFormation,
minusEquality,
approximateComputation,
int_eqEquality,
multiplyEquality,
applyLambdaEquality,
imageElimination,
universeEquality
Latex:
\mforall{}[p:\{p:\{2...\}| prime(p)\} ]. \mforall{}[k:\mBbbN{}]. \mforall{}[a,b:p-adics(p)].
(a = b) supposing ((k(p) * a = k(p) * b) and CoPrime(k,p))
Date html generated:
2018_05_21-PM-03_22_50
Last ObjectModification:
2018_05_19-AM-08_21_15
Theory : rings_1
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