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) k(p) b ∈ p-adics(p)) and CoPrime(k,p))


Definitions occuring in Statement :  p-int: k(p) p-mul: y p-adics: p-adics(p) prime: prime(a) coprime: CoPrime(a,b) int_upper: {i...} nat: uimplies: supposing a uall: [x:A]. B[x] set: {x:A| B[x]}  natural_number: $n equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a all: x:A. B[x] prop: int_upper: {i...} nat_plus: + nat: le: A ≤ B and: P ∧ Q decidable: Dec(P) or: P ∨ Q iff: ⇐⇒ Q not: ¬A rev_implies:  Q implies:  Q false: False uiff: uiff(P;Q) subtype_rel: A ⊆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 ≡ mod m divides: a exists: x:A. B[x] cand: c∧ B ge: i ≥  satisfiable_int_formula: satisfiable_int_formula(fmla) crng: CRng rng: Rng p-adic-ring: (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: 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

\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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