### 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:
(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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