### Nuprl Lemma : p-mul-int-cancelation-1

`∀[p:{2...}]. ∀[k:ℕ]. ∀[a,b:p-adics(p)].  ((p^k(p) * a = p^k(p) * b ∈ p-adics(p)) `` (a = b ∈ p-adics(p)))`

Proof

Definitions occuring in Statement :  p-int: `k(p)` p-mul: `x * y` p-adics: `p-adics(p)` exp: `i^n` int_upper: `{i...}` nat: `ℕ` uall: `∀[x:A]. B[x]` implies: `P `` Q` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` implies: `P `` Q` all: `∀x:A. B[x]` nat_plus: `ℕ+` nat: `ℕ` int_upper: `{i...}` le: `A ≤ B` and: `P ∧ Q` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` not: `¬A` rev_implies: `P `` Q` false: `False` uiff: `uiff(P;Q)` uimplies: `b supposing a` subtype_rel: `A ⊆r B` less_than': `less_than'(a;b)` true: `True` prop: `ℙ` rev_uimplies: `rev_uimplies(P;Q)` p-int: `k(p)` p-mul: `x * y` int_seg: `{i..j-}` p-adics: `p-adics(p)` guard: `{T}` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` eqmod: `a ≡ b mod m` divides: `b | a` int_nzero: `ℤ-o` nequal: `a ≠ b ∈ T ` lelt: `i ≤ j < k`
Lemmas referenced :  p-adics-equal decidable__lt istype-false not-lt-2 add_functionality_wrt_le add-commutes zero-add le-add-cancel less_than_wf nat_plus_wf p-mul_wf p-int_wf exp_wf2 p-adics_wf nat_wf int_upper_wf nat_plus_subtype_nat p-reduce_wf eqmod_functionality_wrt_eqmod eqmod_transitivity p-reduce-eqmod multiply_functionality_wrt_eqmod eqmod_weakening nat_plus_properties nat_properties int_upper_properties full-omega-unsat intformand_wf intformnot_wf intformless_wf itermConstant_wf itermAdd_wf itermVar_wf intformle_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_le_lemma int_formula_prop_wf p-adic-property decidable__le le_wf eqmod_inversion subtype_base_sq set_subtype_base int_subtype_base exp-positive exp_add subtract_wf mul_cancel_in_eq exp_wf3 subtype_rel_sets nequal_wf intformeq_wf int_formula_prop_eq_lemma exp_wf_nat_plus int_seg_properties decidable__equal_int itermMultiply_wf itermSubtract_wf int_term_value_mul_lemma int_term_value_subtract_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut lambdaFormation_alt hypothesis extract_by_obid sqequalHypSubstitution dependent_functionElimination thin dependent_set_memberEquality_alt because_Cache setElimination rename productElimination natural_numberEquality hypothesisEquality unionElimination independent_pairFormation voidElimination independent_functionElimination independent_isectElimination isectElimination sqequalRule applyEquality universeIsType equalityIsType1 inhabitedIsType lambdaEquality_alt axiomEquality functionIsTypeImplies isect_memberEquality_alt isectIsTypeImplies multiplyEquality equalityTransitivity equalitySymmetry addEquality approximateComputation dependent_pairFormation_alt int_eqEquality intEquality instantiate cumulativity closedConclusion equalityIsType4 baseApply baseClosed setIsType

Latex:
\mforall{}[p:\{2...\}].  \mforall{}[k:\mBbbN{}].  \mforall{}[a,b:p-adics(p)].    ((p\^{}k(p)  *  a  =  p\^{}k(p)  *  b)  {}\mRightarrow{}  (a  =  b))

Date html generated: 2019_10_15-AM-10_35_20
Last ObjectModification: 2018_10_15-PM-08_54_21

Theory : rings_1

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