Nuprl Lemma : p-shift-mul

`∀[p:ℕ+]. ∀[a:p-adics(p)]. ∀[k:ℕ+].  p^k(p) * p-shift(p;a;k) = a ∈ p-adics(p) supposing (a k) = 0 ∈ ℤ`

Proof

Definitions occuring in Statement :  p-shift: `p-shift(p;a;k)` p-int: `k(p)` p-mul: `x * y` p-adics: `p-adics(p)` exp: `i^n` nat_plus: `ℕ+` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` apply: `f a` 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` prop: `ℙ` p-adics: `p-adics(p)` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` nat_plus: `ℕ+` all: `∀x:A. B[x]` uiff: `uiff(P;Q)` and: `P ∧ Q` rev_uimplies: `rev_uimplies(P;Q)` p-shift: `p-shift(p;a;k)` p-int: `k(p)` p-mul: `x * y` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` int_nzero: `ℤ-o` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nequal: `a ≠ b ∈ T ` nat: `ℕ` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` int_upper: `{i...}` sq_type: `SQType(T)` guard: `{T}` eqmod: `a ≡ b mod m` divides: `b | a` lelt: `i ≤ j < k`
Lemmas referenced :  equal-wf-T-base int_seg_wf exp_wf2 nat_plus_wf p-adics_wf p-adics-equal p-mul_wf p-int_wf nat_plus_subtype_nat p-shift_wf p-reduce_wf nat_plus_properties decidable__lt full-omega-unsat intformand_wf intformnot_wf intformless_wf itermConstant_wf itermAdd_wf itermVar_wf int_formula_prop_and_lemma 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_wf less_than_wf exp_step mul_nzero subtype_rel_sets nequal_wf intformeq_wf int_formula_prop_eq_lemma equal-wf-base int_subtype_base exp_wf3 subtract_wf decidable__le intformle_wf itermSubtract_wf int_formula_prop_le_lemma int_term_value_subtract_lemma le_wf eqmod_functionality_wrt_eqmod p-reduce-eqmod eqmod_weakening multiply_functionality_wrt_eqmod p-adic-property subtype_base_sq set_subtype_base lelt_wf decidable__equal_int multiply-is-int-iff subtract-is-int-iff itermMultiply_wf int_term_value_mul_lemma false_wf exp_wf_nat_plus int_seg_properties div-cancel2
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis extract_by_obid sqequalHypSubstitution isectElimination thin intEquality applyEquality setElimination rename hypothesisEquality lambdaEquality natural_numberEquality because_Cache sqequalRule baseClosed isect_memberEquality axiomEquality equalityTransitivity equalitySymmetry dependent_functionElimination independent_isectElimination productElimination lambdaFormation multiplyEquality divideEquality dependent_set_memberEquality addEquality unionElimination approximateComputation independent_functionElimination dependent_pairFormation int_eqEquality voidElimination voidEquality independent_pairFormation setEquality instantiate cumulativity universeEquality pointwiseFunctionality promote_hyp baseApply closedConclusion applyLambdaEquality

Latex:
\mforall{}[p:\mBbbN{}\msupplus{}].  \mforall{}[a:p-adics(p)].  \mforall{}[k:\mBbbN{}\msupplus{}].    p\^{}k(p)  *  p-shift(p;a;k)  =  a  supposing  (a  k)  =  0

Date html generated: 2018_05_21-PM-03_21_52
Last ObjectModification: 2018_05_19-AM-08_19_07

Theory : rings_1

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