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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