Nuprl Lemma : p-reduce-eqmod-exp
∀p:ℕ+. ∀n:ℕ. ∀m:{n...}. ∀z:ℤ. (z mod(p^m) ≡ z mod p^n)
Proof
Definitions occuring in Statement :
p-reduce: i mod(p^n),
eqmod: a ≡ b mod m,
exp: i^n,
int_upper: {i...},
nat_plus: ℕ+,
nat: ℕ,
all: ∀x:A. B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
uimplies: b supposing a,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
nat_plus: ℕ+,
ge: i ≥ j ,
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,
prop: ℙ,
and: P ∧ Q,
sq_type: SQType(T),
p-reduce: i mod(p^n),
label: ...$L... t,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
int_seg: {i..j-},
lelt: i ≤ j < k,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
eqmod: a ≡ b mod m,
divides: b | a,
le: A ≤ B,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T
Lemmas referenced :
subtype_base_sq,
int_upper_wf,
set_subtype_base,
le_wf,
int_subtype_base,
int_upper_properties,
nat_properties,
nat_plus_properties,
decidable__equal_int,
subtract_wf,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermAdd_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
int_formula_prop_wf,
decidable__le,
intformand_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
itermConstant_wf,
int_term_value_constant_lemma,
equal_wf,
nat_wf,
nat_plus_wf,
exp_wf2,
multiply-is-int-iff,
itermMultiply_wf,
int_term_value_mul_lemma,
false_wf,
exp-positive-stronger,
less_than_wf,
int-subtype-int_mod,
eqmod_wf,
squash_wf,
true_wf,
modulus_wf_int_mod,
int_mod_wf,
exp_add,
int_seg_wf,
int_seg_properties,
subtype_rel_self,
iff_weakening_equal,
mod-eqmod,
mul_nat_plus,
exp_wf_nat_plus,
subtract-is-int-iff,
mod_bounds_1,
mul_nzero,
exp_wf3,
subtype_rel_sets,
nequal_wf,
intformless_wf,
int_formula_prop_less_lemma,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
setElimination,
rename,
because_Cache,
hypothesis,
independent_isectElimination,
sqequalRule,
lambdaEquality,
hypothesisEquality,
intEquality,
dependent_functionElimination,
addEquality,
unionElimination,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
productElimination,
applyEquality,
imageElimination,
imageMemberEquality,
applyLambdaEquality,
universeEquality,
multiplyEquality,
setEquality
Latex:
\mforall{}p:\mBbbN{}\msupplus{}. \mforall{}n:\mBbbN{}. \mforall{}m:\{n...\}. \mforall{}z:\mBbbZ{}. (z mod(p\^{}m) \mequiv{} z mod p\^{}n)
Date html generated:
2018_05_21-PM-03_18_06
Last ObjectModification:
2018_05_19-AM-08_09_10
Theory : rings_1
Home
Index