Nuprl Lemma : p-reduce-self
∀p:ℕ+. ∀n:ℕ. (p^n mod(p^n) = 0 ∈ ℤ)
Proof
Definitions occuring in Statement :
p-reduce: i mod(p^n),
exp: i^n,
nat_plus: ℕ+,
nat: ℕ,
all: ∀x:A. B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
p-reduce: i mod(p^n),
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q
Lemmas referenced :
nat_wf,
nat_plus_wf,
modulus-is-rem,
exp_wf4,
nat_plus_subtype_nat,
exp_wf3,
subtype_rel_sets,
less_than_wf,
nequal_wf,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
rem_eq_args,
exp_wf_nat_plus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalRule,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
intEquality,
because_Cache,
lambdaEquality,
natural_numberEquality,
independent_isectElimination,
setElimination,
rename,
setEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
baseClosed
Latex:
\mforall{}p:\mBbbN{}\msupplus{}. \mforall{}n:\mBbbN{}. (p\^{}n mod(p\^{}n) = 0)
Date html generated:
2018_05_21-PM-03_17_57
Last ObjectModification:
2018_05_19-AM-08_08_55
Theory : rings_1
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