Nuprl Lemma : exp-is-zero
∀[x:ℤ]. ∀[n:ℕ]. uiff(x^n = 0 ∈ ℤ;0 < n ∧ (x = 0 ∈ ℤ))
Proof
Definitions occuring in Statement :
exp: i^n,
nat: ℕ,
less_than: a < b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
sq_type: SQType(T),
guard: {T},
true: True,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
decidable: Dec(P),
or: P ∨ Q,
nat_plus: ℕ+
Lemmas referenced :
nat_wf,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
itermMultiply_wf,
intformeq_wf,
decidable__equal_int,
le_wf,
exp_wf2,
int_entire,
exp_step,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
subtype_base_sq,
exp0_lemma,
equal_wf,
and_wf,
int_subtype_base,
equal-wf-base,
member-less_than,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
productElimination,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
because_Cache,
instantiate,
cumulativity,
promote_hyp,
imageElimination,
unionElimination,
dependent_set_memberEquality,
multiplyEquality
Latex:
\mforall{}[x:\mBbbZ{}]. \mforall{}[n:\mBbbN{}]. uiff(x\^{}n = 0;0 < n \mwedge{} (x = 0))
Date html generated:
2016_05_14-PM-04_26_54
Last ObjectModification:
2016_01_14-PM-11_36_53
Theory : num_thy_1
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