Nuprl Lemma : exp-equal-one
∀x:ℤ. ∀n:ℕ. (x^n = 1 ∈ ℤ ⇐⇒ (x = 1 ∈ ℤ) ∨ (n = 0 ∈ ℤ) ∨ ((x = (-1) ∈ ℤ) ∧ ((n mod 2) = 0 ∈ ℤ)))
Proof
Definitions occuring in Statement :
exp: i^n,
modulus: a mod n,
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
minus: -n,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
or: P ∨ Q,
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
decidable: Dec(P),
sq_type: SQType(T),
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
false: False,
nat_plus: ℕ+,
le: A ≤ B,
not: ¬A,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
true: True,
subtract: n - m,
cand: A c∧ B,
squash: ↓T,
less_than: a < b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
bool: 𝔹,
unit: Unit,
it: ⋅,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
equal-wf-T-base,
exp_wf2,
or_wf,
equal-wf-base,
modulus_wf_int_mod,
subtype_rel_set,
int_mod_wf,
le_wf,
int-subtype-int_mod,
nat_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
nat_properties,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
exp-assoced-one,
decidable__lt,
false_wf,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
less_than_wf,
assoced_weakening,
assoced_elim,
subtype_rel-equal,
intformand_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
equal_wf,
squash_wf,
true_wf,
exp-minusone,
iff_weakening_equal,
eq_int_wf,
bool_cases,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
exp-one,
exp0_lemma,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
hypothesisEquality,
hypothesis,
baseClosed,
unionElimination,
because_Cache,
productEquality,
applyEquality,
sqequalRule,
natural_numberEquality,
lambdaEquality,
independent_isectElimination,
dependent_functionElimination,
setElimination,
rename,
instantiate,
cumulativity,
independent_functionElimination,
inrFormation,
inlFormation,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
dependent_set_memberEquality,
productElimination,
addEquality,
minusEquality,
applyLambdaEquality,
int_eqEquality,
imageElimination,
universeEquality,
equalityUniverse,
levelHypothesis,
imageMemberEquality,
promote_hyp,
impliesFunctionality,
equalityElimination
Latex:
\mforall{}x:\mBbbZ{}. \mforall{}n:\mBbbN{}. (x\^{}n = 1 \mLeftarrow{}{}\mRightarrow{} (x = 1) \mvee{} (n = 0) \mvee{} ((x = (-1)) \mwedge{} ((n mod 2) = 0)))
Date html generated:
2018_05_21-PM-01_06_14
Last ObjectModification:
2018_01_28-PM-02_02_12
Theory : num_thy_1
Home
Index