Nuprl Lemma : rnexp-rminus
∀x:ℝ. ∀n:ℕ. (-(x)^n = if isOdd(n) then -(x^n) else x^n fi )
Proof
Definitions occuring in Statement :
rnexp: x^k1,
req: x = y,
rminus: -(x),
real: ℝ,
isOdd: isOdd(n),
nat: ℕ,
ifthenelse: if b then t else f fi ,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
isEven: isEven(n),
rev_uimplies: rev_uimplies(P;Q),
not: ¬A
Lemmas referenced :
isOdd_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
nat_wf,
real_wf,
req_wf,
rnexp_wf,
rminus_wf,
rmul_wf,
int-to-real_wf,
eq_int_wf,
modulus_wf_int_mod,
less_than_wf,
subtype_rel_set,
int_mod_wf,
le_wf,
int-subtype-int_mod,
assert_of_eq_int,
neg_assert_of_eq_int,
exp_wf2,
isEven_wf,
req_weakening,
rmul-identity1,
uiff_transitivity,
req_functionality,
rnexp_functionality,
rminus-as-rmul,
rnexp-rmul,
rmul_functionality,
req_transitivity,
rnexp-int,
squash_wf,
true_wf,
exp-minusone,
even-iff-not-odd
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
because_Cache,
independent_functionElimination,
voidElimination,
minusEquality,
natural_numberEquality,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
applyEquality,
intEquality,
lambdaEquality,
cumulativity,
imageElimination
Latex:
\mforall{}x:\mBbbR{}. \mforall{}n:\mBbbN{}. (-(x)\^{}n = if isOdd(n) then -(x\^{}n) else x\^{}n fi )
Date html generated:
2017_10_03-AM-08_39_52
Last ObjectModification:
2017_07_28-AM-07_31_00
Theory : reals
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